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m as an equivalent logarithmic 2 equation. 9. Solve for x by converting the logarithmic equation 10. Evaluate log(10,000,000) without using a calculator. log 1 _ 7 (x) = 2 to exponential form. 11. Evaluate ln(0.716) using a calculator. Round to the 12. Graph the function g (x) = log(12 β 6x) + 3. nearest thousandth. 13. State the domain, vertical asymptote, and end 14. Rewrite log(17a Β· 2b) as a sum. behavior of the function f (x) = log5(39 β 13x) + 7. 15. Rewrite logt(96) β logt(8) in compact form. 17. Use properties of logarithm to expand ln (y 3z 2 Β· 3 β β x β 4 ). 19. Rewrite 163x β 5 = 1000 as a logarithm. Then apply the change of base formula to solve for x using the natural log. Round to the nearest thousandth. 21. Use logarithms to find the exact solution for β9e10a β 8 β5 = β41. If there is no solution, write no solution. 16. Rewrite log8 ( a 1 __ b ) as a product. 18. Condense the expression 4ln(c) + ln(d) + logarithm. ln(a) _ 3 + ln(b + 3) _ 3 to a single 20. Solve ( 1 ) = ( _ 9 side with a common base. 1 _ 243 1 ) _ 81 x Β· β3x β 1 by rewriting each 22. Find the exact solution for 10e 4x + 2 + 5 = 56. If there is no solution, write no solution. 23. Find the exact solution for β5eβ4x β 1 β 4 = 64. If 24. Find the exact solution for 2x β 3 = 62x β 1. If there is there is no solution, write no solution. no solution, write no solution. 25. Find the exact solution for e2x β e x β 72 = 0. If there is no solution, write no solution. 26. Use the definition of a logarithm to find the exact solution for 4log(2n) β 7 = β11. Download the OpenStax text |
for free at http://cnx.org/content/col11759/latest. 574 CHAPTER 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 27. Use the one-to-one property of logarithms to find an exact solution for log(4x2 β 10) + log(3) = log(51) If there is no solution, write no solution. 28. The formula for measuring sound intensity in decibels D is defined by the equation I ) D = 10log ( __ I0 where I is the intensity of the sound in watts per square meter and I0 = 10β12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7 Β· 10β1 watts per square meter? 30. Write the formula found in the previous exercise as an equivalent equation with base e. Express the exponent to five significant digits. 32. The population of a wildlife habitat is modeled __ 1 + 6.2eβ0.35t, where t is by the equation P(t) = 360 given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity? 29. A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance? 31. A bottle of soda with a temperature of 71Β° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35Β° F. After ten minutes, the internal temperature of the soda was 63Β° F. Use Newtonβs Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour? 33. Enter the data from Table 2 into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic. x f (x) 1 3 2 8.55 3 11.79 4 14.09 5 15.88 Table 2 6 17.33 7 18.57 8 19.64 9 20.58 10 21.42 34. The population of a lake of fish is modeled by the logistic equation |
P(t) = __ 1 + 25eβ0.75t, where t is time in years. To the nearest hundredth, how many years will it take the lake to reach 80% of its carrying capacity? 16, 120 For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. 35. 36. 37. x f (x) x f (x) x f (x) 1 20 2 21.6 3 29.2 4 36.4 5 46.6 6 55.7 7 72.6 8 87.1 9 107.2 10 138.1 3 13.98 4 17.84 5 20.01 0 2.2 0.5 2.9 1 3.9 6 22.7 1.5 4.8 7 24.1 2 6.4 8 26.15 9 27.37 10 28.38 11 29.97 12 31.07 13 31.43 3 9.3 4 12.3 5 15 6 16.2 7 17.3 8 17.9 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. Try It Answers Chapter 1 Section 1.1 4 3 11 __ __ _ c. β 2. a. 4 (or 4.0), terminating b. 0. b. 1 1 1 1. a. _ 615384, 3. a. rational and repeating repeating c. β0.85, terminating b. rational and terminating c. irrational d. rational and repeating e. irrational 4. a. positive, irrational; right b. negative, rational; left c. positive, rational; right d. negative, irrational; left e. positive, rational; right 5. N W I Q Q' 6. a. 10 b. 2 c. 4.5 d. 25 e. 26 a. β 35 __ 7 b. 0 c. β β 169 β 24 d. β e. 4.763763763... Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ 7. a. 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property |
of multiplication; b. 33, distributive property; 4 _ c. 26, distributive property; d., commutative property of 9 addition, associative property of addition, inverse property of addition, identity property of addition; e. 0, distributive property, inverse property of addition, identity property of addition 8. 9. a. 5 b. 11 c. 9 Constants Variables a. 2Οr(r + h) b. 2(L + W) c. 4y3 + y 2, Ο 2 4 r, h L, W y d. 26 121 _ Ο 3 10. a. 4 b. 11 c. d. 1,728 e. 3 11. 1,152 cm2 2 _ 12. a. β2y β 2z or β2(y + z) b. β 1 c. 3pq β 4p + q d. 7r β 2s + 6 t 13. A = P(1 + rt) Section 1.2 1. a. k15 b. ( 2 _ y ) 3. a. (3y)24 b. t35 c. (βg)16 2 _ 5k 3 5. a. 5 c. t 14 2. a. s7 b. (β3)5 c. (ef 2)2 1 _ 4. a. 1 b. c. 1 d. 1 2 _ b. c. β 8. a. 1 _ 1 1 _ _ 6. a. t β5 = t 5 b. 25 a18b21 e. r12 s8 1 1 _ _ f 3 c. (β3t)6 b. 7. a. g10h15 b. 125t3 c. β27y15 d. q24 _ p32 e. 1 f. b15 _ c3 v 6 _ 8u3 b. 625_ u32 e 4 1 _ _ f 4 d. x 3 c. 1 _ w105 d. 27r _ s b. 7.158 Γ 109 c. $8.55 Γ 1013 d. 3.34 Γ 10β9 e. 7.15 Γ 10β8 11. a. 703,000 b. β816,000,000,000 c. β0.00000000000039 12. a. β8.475 Γ 106 b. 8 Γ 10β8 c. 2.976 Γ 1013 d. 0.000008 |
d. β4.3 Γ 106 e. β 1.24 Γ 1015 13. Number of cells: 3 Γ 1013; length of a cell: 8 Γ 10β6 m; total length: 2.4 Γ 108 m or 240,000,000 m 10. a. $1.52 Γ 105 16h10 _ 49 1 _ c20d12 9. a. e. β 4. Section 1.3 1. a. 15 b. 3 c. 4 d. 17 2. 5β£ x β£ β£ y β£ β 2yz Notice the absolute value signs around x and y? That's because their value must be positive. β x β 2 3. 10 β£ x β£ _ 3y2 We do not need the absolute value signs for y 2 because that term will always be nonnegative. 5. b4 β 10. a. β6 b. 6 c. 88 13. 28 x 23 __ 15 6 9. 14 β 7 β 9 ) 5 = 35 = 243 12. x (5y) 9 __ 2 5 7. 0 8. 6 β 9 11. ( β 3ab 6. 13 β 3 β β 3 β β β β β Section 1.4 1. The degree is 6, the leading term is βx 6,and the leading coefficient is β1. 2. 2x 3 + 7x 2 β 4x β 3 3. β11x 3 β x 2 + 7x β 9 4. 3x 4 β 10x 3 β 8x 2 + 21x + 14 7. 4x 2 β 49 6. 16x 2 β 8x + 1 8. 6x 2 + 21xy β 29x β 7y + 9 5. 3x 2 + 16x β 35 Section 1.5 1. (b 2 β a)(x + 6) 2. (x β 6)(x β 1) 3. a. (2x + 3)(x + 3) b. (3x β 1)(2x + 1) 5. (9y + 10)(9y β 10) 6. (6a + b)(36a2 β 6ab + b2) 8. (5a β 1 ) β 1 __ (17a β 2) 4 7. (10x β 1)(100x 2 + 10x + 1) 4. (7x β 1)2 1. Section 1. x |
y 2 5. 2. (x + 5)(x + 6) __ (x + 2)(x + 4) Chapter 2 Section 2.1 1. x y = 1 __ x + 2 (x, y) 2 β2 y = 1 __ (β2) + 2 = 1 (β2, 1) 2 (β1) + 2 = 3 β1 y = 1 ( β1, 3 __ __ __ ) 2 2 2 0 y = 1 __ (0) + 2 = 2 2 (1) + 2 = 5 1 y = 1 __ __ 2 2 2 y = 1 __ (2) + 2 = 3 2 (0, 2) ( 1, 5 __ 2 ) (2, 3) 3. 1 4. 2(x β 7) __ (x + 5)(x β 3) y 5 4 3 2 1 β1β1 β2 (β2, 1) β5 β4 β3 β2 (2, 3) (0, 2) 21 3 4 5 x β 5 β 125 = 5 β 3. β 5 4. ( β5, ) _ 2 2. x-intercept is (4, 0); y-intercept is (0, 3). y 5 4 3 2 1 21 3 4 5 x β5 β4 β3 β2 β1β1 β2 β3 A-1 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. A-2 Section 2.2 TRY IT ANSWERS Chapter 3 Section 3.1 3. x = 2. x = β3 10 _ 4. x = 1 3 1 1 _ _ and β, excluded values are β. 2 3 8. y = 4x β 3 1. x = β5 7 _ 17 5. x = β 1 2 _ _ 6. x = 7. m = β 3 3 10. Horizontal line: y = 2 11. y 9. x + 3y = 2 Parallel lines: equations are written in slope-intercept form. 12. y = 5x + 3 5 4 3 2 1 21 3 4 5 x β5 β4 β3 β2 β1β1 β2 Section 2.3 1. 11 and 25 4. L = 37 cm, W = 18 cm 5. 250 ft2 2. C = 2.5x + 3,650 3. 45 mi/h Section 2.4 β β β24 |
= 0 + 2i β 1. β 3. (3 β 4i) β (2 + 5i) = 1 β 9i 6 2. 5 _ β i 4. 2 5. 18 + i 6. β3 β 4i 7. β1 β5 β4 β3 β2 i 5 4 3 2 1 β1β1 β2 β3 β4 β5 21 3 4 5 r Section 2.5 1. (x β 6)(x + 1) = 0; x = 6, x = β1 3. (x + 5)(x β 5) = 0; x = β5, x = 5 x = 7, x = β3 2 1 _ _ 4. (3x + 2)(4x + 1) = 0; x = β 5. x = 0, x = β10,, x = β 4 3 2 22 8. x = β _ 7. x = 3 Β± β 3 2. (x β 7)(x + 3) = 0; β 5 β 6. x = 4 Β± β 9. 5 units x = β1 1 _ x = 3 Section 2.6 2. 25 3. {β1} 1 1 1 __ _ _, x = β 4. x = 0, x = 1. 4 2 2 2 _ 5. x = 1; extraneous solution: β 6. x = β2; extraneous 9 3 _ 8. x = β3, 3, βi, i solution: β1 7. x = β1, x = 2 10. x = β1, 0 is not a solution. 9. x = 2, x = 12 Section 2.7 1. [β3, 5] 2. (ββ, β2)βͺ[3, β) 3. x < 1 3 6. [ β _ 5. (2, β) 14 1 1 8. ( β 9 10., β#) 7. 6 < x β€ 9 or (6, 9] 4. x β₯ β5 k β€ 1 or k β₯ 7; in interval notation, this would be (ββ, 1]βͺ[7, β). 9 8 7 6 5 4 3 2 1 21 3 4 5 876 9 10 x β2 β1β1 β2 β3 β 4. g(5) = 1 7. g(1) = 8 (Note: If two players |
had been tied for, say, 4th 5. m = 8 8. x = 0 or x = 2 1. a. Yes b. Yes place, then the name would not have been a function of rank.) 2. w = f (d) 3. Yes 3 x ___ 6. y = f (x) = β 2 9. a. Yes, because each bank account has a single balance at any given time; b. No, because several bank account numbers may have the same balance; c. No, because the same output may 10. a. Yes, letter grade correspond to more than one input. is a function of percent grade; b. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade. 12. No, because it does not pass the horizontal line test. 11. Yes 5. a. Values that are less than or equal to β2, or Section 3.2 1 1 __ __ 1. {β5, 0, 5, 10, 15} 2. (ββ, β) 3. ( ββ,, β#) ) βͺ ( 2 2 5, β#) 4. [ β __ 2 values that are greater than or equal to β1 and less than 3; b. { x |x β€ β2 or β1 β€ x < 3 } c. (ββ, β2] βͺ [β1, 3) 6. Domain = [1950, 2002]; Range = [47,000,000, 89,000,000] 7. Domain: (ββ, 2]; Range: (ββ, 0] 8. y 21 1 β1 β2 β3 β4 β5 β6 β6 β5 β4 β3 β2 Section 3.3 1. $2.84 β $2.31 ___________ = 5 years $0.53 _____ 5 years = $0.106 per year. 1 __ 2. 2 4. The local maximum appears to occur at (β1, 28), 3. a + 7 and the local minimum occurs at (5, β80). The function is increasing on (ββ, β1) βͺ (5, β) and decreasing on (β1, 5). f (x) 321 4 5 6 7 8 x (5, β80 |
) (β1, 28) 40 20 β4 β3 β2 β1 β20 β40 β60 β80 β100 β120 Section 3.4 1. a. (fg)(x) = f (x)g(x) = (x β 1)(x 2 β 1f β g)(x) = f (x) β g(x) = (x β 1) β (x2 β 1) = x β x 2 b. No, the functions are not the same. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. TRY IT ANSWERS A-3 2. A gravitational force is still a force, so a(G(r)) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F)) does not make sense. 3. f ( g (1)) = f (3) = 3 and g ( f (4)) = g (1) = 3 4. g ( f (2)) = g (5) = 3 5. a. 8; b. 20 7. Possible answer: g(x) = β 6. [β4, 0) βͺ (0, β) 4 + x2 ; h(x ____ 3 β x Section 3.5 1. b(t) = h (t) + 10 = β 4.9t 2 + 30t + 10 2. y 5 4 3 2 1 β1 β1 β2 g(x) f(x) 21 3 4 5 β5 β4 β3 β2 The graphs of f (x) and g (x) are shown here. The transformation is a horizontal shift. The function is shifted to the left by 2 units. x 3. h(x) 4. g(x) = 1 ____ x β 1 + 1 h(x)! |x β 2|+ 4 10 8 6 4 2 β6 β5 β4 β3 β2 β1 β2 21 3 4 5 6 x 5. a. y 3 2 1 β1 β1 β2 β3 β4 β3 β2 b. 21 3 4 x y 4 3 2 1 21 3 4 x β4 β3 β2 β1 β1 β2 6. a. g(x) = βf (x) b. h(x) = f (βx) |
7. x β2 0 g(x) β5 β10 β15 β20 4 2 x β2 h(x) 15 0 10 2 4 5 unknown 7. y f (x) = x2 h(x) = f (β x)= (β x)2 Notice: h(x) = f (βx) looks the same as f (x). β10 β8 β5 β4 β3 β2 5 4 3 2 1 0 β1β1 β2 β3 β4 β5 8. Even 21 3 4 5 9. x x g(x) 6 4 2 9 12 15 8 0 10. g(x) = 3x β 2 g(x) = βf (x) = βx2 1 __ 11. g(x) = f ( x ) so using the square root function we get ___ 3 g(x) = β 1 _ x 3 Section 3.6 1. Using the variable p for passing, | p β 80 | β€ 20 3. x = β1 or x = 2 2. f (x) = β| x + 2 | + 3 Section 3.7 3. Yes 2. Yes 1. h(2) = 6 f β1 is ( ββ, β2) and the range of function f β1 is (1, β). 5. a. f (60) = 50. In 60 minutes, 50 miles are traveled. b. f β1(60) = 70. To travel 60 miles, it will take 70 minutes. 4. The domain of function 7. x = 3y + 5 6. a. 3 b. 5.6 domain of f : [0, β); domain of f β1: (ββ, 2] 9. y 8. f β1(x) = (2 β x)2; y=x f β1(x) 21 3 4 5 76 (x) β3 β2 β1β1 β2 β3 Chapter 4 Section 4.1 4 β 3 1 _ β2 _ 0 β 2 1,868 β 1,442 __ 2,012 β 2,009 1. m = 2. m = = 1 __ ; decreasing because m < 0. = β 2 426 _ 3 = 142 people per year = 3. y = β7x + 3 5. 4. H(x) = 0.5x + 12.5 y 6. Possible answers include (β3 |
, 7), (β6, 9), or (β9, 11) (0, 6) (4, 3) (8, 0) 8 10 x 42 6 y = 2x + 4 y = x 8. (16, 0) 9. a. f (x) = 2x; b. g(x) = β 1 __ x 2 x + 6 10. y = β 1 __ 3 42 6 8 10 x β10 β8 β6 β4 10 8 6 4 2 β2 β2 β4 β6 β8 β10 y 10 8 6 4 2 β6 β4 β2 β2 β4 β6 β8 β10 y = 2x Section 4.2 1. C(x) = 0.25x + 25,000; The y-intercept is (0, 25,000). If the company does not produce a single doughnut, they still incur a cost of $25,000. 2. a. 41,100 b. 2020 3. 21.15 miles Section 4.3 1. 54Β° F 2. 150.871 billion gallons; extrapolation Chapter 5 Section 5.1 1. The path passes through the origin and has vertex at (β4, 7), so (h)x = β 7 __ 16 need to be about 4 but h(β7.5) β 1.64; he doesnβt make it. 2. g(x) = x 2 β 6x + 13 in general form; g(x) = (x β 3)2 + 4 in standard form (x + 4)2 + 7. To make the shot, h(β7.5) would Download the OpenStax text for free at http://cnx.org/content/col11759/latest. A-4 TRY IT ANSWERS 8 _ 11 4. y-intercept at (0, 13), No x-intercepts 3. The domain is all real numbers. The range is f (x) β₯ 8 [ _ 11 5. 3 seconds; 256 feet; 7 seconds, β#)., or 3. The degree is 6. The leading term is βx 6. Section 5.2 1. f (x) is a power function because it can be written as f (x) = 8x 5. The other functions are not power functions. 2. As x approaches positive or negative infinity, f (x) decreases without bound: |
as x β Β±β, f (x) β ββ because of the negative coefficient. The leading coefficient is β1. 4. As x β β, f (x) β ββ; as x β ββ, f (x) β ββ. It has the shape of an even degree power 5. The leading term is function with a negative coefficient. 0.2x 3, so it is a degree 3 polynomial. As x approaches positive infinity, f (x) increases without bound; as x approaches negative infinity, f (x) decreases without bound. 6. y-intercept (0, 0); x-intercepts (0, 0), (β2, 0), and (5, 0) 7. There are at most 12 x-intercepts and at most 11 turning points. 8. The end behavior indicates an odd-degree polynomial function; there are 3 x-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know 9. The x-intercepts that the lead coefficient must be negative. are (2, 0), (β1, 0), and (5, 0), the y-intercept is (0, 2), and the graph has at most 2 turning points. Section 5.3 1. y-intercept (0, 0); x-intercepts (0, 0), (β5, 0), (2, 0), and (3, 0) 2. The graph has a zero of β5 with multiplicity 1, a zero ofβ1 with multiplicity 2, and a zero of 3 with even multiplicity. Section 5.6 1. End behavior: as x β Β±β, f (x) β 0; Local behavior: as x β 0, f (x) β β (there are no x- or y-intercepts). 2. y The function and the asymptotes are shifted 3 units right and 4 units down. As x β 3, f (x) β β, and as x β Β±β, f (x) β β4. The function is f (x) = ______ (x β 3)2 β 4. 1 β5 β4 β3 β2 5 4 3 2 1 β1 β1 β2 β3 β4 β5 21 3 4 5 x y = β4 x = 3 3 |
. 4. The domain is all real numbers except x = 1 and x = 5. 12 __ 11 5. Removable discontinuity at x = 5. Vertical asymptotes: x = 0, x = 1. 6. Vertical asymptotes at x = 2 and x = β3; horizontal asymptote at y = 4. squared function, we find the rational form. 7. For the transformed reciprocal f (x) = 1 ______ (x β 3)2 β 4 = 1 β 4(x β 3)2 ___________ (x β 3)2 = 1 β 4(x2 β 6x + 9) _______________ (x β 3)(x β 3) = β4x2 + 24x β 35 ______________ x2 β 6x + 9 Because the numerator is the same degree as the denominator we know that as x β Β±β, f (x) β β4; so y = β4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x = 3, because as x β 3, f (x) β β. We then set the numerator equal to 0 and find the x-intercepts are at (2.5, 0) and (3.5, 0). Finally, we evaluate the function at 0 and find 3. y 10 β4 β3 β2 β1 1 2 x β10 β20 β30 β40 β50 4. Because f is a polynomial function and since f (1) is negative and f (2) is positive, there is at least one real zero between x = 1 and x = 2. 8. 5. f (x) = β 1 _ (x β 2)3(x + 1)2(x β 4) 6. The minimum occurs 8 at approximately the point (0, β6.5), and the maximum occurs at approximately the point (3.5, 7). β10 β8 β6 β4 Section 5.4 y 6 5 4 3 2 1 β2 β1 β2 β3 β4 β5 642 8 10 the y-intercept to be at ( 0, β 35 ). _ 9 1 _ Horizontal asymptote at y =. 2 Vertical asymptotes at x = 1 4 and x = 3. y-intercept at ( 0, _ |
). 3 x-intercepts at (2, 0) and (β2, 0). (β2, 0) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2, 0) is a single zero and the graph crosses the axis at this point. x 1. 4x 2 β 8x + 15 β 2. 3x 3 β 3x 2 + 21x β 150 + 78 _ 4x + 5 1,090 _ x + 7 3. 3x 2 β 4x + 1 Section 5.5 1. f (β3) = β412 2. The zeros are 2, β2, and β4. 1 _ 3. There are no rational zeros. 4. The zeros are β4,, and 1. 2 5. f (x positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros. 7. 3 meters by 4 meters by 7 meters 6. There must be 4, 2, or x 2 β 2x + 10 Section 5.7 1. f β1( f(x)) = f β1 ( ) β 5 = (x β 5) + 5 = x x + 5 _____ 3 ) = 3 ( x + 5 _____ 3 (3x β 5) + 5 __________ = 3 3. f β1(x) = β x β 1 β 3x __ 3 = x, x β₯ 0 5. f β1(x) = 2x + 3 ______ x β 1 and f ( f β1 (x)) = f (3x β 5) = 2. f β1(x ______ 2 4. f β1(x) = Section 5.8 1. 128 ____ 3 9 __ 2. 2 3. x = 20 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. TRY IT ANSWERS A-5 Chapter 6 Section 6.1 β 3. About 1.548 billion people; by the year 2031, 1. g(x) = 0.875x and j(x) = 1095.6β2x represent exponential functions. 2. 5.5556 India's population will exceed China's by about 0.001 billion, or 1 4. (0, 129) and (2, 236); N(t) = |
129(1.3526)t million people. β 2 )x ; Answers may vary due to 5. f (x) = 2(1.5)x 6. f (x) = β 2 ( β round-off error. the answer should be very close to 1.4142(1.4142)x. 7. y β 12 Β· 1.85x 8. About $3,644,675.88 10. eβ0.5 β 0.60653 11. $3,659,823.44 12. 3.77E-26(This is calculator notation for the number written as 3.77 Γ 10β26 in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.) 9. $13,693 Section 6.2 1. f(x) f(x (β1, 0.25) β5 β4 β3 β2 β1β1 β2 (1, 4) (0, 1) 21 3 4 5 x The domain is ( ββ, β); the range is (0, β); the horizontal asymptote is y = 0. The domain is ( ββ, β); the range is (3, β); the horizontal asymptote is y = 3. 2. f(x) (β1, 3.25) 10 8 6 4 2 β5 β4 β3 β2 β1β2 β4 f(x) = 2 x β 1 + 3 (1, 4) y = 3 (0, 3.5) 21 3 4 5 x 3. x β β1.608 4. f(x) The domain is ( ββ, β); the range is (0, β); the horizontal asymptote is y = 0. 1 f(x) = (4)x 2 (1, 2) (0, 0.5) y = 0 21 1β1 β2 (β1, 0.125) β5 β3 β2 β4 5. g(x) = 1.25 βx (β1, 1.25) g(x) 5 4 3 2 1 (0, 1) (1, 0.8) β10 β8 β6 β4 β2β1 42 6 8 10 |
y = 0 x The domain is ( ββ, β); the range is (0, β); the horizontal asymptote is y = 0. 1 __ 6. f(x) = β ex β 2; the domain is ( ββ, β); the range is 3 (β β, 2); the horizontal asymptote is y = 2. Section 6.3 1. a. log10(1,000,000) = 6 is equivalent to 106 = 1,000,000 b. log5(25) = 2 is equivalent to 52 = 25 2. a. 32 = 9 is equivalent to log3(9) = 2 b. 53 = 125 is equivalent to log5(125) = 3 1 1 is equivalent to log2 ( _ _ ) = β1 c. 2β1 = 2 2 1 121 = 121 1 _ 3. log121(11) = (recalling that β 2 _ = 11) 2 β 1 4. log2 ( _ ) = β5 5. log(1,000,000) = 6 6. log(123) β 2.0899 32 8. It is not 7. The difference in magnitudes was about 3.929. possible to take the logarithm of a negative number in the set of real numbers., 11 5 (1, 0) 42 6 8 f (x) = log (x) x = 0 10 1 5 Section 6.4 1. (2, β) 3. 2. (5, β) f(x) 4 3 2 1 β2β1 β2 β3 β10 β8 β6 β4 4. x = β4 (β1, 1) y 5 4 3 2 1 β6 β5 β4 β3 β2 β1 (β3, 0) β1 β2 β3 β4 β5 The domain is (0, β), the range is (ββ, β), and the vertical asymptote is x = 0. x The domain is (β4, β), the range (ββ, β), and the asymptote x = β4. x = 0 f (x) = log3(x + 4 ) y = log3(x) 5 4 6 (3, 1) x 21 3 (1, 0) The domain is (0, β), the range is (ββ, |
β), and the vertical asymptote is x = 0. 5. y f(x) = log2(x) + 2 x = 0 (0.5, 1) (0.25, 0) (2, 1) (1, 0) y = log2(x) x 6. y x = 0 y = log4(x) (4, 1) (1, 0) 1 f(x) = log4(x) 2 (16, 1) x The domain is (0, β), the range is (ββ, β), and the vertical asymptote is x = 0. 7. 8 10 β10 β8 β6 β4 β2β1 β2 β3 β4 β5 y 4 3 2 1 β10 β8 β β 46 β2β1 β2 42 x = 0 6 8 10 The domain is (2, β), the range is (ββ, β), and the vertical asymptote is x = 2. x The domain is (ββ, 0), the range is (ββ, β), and the vertical asymptote is x = 0. 9. x β 3.049 11. f (x) = 2ln(x + 3) β 1 10. x = 1 x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. A-6 Section 6.5 TRY IT ANSWERS 1. logb(2) + logb(2) + logb(2) + logb(k) = 3logb(2) + logb(k) 2. log3(x + 3) β log3(x β 1) β log3(x β 2) 4. β2ln(x) 5. log3(16) 7. 2 1 __ __ ln(x β 1) + ln(2x + 1) β ln(x + 3) β ln(x β 3) ln(x) 8. 2 3 6. 2log(x) + 3log(y) β 4log(z) 3. 2ln(x) 9. log ( 3 β
5 ____ 4 β
6 5 __ ) ; can also be written log ( ) by reducing the fraction 8 β to lowest terms. x 10. log ( |
) (2x + 3)4 ; this answer could also be written log ( 5(x β 1)3 β ___________ (7x β 1) x12(x + 5)4 ________ 11. log 12. The pH increases by about 0.301. 13. ln(8) ____ ln(0.5) 14. ln(100) _____ β ln(5) 4.6051 _____ 1.6094 = 2.861 4 x3(x + 5) ) _______. (2x + 3) Section 6.6 1. x = β2 2. x = β1 2 4. The equation has no solution. 11 11 __ __ ) or ln ( 6. t = 2ln ( ) 3 3 7. t = ln ( 1 1 __ _ ) = β ln(2) β 2 2 β 10. x = e5 β 1 11. x β 9.97 ln(0.8) ______ ln(0.5) 1 __ 3. x = 2 5. x = ln(3) _ 2 _ ) ln ( 3 8. x = ln(2) 9. x = e4 12. x = 1 or x = β1 13. t =#703,800,000 Γ years β 226,572,993 years. Section 6.7 1. f (t) = A0 e β0.0000000087t 2. Less than 230 years; 229.3157 to be exact 3. f (t) = A0 e ( ln(2) 5. 895 cases on day 15 6. Exponential. y = 2e 0.5x 4. 6.026 hours 7. y = 3e (ln 0.5)x ) t ____ 3 Section 6.8 1. a. The exponential regression model that fits these data is y = 522.88585984(1.19645256)x. b. If spending continues at this rate, the graduateβs credit card debt will be $4,499.38 after one year. 2. a. The logarithmic regression model that fits these data is y = 141.91242949 + 10.45366573ln(x) b. If sales continue at this rate, about 171,000 games will be sold in the year 2015. 3. |
a. The logistic regression model that fits these data is y = 25.65665979 _____________________ 1 + 6.113686306eβ0.3852149008x. b. If the population continues to grow at this rate, there will be c. To the nearest whole number, about 25,634 seals in 2020. the carrying capacity is 25,657. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. Answers CHAPTER 1 Section 1.1 3. Th 1.. Th d diffff 9. β 11. 23. β 35. 43. zβ 51. x 49. x_ 55. 45. y+ β+ 31. β 41. βb+ 13. β 25. 17. 29. β 39. βyβ 15. 27. 47. βb+ 33. Β± 21. 7. β 5. β 37. 19. 53. 57. g+β= 59. 65. 61. 67. 63. Section 1.2 1. ΓΓΓ 3. 5. 7. 17. 25. a 35. y x 19. 27. b c 29. ab d 37. 39. a 9. 11. 21. Γβ 13. 15. 23. 31. m c b 41. 33. q p y z 43. 45. 47 Γ 49. 51. m 53. a 55. n a c 57. a b c 59. Section 1.3 1. t. Th β 13. β _____ 3. Th β 11. β β 19. β 21. β 15. 17. β 9. 5. 7. β β 23. β β 31. β β 25. 33. β β 27. β β 35. x 29. β β +β __________ 37. β β p 45. y β β 39. m β β m 41. bβ β a 43. x β 47. β d ______ d β xββ β x 53. β 49. β β β +β x ____________ β#x 55. nβ β 51. βwβ β w 57. β β m m 59. d 61. β x β ______ β β mnc ______ acmn 69. 63. z β β 65. 67 β #β# |
ββ _________ 71. β β β x+β ___________ 73. β β ____ Section 1.4 9. 5. 11. x +x+ 15. bβ b+bβb+ 1. Th Th 3. 7. 13. w +w+ 17. x βxβ 21. v βv+ 25. y βy+ 29. y βy+ 35. βm+ 41. y βy βy+ 45. aβb 47. t βtu+u 49. t +x +tβtxβx 51. r +rdβd 55. t βt +t+ 37. q β 39. t +t βt βt+ 27. p+p+ 31. c β 57. a +acβac βc 23. nβn+nβ 43. pβp βp+ 19. bβb+ 53. x βxβm 33. nβ Section 1.5 15. p+pβ 5. m 7. m 9. y 11. aβa+ 1. Th x βy x βy =x+yxβy 3. x 13. n β n+ 17. h+hβ 21. t+t β 27. d+dβ 25. p+pβ 31. n+ 29. b+cbβc 35. pβ 37. x+x βx+ 39. a+aβa+ 41. xβx +x+ 43. r+sr βrs+s 45. c+β βcβ 47. x+ x+ 51. xβx+ 55. x+xβ 59. 53. x+xβ 57. z +az+azβa 19. dβdβ 23. x+xβ ___ β zβ 49. zβ 33. y+ x+xβx+ Section 1.6 1. 3. B-1 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-2 5. 7. b+ 9. x+ x+ 11. a+ aβ d β d β 17. 19. 15. cβ c+ xβ x+ 31. yβ y+ 23. y+ y+ nβ nβ t+ t+ b+ bβ y βy+ y βyβ b+a ab x+ |
xβ 13. 21. 29. 37. 43. 51. 25. p+ p+ x+y xy 27. d+ d+ 35. aβ aβaβ 33. 39. z +z+ z βzβ 41. x+xy+y x+xy+y+ 45. 53. +ab b x+ xβ 47. aβb 49. c +cβ c +c+ 55. y+ 57. ANSWERS 31. 33. β x y β β β β β β β β β β β 35. x β y 37. x β y Chapter 1 Review Exercises 1. β 3. 5. y= 7. m 11. 13. 15. a 17. 19. a 9. x y 27. 25. β β 33. β β β 35. x +x + β β β β β β 39. 23. β β 31. β 21. Γ 29. β 37. x βx+ 43. a+abβb 49. aβa+ 55. p+pβp+ βpβ 59. p+ 39. kβkβ 45. p 51. x+ 47. a 53. hβk 41. x +x +x+ 57. qβpq+pq+p 61. x+ 65. m+ 63. xβ m 41 β β β β β β β 67. x+y xy 69. Chapter 1 Practice Test 1. 3. x= 9. 11. x 13. 5. 15. β x β 7. 17. β β 19. q βq βq 23. x+xβ 21. nβn +nβ 25. cβc +c+ 27. zβ zβ 29. a+b b CHAPTER 2 Section 2.1 1. xy 3. y 5. x y 7. x y yβ y( ) 15. y=xβ 17. d=β 19. d=β ) 25. β 27. 29. y= 21. dβ 23. ( β 9. x #β#x 11. y=βx 13. y= = β β 43. d= 45. d= 47. β 49. x=y=β 51. x=y= 55. β= 61. 53. x=βy= 57. 59. 63. 54 ft 3. |
Th Section 2.2 1. 5. x 7. x= 9. x= 11. x= 13. x= 17. xβ βx=β 19. xβ x= 21. xβ x=β ) :(, = β ) 31. :(,) :(, m= 15. x=β ; ; 23. 29. 33. ; 27. :(, 35. = = 39. =; :(, x+ ___ : 43. y=β y=β :(,) :(,) ) 51. y=β 49. ) x+ y= 25. m=β =; = ; = β 41. :; ___ y=β x+ 45. ; 37. 47. 53. x+y= Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 55. y 57. y ANSWERS B-3 β β β β ββ β β β β x β β β β ββ β β β β x 61. y = β3x β 5 5. xx = ax+b=d 9. x=β x=β 15. x=β 11. x=x=β 7. x=x= 13. x=β x= ___ 17. x=x=β 19. x=βx= 21. x=x=β 27. x=Β±β β 23. x=x=β 29. z= z=β m= C x+ B Th A 67.y=β B 65. = 69. 71. 30 ft m=β 58. 63. = β + Section 2.3 1. 3. βx 9. +m 7. 17. βx 13. +P 15. 21. 23. 27. 25. B=+x 31. r= 11. 29. R= 5. v+ 19. 33. W= 35. f= 39. h= PβL = pq p+q A b+b = = β + = 37. m=β 41. = =160 ft 47. 49. h= V Οr 45. A= 53. C=Ο 43. 51. r =β V Οh Section 2.4 33. 35. 39. x= β β |
Β±β β βΒ±β Β± xββxβ 45. = 53. Β± 25. x=βx= β Β±β 31. x= 37. β Β±β Β± 41. x= 43. x= 47. = 49. = 51. xβxβ 55. ax +bx+c= b x=β c x + a a c b b =β x+ + x+ a a a ) ( x+ b a x+b a =Β± β = b a bβ ac a bβ ac a β βbΒ±βb ac a β x= xx+= Th 57. 61. xβxβx+=300. Thx x= 63. 59. 1. 3. ii Section 2.6 5. β+i 7. +i 11. i 13. 9. β + i ββββ ββ β β β β r ββββ 15. βi 17. β+i 19. βi 23. β+i 25. ββi 27. 31. βi 33. + i 39. 41. β 43. i 47. i 49. 51. βi 35. i 45. ( β i ) β + 53. βi i ββ β β β β r 21. +i 29. β i β 37. +iβ =β β i 55. 1. 3. β. Th β 5. 7. x= 9. x= 11. x=x= 15. y= β 23. t= 21. x= 17. m=β 13. x=ββ 19. x= 27. x=β 33. x=β 39. x=β 45. x=ββ 29. x=β 35. x=β 41. x= 47. 25. x= 31. x=β 37. x=ββ 43. 49. Section 2.5 1. 3. a β b= ff a=b= Section 2.7 1. 3. βββ Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-4 ANSWERS 5. x= y = xβe left y ____ __ ] 9. [ β 7. ( ββ β) 11. ββ 13. ( βββ 16. x |
> β6 and x > β2 x > β2, (β2, +β) x < β 3 or x β₯ 1 (ββ, β3) βͺ [1, β) All real numbers (ββ, β) Take the intersection of two sets. Take the union of the two sets. 5,9) 18. 23. 21. ] ( [10,,9/2] ( [ 1 All real numbers (ββ, β) (ββ, β4] βͺ [8, +β) 1 ) 27. [2,5] 29. No solution [β11, β3] y 45. 65. ββ+β 67 69. x=β 71. β 73. β€Tβ€β€Tβ€ Chapter 2 Review Exercises 1. xyβ 3. y= x+ 9. ( ) 5. β β =β β 7. 11 β β β β β β β 13. x= 15. x= 17. 19. y= x+ 21. y= x+ 23. 25. 27. x=β Β± β iβ 12 10 8 6 4 2 β10 β8 β6 β4 β2β2 β4 β6 β8 β10 β12 42 6 8 10 x 47. It is never less than zero. No solution. y 12 10 8 6 4 2 β10 β8 β6 β4 β2β2 β4 β6 β8 β10 β12 31. +i 33. i 35. ββi 42 6 8 10 x 39. x=βi 45. x=β 51. x=Β±β β 41. x=ββ β βΒ±β 47. x= 53. x= 29. β β β 37. ββiβ 43. x= β 49. x= β 55. x=Β±β 57. x=β ____ ] [ β 63. β 59. x= 65. ____ 61. ββ ) 67. ( β 69. x=β 2 4 6 8 10 x 51. Where the blue line is above the red line; always. All real numbers. ( ββ, ββ) y 12 10 8 6 4 2 β10 β8 β6 β4 β2β2 β4 β6 β8 β10 β12 42 6 8 10 25. |
30. 34. 40. 43. [β10, 12] 42. [6, 12] (ββ, β1) βͺ (3, β) y 12 10 8 6 4 2 β10 β8 β6 β4 β2β2 β4 β6 β8 β10 β12 2 4 6 8 10 x 32. ( ββ, β _ ) βͺ (4, β) 36. No solution 38. (β5, 11) 49. Where the blue line is above the red line; point of intersection is x = β3. (ββ, β3) y 12 10 8 6 4 2 β10 β8 β6 β4 β2β2 β4 β6 β8 β10 β12 53. 55. 57. 59. 61. 63. (β1, 3) (ββ, 4) {x|x < 6} {x|β3 β€ x < 5} (β2, 1] (ββ, 4] Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-5 Chapter 2 Practice Test 1. y= x. β y __ __ 73. fβ= f=f=f= 75. fβ= 77. Th 79. Th β 13. β 5. ββ 7. x=β β Β±β 11. x= 17. y= xβ Β± β 25. x= 19. i β 27. 9. xβ βx=β 15. y=β xβ i β 29. x= 21. β 23. x=β β β β β β β β 81. Th β y 83. Th 0, 10 y x β x CHAPTER 3 Section 3.1 1. 3. 5. 9. 11. 7. 13. 17. 15. 23. 21. 27. fβ=βf=βfβa=βaββfa=βa+ fa+h=a+hβ fβa=β +f= βa βfa+h= 19. 25. +a +βfa=ββ 29. fβ=β β β β β βaβh + 31. fβ=f=β β fβa=β£ βaββ£ββ£ βa+β£βfa=ββ£ aββ£+ |
β£ a+β£ fa+h=β£ a+hββ£ββ£ a+h+β£ 33. gxβga _ xβa 37. a. f = b. x=β =x + a+ x β a 35. a. f β= b. x= __ 39. a. r=β t 43. 49. 41. b. fβ= c. t= 45. 47. 53. a. f = b. f x=βx=β 51. 55. 57. 59. 65. 69. fβ=fβ=f=f=f= 71. fβ=fβ=f=f=f= 67. f x=x= 61. 63. 85. Th β 87. Th β β 89. a.g=b.Th f 100 s 91. a. fts 200 ft. b. Thfts 350 ft. Section 3.2 β β 1. Th β 3. Thxfx= x βββ x fx= β β xy βββ x 5. 9. ββ )βͺ ( β 15. ( βββ β#) 17. ββββͺββͺβ 19. ββββͺββͺβ 11. βββ 13. βββ 7. βββ Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-6 ANSWERS 23. β 21. ββ 25. ββββͺββͺβ 27. 29. β 31. β 33. βββ ] [ ββ 35. [ ββ ] ] βͺ[ ] βͺ[ 37. βββ 39. βββ y 41. βββ y β β β β β β β β β β 43. βββ y x x x β β β β β β β 45. βββ y x β β β β β β β β β β β β β β β β β 47. fβ=fβ=fβ=f= 49. fβ=βf=f=f= 51. f β=βf=f=f= 53. ββοΏ½ |
οΏ½β 55 Th Th 57. 59. fx= β xβ β 61. a. Th b. Th c. Th Section 3.3 8. β 6. x+h 1. 3. Th 10. _________ 14. +h x+hβ 31. x+a 33. 25. 29. + ββββͺβ h+h+ ββ b+ β + 23. 20. 18. 12. 26. 16. 35. 27. β 37. βββͺ βͺβ β 40. 42. ββ 44. a. β b. β 45. ββββ 47. ββββ ββ βββββ βββββ β 49. 51. 57. 53. 55. b= ββ β Section 3.4 1. g f g 3. f x= x+gx= xβ Thf (gx=f xβ=xβ+=x g ( f x=g x+=x+β=x fβg=gβf 5. f+gx=x+βββ fβgx=x +xββββ fgx=βx βx +x +xβββ f ( g ) x= _ 7. f+gx= x +x βx βββ β x+x+ βββͺβ x βͺββ βͺβ β β β β β β fβgx= x+xβ βββͺβ x β β fgx=x+βββͺβ f ( g ) x=x +x βββͺβ _ xββ 9. f+gx=x +β fβgx=x ββ fgx=x β f ( x g ) x= _ xβ β b. fgx=#x βx+ d. g β gx=xβ e.f β fβ= xββ xββ β β β 11. a.fg= c.gfx=x β 13. fgx=β β x++gfx=x+β β x |
+ β β x + β ; gfx= x x+ β 15. fgx= x x __ xβ gfx=xβxβ 17. fgx= x++ β βx β __ ) b. ( ββ 21. a.gβf x=β 19. fghx= 23. a. βͺβx=βb.βc.β 25. β 31. fx= 27. fx=xgx=xβ x ; gx=x+ 29. fx= x ; gx= β xβ xβ x+ x ; gx=x+ 35. fx=β 33. fx= x ; gx= β β β β Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-7 37. fx= β β x gx=xβ 29. 39. fx=x gx= xβ 41. fx=β x ; gx= β xβ x+ 43. 55. 67. 45. 57. 69. 75. fg= gf= 47. 59. 71. 51. 63. 49. 61. 73. fg=gf=β 53. 65. 77. fgx=x+x+ β β β β β 87. fβg =gβf = 85. 81. fβgx=gβf x= 79. gβgx=x+ 83. βββ 89. fβg =gβf = 93. At=Ο ( β 97. a.NTt=t +tβ b. β t+)A=Ο ( β 95. A=Ο 91. β )=Ο β Section 3.5 1. 3. 5. f βxxfx fβx= fx fβx= βfx x++ 9. gx= 7. gx= β£ xββ£β 11. Thfx+ f 13. Thfxβ 15. Thfx+ f f 19. Th f fx+βhift 21. βββ f ββ 17. Thfxβ 23. β 25. y 27 β β β β β β β β β β β β |
β β β β β fx= β£ xββ£β fx= β x+β fx= xβ fx= β£ x+β£β β fx= β β x fx= βx++ fx= β 31. 33. 35. 37. 39. 41. 43. 45. 49. g x Thg f f βx + 47. 51. Th 53. 55. Thg Thg f f y Thg 59. 57. f 63. gx= x+β hift 67. hift y gx= β£ βx β£ 61. gx= xβ+ 65. 69. hift hift β β β β β β β β β β β 71. y Th 73. hift β β β β β β β β β Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-8 ANSWERS Th 25. Th Th β β β β β 29. β β β β 27 31 β β β β β β β 75. Th fx= βxhift y 77. 79. 81. y a x Section 3.6 1. β£ A β£ =B A B A βB 3. x 5. Thx x β£ xββ£= 7. β£ xββ£β₯ 9. Thx 11. β 13. ββ 15. β 17. y 19. y β β x β β β β β 23. x β β β β β 21 33. β y 35 37. Tha yTh yx= 39. β£ pββ£β€ 41. β£ xββ£β€ SECTION 3.7 1. y y 5. y=f βx 7. 11. 9. x 3. f x= f βx=xβ 13. f βx=βx f βx= 17. f βx= 15. 19. f βx= 23. f βx= 29. f βxx= p+ + f βx= f βx= 21. 25. 31. r + 27. f βx= + 33. 35. 37. f xββf βx=βx β β f xβf βx=βx+ fgx= |
xgfx=x β 39. 41. 43. 45. 47. 49. 51. 53. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-9 55. y f β f ββ β β ββ β 69. f βx=+ β β β β β 57. 59. 61. β 63. x 65. 67. x f β1(x) 71. 73. f βx = x β d _ t= td= Th x 49. β β β β 53. y β β β β β β y Chapter 3 Review Exercises 1. 3. 5. fβ=βf=β fβa=βaβa; βfa=aβa; fa +h=βaβah βh+a +h 7. 11. 9. 13. β β β β 23 15. 19. 17. β β+aβa β+a =βa +aβ 21. ββββͺ(ββͺ(β 25. 27. β ββ 29. βββββ 31. ββ 33. 35. fβg x=βx, gβf x=ββx 37. fβg x= β x + gβf x= _ β x+ β 39. fβg x= 41. fβg x= +x _ +x x + = x + β) β x β )βͺ ( β ( βββ )βͺ(β) 43. gx= 47. fx=β β x 45 51. y β β β β β β β β β β β β β β β β 55. fx=β£ x ββ£ 57. 59. 61. x β£ x +β£+ 63. fx= 65. fx=ββ£ xββ£+ x β 69. f βx= 71. f βx=β β x β 73. Th β β β β β β β 67. y β β β β β β β β β 75. 5. Th y 3. β 15. 9. βa+b+bβ a Chapter 3 Practice Test 1. β 7. aβa 11. β 13. 17. x+ |
21. ββββ 23. β 27. fx={ |x| x β€ x > 29. x = 31. x β 33. f βx=β 19. fβx 25. f= βx_ CHAPTER 4 Section 4.1 3. dt = βt 1. 5. Thaa ya xa. Th Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-10 ANSWERS 9. 11. 27. β 31. y = xβ 7. 17. 23. 13. 15. 21. 29. _ x β 19. 25. 33. y = β x+ 37. 39. 41. yx 43. yβx ( ) 45. yxβ 47. m=βm=β 49. m=βm= 35. y = βxβ 51. m=β m=β 55. y=β __ t+ 57. 53. y =xβ 61. y =xβ 63. y =β 65. 59. y=β __ x+ 67. 69. x 99. y β βββ ββ β β β β β 103. y β β x β β β x 101. aa = b = b. qp = pβ 107. x = a 105. x = β __ d ____ cβa 109. y= xβ ad ____ cβa 111. y=xβ 113. x< x> 115. 71. β β β β β 75. β β β β β 79. β β β β β 83 β β β β β β β 73. x β β β β β 77. x β β β β β 81 β β β β β β β 85. y = 87. x = β 89. gx = βx+ 91. fx = xβ 93. gx = β __ 95. fx = xβ 97. fx = βx+ x+ x 117. Th ts. Th 119. Thβ 121. Section 4.2 9. 11. 5. 7. 19. Wt=t+ 25. Ct=βt 23. 27. 1. 3. 15. β 13. Pt=t + 17. ft 21. β15, 0) Th x 29. 33. fi 35. y =t β 37. fi 39. 43. 41. 45. a. |
b. c. d. e. Pt=+t f. 47. a. Cx=x + b. The fl c. 49. a. Pt=t + 51. a. Rt= βt + b. b. c. 53. 55. 57. 31. y =βt + Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-11 Section 4.3 1. ft 3. 5. Th 9. 7. 11. 13. 15. 17. 19. 21. 29. y=βx+r=β 23. 25. y=x+r= 27. y=βx+ r=β 31. y=xβr= β(ββ fi 37. y=x+ 41. y=βx+ 33. βββ 35. 39. y=xβ Chapter 4 Review Exercises 3. 1. 9. y=xβ 11. 13. 15. ββ 17. mβm=β 19. y=βx+ 5. y=βx+ 7. 23. 25. 27. y=βx+ 29. a. b. c. Pt= t+ 31. 33. y= x 21. β β β β β y β β β β β β β 37. y 35 Year 39. 41. y=βx+ r=β 43. 45. Chapter 4 Practice Test 1. 3. 5. y=βxβ 7. y=βxβ 9. 15. y=βx+ 13. ββ 11. 21. 19. 23. y=x+ 25. a. b. c. y=βt+ 17. =β y= y β β ββ β β β x 29. 31. y=x+ 33. r= 27. y x Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-12 ANSWERS CHAPTER 5 Section 5.1 ) β 17. β 1. 3. a= 5. 7. gx=x+ β ββ β ) ( β 9. fx=( x+ ) β 11. kx= xβββ 13. fx= ( xβ ( ) β 15. β x= x=β β 19. β βx=β 23. ββ |
β 21. ββββ ββ 27. fx=x+x+ 29. fx=xβx+ 31. fx=β __ __ 35. β) x= +β 33. fx=xβx+ 37. ( ) β x= β +β )( ( 25. βββββ β β __ x+ x β β β β β β) 39. ( x= +β )( ( β# β ) β β y βββββ β β β β β x 41. fx=x+x+ 43. fx=βxβxβ 45. fx=β __ xβ x+ 47. fx=x+x+ 49. fx=βx+x 51. Th h. Th 53. Thhift 55. Th 57. βββ ββ 59. βββ β 61. fx=x+ 63. fx=βxβ 65. fx=x+xβ 67. 71. Th 69. 73. 75. Section 5.2 13. β 15. β 17. xββfxββxβββfxββ 1. Th. Th 3. x fxx fx 5. Th 7. 9. 11. 19. xβββfxβββxββfxβββ 21. xβββfxβββxββfxβββ 23. xββfxββxβββfxβββ 25. ytβ 27. yβxβ 29. yxβ 31. 33. 37. 39. 41. 43. 45. 47. xβββfxββ xββfxββ 49. xβββfxββ xββfxβββ 35. x fx x fx β β β β 51. y x xβββfxββ xββfxββ β β 53. y x xβββfxβββ xββfxββ y ββ β β β β β x x ββ β β β β β β ββ β 55. y xβ xβββfx |
βββ xββfxββ 57. yβ xβ xβββfxββ xββfxββ y y ββββ β β β β β β x ββββββ β β β β β x x y Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-13 59. yx β xβββfxβββ xββfxββ 61. fx=xβ 63. f x=xβx+ x 65. f x=x+ 67. Vm=m+ m+ m+ 69. Vx=xβx+ x y βββββ β β β β β Section 5.3 x 9. β 7. ββ 5. Th 3. a b 1. x x fx=$ a b 11. β 15. ββ 19. β 23. ββ 25. f =βf = 27. f =f =β 29. f =f =β 31. β 33. β 35. β β 13. β 17. β( _ ) )( ββ 21. ( β 37. ) β β 39. 41. 43. x1, 0 β4, 0 y (0, 4); axβββg xβ ββxββg xββ 45. x3, 0 2, 0 y (0, βxβββk xβ ββxββk xββ kx gx βββββ β β x ββββββ β β β β β β β β β β x 47. x0, 0β2, 0 y(0, 0); axβββ n xββxββ n xβββ nx βββββ β β β β β x x+ xβ xβx+ x+ 51. f x= 49. f x=β 53. ββ 55. β 33. 35. 57. f x=β x+ xβxβ 59. f x= xβxβx+ 61. f x=βxβxβ 63. f x=βx+ x+ xβ 65. f x=β xβxβx+ 67. |
βββ 69. : ββ 71. β 73. f x=xβx+ 77. f x=xβx+ x+ Ο x+ x+ x+ 79. f x= 75. f x=xβx+ x Section 5.4 1. Th 3. x+ + x+ xβ 5. x+ x+ 9. xβ+ xβ 7. xβ xβ x+ 11. xβ+ xβ x+ 13. x βx+ xβx+ 15. x+ x+ + xβ 17. xβx+ β x+ 19. xβx+ β 23. xβx+ β 27. x+ x+ xβ x+ x+ 21. x+ x+ 25. xβx+ 29. xβx+ 31. xβx+ 41. 35. xβx+ xβ 39. xβxβ 43. 33. xβx+ xβ 37. xβx+ xβx+ x+ x+ 45. xβx+ x+ 49. x+ x+ β 51. x+ x+ 53. xβx+ xββ 57. xβx+ xβ+ 55. xβx+ xβx+ xβx+ + i xβi 47. xβx+ x+ x+ 61. + 59. + βi x+ i βi xβi 71. xβ 63. x+ ixβ+ 69. x+ Section 5.5 65. x + 67. x+ 73. x β 1. Th 3. 7. 9. 5. p + p 19. .+,.. p,, 15. + p + 23. 1 1 27. Β± Β± p, 11. 13. ,+, + ββ β,, Β± Β± Β± 3 5 5 2 1 8 Β± (+)( Β± 5 4 Β± ( Β±,,,,, 17. 21. 25. Β± 29. 31. ( ) + (+)( 3p β,,, 1, 2 Β± Β± 15 15 2 1 Β± (+)) p! ), Β± 5 8 Β± ))( p! 3p )(+ , 1 4 15 4, 1 8 Β± |
15 8, Β± Β± β (+) β ( ( ))( (+)) β 3 1, 3 2, Β± 3 4, Β± Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-14 ANSWERS 44. f x y= β βββ 38. f x 40. f x ββ β β β β β β x ββββββ β β β β β x ββ β 42. f x x β β β β β β ββ β β β β 46. f x ββββββ β β β β β x ββββββ β β β β β x 47. 49. 53. 55. 51. β _, β _3 1 2, 4 2 fx= x +x βxβ fx=β __ xβx 57. 61. 59. Section 5.6 3. Th 5. es. Th 1. Th 7. x=β 9. x=ββ 11. x=β y= x=β x=βy= x=β 15. x=β y=x=β 17. x=βy= x=β 13. 19. x= y=β x= 21. ) 23. xy( β + 25. xββ f xβββxββ f xββ xβΒ±βf xβ 27. xβ+ f xβββ xββ f xββ; 29. xβΒ±βf xββ β f xβββ xββ f xββ β + xββ f xββ xββ + xββ f xβββxβΒ±βf xβ 31. y=x+ 35. x= y= 37. x= y= y 33. y=x y ββ β β β β ββ β β β β βββ y= x β x= x x= 39. x=β ) )( β y=( px ββ β β β β β βββ x= β y= x 43. x=β y= )( ( β ) f x ββ β β β β β β ββββ x= β x= y= x 47. x= y=x+β( ) |
) ( hx 41. x= y= sx x= ββ β β βββ y= x 45. x=β y= β ax β βββ y= x x=β ββ β β β β 49. x=β y= β( β ) wx x= x= β y=x+ x= x β ββββ y= x β β β β ββ β β β β ββ β β β β Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-15 51. fx= 55. fx= β
xβ xβ ________ xβ xβ x+ x+ 53. fx= x+xβ x+x+ 57. fx= xβ xβ xβ 59. fx=β xβ xβ 61. fx= β x+xβ xβ 63. fx=β xβ ____________ x+xβ 65. x=y= x y x y β β 31. fβx=β y β xβ 33. fβx=β y β x β β ββββ β β β β β x β ββββ β β β β β x β xβ 37. fβx=β y β x+β x β ββ ββ β β β β β x 35. fβx 67. x=βy= β β β x y x y β β β β β β β 39. 69. x=βy= β0 β β0 β0 β x y x y β#) 71. ( fx 73. βββͺβ fx β ββββ β β β β β 43. ββͺ β f x β β β ββ β β β β β 47. β y βββ β β β β β β β β β β β ββ β β β β x β β β β ββ β β β β x 75. () 77. () 79. (β) 81. Ct= +t +t 83. ft 87. 85. Section 5.7 1. diffict oxy 3. 5. fβx=β 7. fβx=β x+β x + β β 9. fβx=β β βx β 13. fβx |
= 17. fβx= β xβ βx ______ β 11. fβx= Β± β β 15. fβx= xβ β βx 19. fβx= 21. fβx=βx 27. fβx= xβ ______ x+ 23. fβx= 29. fβx=β x+ ______ x β x+β xβ+ __________ 25. fβx= x+ ______ βx x x x 41. ββͺβ f x βββββ β β β β β 45. βββΔβ f x β β β ββ β β β β β 49. βββ y βββ β β β β β β β xβb a βh x x x 53. fβx= β 57. th = β xβb 51. fβx= a cxβb 55. fβx= aβx A, β Ο A+Ο β2, 3.99 ft Ο 59. r A=β 63. rA=β T 61. lT=( ) __ Ο 65. rV=β β 3.26 ft V β 5.64 ft Ο Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-16 ANSWERS Section 5.8 1. Th 5. y=x 3. x 19. y=xβ 17. y=xzw 9. y=x 11. y= 13. y= β z x 23. y= xz β wt β 31. y= 25. y= 33. y= 27. y= 29. y= 35. y= 37. y= 31. β( β ) x=y= y y = β β β ββ β β β β β x x = 33. β ( )x=β y= y β β βββ β β β β β x = β y = x x = 35. y=xβ 37. fβx=β β x + 39. f βx=β β x+β 41. fβx= x+β 47. β xβ₯ β x Chapter 5 Practice Test 43. y= 45. y= 3. xβββ 7. y=x 15 |
. y= β x β xz w 21. y= 41. y= x y ββββ β β β β β β __ 45. y= x y β ββββ β β β β β 39. y= __ 43. y= β β x y β β β ββ β β β β β x 47. β 49. β 51. 53. 55. x 1. f xββ xββf xββ β 5. f x=xβ 7. 9. β 13. { ββ } 11. x+x+x++ xβ 15. ββ 17. f x=β xβxβx+ 19. 21. β( ) x=βy= y 23. fβx=xβ+xβ₯ x+ xβ 25. fβx= ββββ β β β β β x = β β 27. y= y = x x = CHAPTER 6 Section 6.1 1. 3. t. Th nominal 11. ft 13. 5. 7. 9. Chapter 5 Review Exercises 1. f x = x β β β ββ fx ββββ ββ β β β β x 3. f x= x++ 5. 7. 9. 11. xβββf xβββ xββf xββ 13. ββ β 15. 17. 19. x + x+ 21. x βx+β 23. x βxβx+x βxβ 25. { ββ 27. { } } 29. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-17 t infl 15. 17. 21. f x=( __ ) 27. r 33. P=Atβ
( + n ) ( __ ) 29. 19. f x=x x βx 35. 25. 23. βnt β 39. 41. 45. f β=β 47. f βββ 51. y=β
x 53. yββ
x 31. 37. 43. 49. f β 55. yββ
x n βa a( + a[ ( + 57. = Atβa a r ___ ) a β] = r ___ ) =( + = a r |
In=( + n ) 59. ff x=aβ
( ) x b b> 1. Thn> fx=aβ
( ) x b 61. 67. x =aeβnx=aeβnx 63. =abβx=aenβ r ) β β 65. Section 6.2 1. x. Th 3. gx=βxy 5. gx) =βx+y 7. gx=( ) y x, β β : ;:= :(,);:= 13. :(,);:= 15. 17. : 19. 21. 0, 1 1024 β β ;:= 9. 11. 23. y 25. y ββ β β β β β β β β fx= x x β β βfx= β x ββ β β β β β β βf x=x β x f x=βx + 27. hx= hx βββββ x 29. xββfxβββ xβββf xββ1 31. x β βfxβ xβ ββfxββ 33. fx=xβ 35. fx=x β 37. fx=βx 39. y=βx+ 41. y=βx+ 43. gβ 47. xββ 45. hβ=β x 51. Thgx) =( ) y b fx=bxb> 49. xββ fx=b ) x( b x yf βx 53. Thgxhx fxn x( bn ) bx b> fx=b fxβn 55. Section 6.3 1. b b. Th ybybx d txby 3. b y=x x 5. Th b ee (x) 7. ac=b 15. e n=w 21. n= 27. x= 35. x=e 43. 45. 47. 49. 51. β 53. 55. β 57. 17. ck=d 23. y ( 25. h=k 31. x= 19. y=x 13. a= 11. b=a 29. x= 41. 9. x y= )=x 33. x= 39. (x) 37. 65. β 63. 71 |
. 61. β β β 67. 59. 73. 69. x=x= fx=x). Thn n=n = n Thx=f x=x 75. x x=x=e x=e. Th x=e= 79. 77. = e Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-18 Section 6.4 5. 1. y=xab ba 3. hift y ff 7. ( ββ, )βββ 9. ( β β)βββ 11. βx= β#) x=β 13. ( β 15. ββx=β 17. ( β#) x= xβ( ) 19. ββx=β xββ+fxβββxββfxββ 21. ββββx= x( )y 23. ββ, 0); raβββ x=xβey 25. ββββx= xey 27. 33. + fxβββs x β βfx β β 31. 29. y y 35. 37. ββ β β β β f x= x x β β gx= x f x= e x β β ββ β β β β x gx= x 39. fx 41. y ββ β β β β β β x ββββββ ββ β β β β β x β β 45. 47. 49. 51. 53. f x=βxβ f x=x+ x= xβ xββ 43. gx ββββ β ββ β β β β x Thfx= 55. bβ bx=β_ 57. xgx=βx b x x+ xβ > x+ xβ fx= > xβββ βThββββͺβ Section 6.5 _ n bx n = 1.. Thbx 3. b +b +bx+by 5. bβb 7. βk 11. b 13. b 15. x+yβz () 19. 17. () 21. ) () x ( xz |
) β y β = b )= () ( ()+ 9. xy () 25. 31. 27. 23. 29. 33. = ( β aβb b 39. a bβ ββ 37. β x= x+βxβ=( x = x+ _____ )= xβ () 35. 41. 42. = = = x+ _____ xβ x+ _____ β xβ xβ x+ _______ _____ β xβ xβ x+βx+ _____________ xβ xβ _____ xβ = x= +β β= βx= bnn 1. Th bn= nn nb = nb Section 6.6 1. 3. Th. Th x= 7. 9. 5. x= x=β x= 19. 13. x= () () 15. 11. 17. x=β x= + 21. () () () Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-19 31. x=β 32. 61. y 39. x= 41. x= 63. 67. 69. 71. 73. 65. β β β ββ _ k ) t=(( y A ) t=(( _ β TβTs k ) TβTs ) x 75. Section 6.7 5. 1. s. Th 3. Th t diff orders of magnitude 7. f β 9. 13. 11. f x=x 15. 25. 33. x= 27. 35. x= x= 37. x= x= 43. x= 45. x= β e y x 47. x ββ β β β β β e+ β x= y x 49. x=β y βββββββ β β β β x β 51. ββ β β β β β 53. 55. y x= β y x β β β β β ββ β β β β β β β ββ β β β β β β 57. x= y β fx x x y x β β β 59. y ββ x 17. Pt β β β β β ββββ ββ t fx=ex x 19. 23. 21. Download the OpenStax text for free at http |
://cnx.org/content/col11759/latest. B-20 ANSWERS 27. y=bx bbβ 1. Th __ __ y=bx y=xb ey=exb y=exb S ) ( 25. M= S S M=( ) S S M =( ) S SM =S 29. A=eβtAβ 33. f t=eβt; 35. rββ 37. f t=etftP β 39. f t=et 41. β 31. 43. β 45. Section 6.8 1. 3. t fi. Th 5. y 7. 13. pβ 19. 21. 11. P = 15. y 17. 9. y x y 31. x 23. 25. 27. f x=x 29. y x 33. f x=eβx 35. f x=xβ 37. y=+x 39. y x 43. f β 45. f x=xβ 47. f x= +eβx 49. 41. y x 51. y 53. y x x 55. f x=xβ g x= y =x 57. f x=x 59. f βx= c aβ( _ β) x b Chapter 6 Review Exercises 5. 1. 3. y =x 7. y 9. y ββββββββ β x 11. gx=βxy 15. ab= β 19. =β 13. x = 17. x = Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-21 33. y 35. y =x y x 37. y = +eβx y x 21. eβ=β 23. gx x βββββ β β β β β β β β 25. x> β x =β xββ+ f xβββ xββf xββ 27. xy z 29. ( xy ) 31. y 33. +b+ b+βbβ 35. ( v w β u β ) 39. x =β 41. 43. 37. x =# 45. x = 51. 47. a=eβ 53. fβx= __ 49. x =Β± xβ β οΏ½ |
οΏ½ 55. f t=tf βg 57. x 59. 61. y x 63. y =x; y =eβx 65. 67. y =βx y Chapter 6 Practice Test x 1. 5. y 3. y fβx= βx fx= βx 7. a= 9. x = 11. ββ 13. x< x = xββf xβββ xβββf xββ βββββ β x 15. t 17. y+z+ xβ 19. x = _______ + __ β 23. 25. x = 21. a= + ________ β 27. x =Β± β ____ 29. f t=eβt; 31. Tt=eβt+TβΒ°F Download the OpenStax text for free at http://cnx.org/content/col11759/latest. Index A absolute maximum 205, 267 absolute minimum 205, 267 absolute value 189, 247 absolute value equation 137, 151, 250 absolute value function 187, 247, 250 absolute value inequality 146, 147 addition method 582, 587, 674 Addition Principle 802, 828 addition property 143 algebraic expression 11, 66 angle of rotation 733, 751 annual interest 797 annual percentage rate (APR) 474, 567 annuity 797, 828 apoapsis 742 area 104, 151 area of a circle 361 arithmetic sequence 771, 772, 774, 775, 790, 791, 828 arithmetic series 791, 828 arrow notation 415, 454 associative property of addition 8, 66 associative property of multiplication 8, 66 asymptote 700 augmented matrix 636, 640, 641, 653, 674 average rate of change 196, 267 axes of symmetry 700 axis of symmetry 344, 347, 454, 722, 723 B base 6, 66 binomial 42, 66, 396, 828 binomial coefficient 812 binomial expansion 813, 815, 828 Binomial Theorem 813, 814, 828 break-even point 589, 674 C carrying capacity 546, 567 Cartesian coordinate system 74, 151 Celsius 254 center of a hyperbola 700, 751 center of an ellipse 685, 751 central rectangle 700 change-of-base formula 525, 526, |
567 circle 607, 609 co-vertex 685, 687, 700 coefficient 41, 42, 66, 361, 405, 454 coefficient matrix 636, 638, 655, 674 column 625, 674 column matrix 626 combination 807, 812, 828 combining functions 210 common base 529 common difference 771, 790, 828 common logarithm 496, 567 common ratio 781, 793, 828 commutative 211 commutative property of addition 8, 66 commutative property of multiplication 8, 66 complement of an event 823, 828 completing the square 124, 125, 151 complex conjugate 116, 151 Complex Conjugate Theorem 409 complex number 112, 113, 151 complex plane 112, 151 composite function 209, 210, 211, 267 compound inequality 145, 151 compound interest 474, 567 compression 292, 486, 508 conditional equation 87, 151 conic 684, 699, 748 conic section 751 conjugate axis 700, 751 consistent system 578, 674 constant 11, 41, 42, 66 constant function 187 constant of variation 448, 454 constant rate of change 309 continuous 376 continuous function 371, 454 coordinate plane 717 correlation coefficient 327, 334 cost function 209, 588, 674 Cramerβs Rule 663, 666, 670, 674 cube root 362 cubic function 187, 439 D decompose a composite function 217 decomposition 615 decreasing function 201, 267, 282 decreasing linear function 283, 334 degenerate conic sections 729 degree 42, 66, 366, 454 dependent system 579, 587, 599, 674 dependent variable 160, 267 Descartesβ Rule of Signs 410, 454 determinant 663, 665, 666, 674 difference of squares 46, 66 directrix 717, 720, 722, 743, 747, 748, 751 discriminant 127, 151 distance formula 80, 151, 701, 717 distributive property 8, 66 diverge 794, 828 dividend 395 Division Algorithm 395, 396, 403, 454 divisor 395 domain 160, 168, 180, 181, 267 domain and range 180 domain and range of inverse functions 258 domain of a composite function 216 domain of a rational function 420 doubling time 539, 543, 567 E eccentricity 743, 751 electrostatic force 199 elimination 608 |
ellipse 608, 685, 686, 687, 689, 692, 716, 743, 747, 751 ellipsis 758 end behavior 363, 424, 454 endpoint 198 entry 625, 674 equation 13, 66, 166 equation in quadratic form 138 equation in two variables 76, 151 equations in quadratic form 151 even function 233, 267 event 819, 828 experiment 819, 828 explicit formula 759, 775, 784, 828 exponent 6, 66 exponential 482 exponential decay 466, 472, 481, 539, 541, 544, 554 exponential equation 528 exponential function 466 exponential growth 466, 469, 539, 543, 545, 554, 567 exponential notation 6, 66 extraneous solution 134, 151, 532, 567 extrapolation 324, 325, 334 F factor by grouping 51, 66 Factor Theorem 404, 454 factorial 766 factoring 119 Fahrenheit 254 feasible region 611, 674 finite arithmetic sequence 776 finite sequence 759, 828 foci 685, 687, 700, 751 focus 685, 717, 720, 722, 742, 747, 748 focus (of an ellipse) 751 focus (of a parabola) 751 FOIL 44, 114 formula 13, 66, 166 function 160, 189, 267 function notation 162 Fundamental Counting Principle 804, 828 Fundamental Theorem of Algebra 408, 409, 454 G Gauss 594, 636 Gaussian elimination 594, 639, 674 general form 345 general form of a quadratic function 345, 347, 454 geometric sequence 781, 783, 793, 828 geometric series 793, 828 global maximum 388, 389, 454 global minimum 388, 389, 454 graph in two variables 76, 151 greatest common factor 49, 66, 119 C-1 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. C-2 H INDEX inverse property of multiplicative inverse of a power rule for logarithms 521, half-life 535, 539, 541, 567 Heaviside method 617 horizontal asymptote 417, 418, 423, 454 horizontal compression 237, 267 horizontal line 95, 297, 298, 334 horizontal line |
test 173, 267 horizontal reflection 229, 230, 267 horizontal shift 225, 267, 484, 505 horizontal stretch 237, 267 hyperbola 699, 702, 703, 704, 707, 708, 711, 717, 744, 746, 751 I identity equation 87, 151 identity matrix 649, 653, 674 identity property of addition 9, 66 identity property of multiplication 9, 66 imaginary number 111, 112, 151, 454 inconsistent equation 87, 151 inconsistent system 579, 586, 598, 674 increasing function 201, 267, 282 increasing linear function 283, 334 independent system 578, 579, 674 independent variable 160, 267 index 36, 66 index of summation 789, 790, 828 inequality 610 infinite geometric series 794 infinite sequence 759, 828 infinite series 794, 828 input 160, 267 integers 2, 5, 66 intercepts 79, 151 Intermediate Value Theorem 386, 454 interpolation 324, 334 intersection 821 interval 142, 151 interval notation 142, 151, 180, 184, 201, 267 inverse function 255, 267, 436, 439 inverse matrix 653, 655 inverse of a radical function 42 inverse of a rational function 444 inverse property of addition 9, 66 multiplication 9, 66 inverse variation 449, 454 inversely proportional 449, 454 invertible function 438, 454 invertible matrix 649, 663 irrational numbers 3, 5, 66 J joint variation 451, 454 L latus rectum 717, 722, 751 leading coefficient 42, 66, 366, 454 leading term 42, 66, 366, 454 least common denominator 60, 66, 89 least squares regression 325, 334 linear equation 87, 151 Linear Factorization Theorem 409, 454 linear function 280, 294, 309, matrix 649, 674 multiplicity 380, 454 mutually exclusive events 822, 828 N n factorial 766, 828 natural logarithm 498, 531, 567 natural numbers 2, 5, 67, 160 Newtonβs Law of Cooling 544, 567 nominal rate 474 nondegenerate conic section 729, 731, 751 nonlinear inequality 610, 674 nth partial sum 789, 828 nth term of a sequence 828 nth term of the sequence 759 O odd function 233, 267 one- |
to-one 482, 494, 519, 525 one-to-one function 170, 257, 334 267 linear growth 466 linear inequality 151 linear model 310, 322 linear relationship 322 local extrema 200, 267 local maximum 200, 267, 389 local minimum 200, 267, 389 logarithm 494, 567 logarithmic equation 533 logarithmic model 557 logistic growth model 546, 567 logistic regression 560 long division 394 lower limit of summation 789, 790 M magnitude 189, 224 main diagonal 638, 674 major and minor axes 687 major axis 685, 689, 751 matrix 625, 626, 630, 636, 674 matrix multiplication 630, 650, 655 matrix operations 626 maximum value 344 midpoint formula 82, 151 minimum value 344 minor axis 685, 689, 751 model breakdown 324, 334 modulus 189 monomial 42, 67 Multiplication Principle 803, 804, 828 multiplication property 143 multiplicative inverse 651 order of magnitude 540, 567 order of operations 6, 67 ordered pair 75, 151, 160, 181 ordered triple 594 origin 75, 151, 248 outcomes 819, 828 output 160, 267 P parabola 344, 350, 611, 716, 721, 723, 742, 745, 751 parallel 96 parallel lines 96, 298, 299, 334 parent function 505 partial fraction 615, 674 partial fraction decomposition 615, 674 Pascalβs Triangle 814 perfect square trinomial 45, 67 periapsis 742 perimeter 104, 151 permutation 804, 828 perpendicular 96 perpendicular lines 97, 299, 334 pH 518 piecewise function 189, 190, 267, 762 point-slope form 285, 334 point-slope formula 98, 705 polar equation 743, 751 polar form of a conic 748 polynomial 41, 42, 67, 405 polynomial equation 133, 151 polynomial function 365, 376, 383, 387, 455 power function 361, 455 525, 567 principal nth root 36, 67 principal square root 31, 67 probability 819, 828 probability model 819, 828 product of two matrices 630 product rule for logarithms 519, 521, 567 profit function 589, 674 properties of determinants 669 Proxima Centauri 540 Pythag |
orean Theorem 80, 127, 152 Q quadrant 74, 152 quadratic 138, 619, 621 quadratic equation 119, 120, 123, 125, 152 quadratic formula 125, 126, 152, 355 quadratic function 187, 347, 349 quotient 395 quotient rule for logarithms 520, 567 R radical 31, 67 radical equation 134, 135, 152 radical expression 31, 67 radical functions 438 radicand 31, 67, 134 radiocarbon dating 542 range 160, 267 rate of change 196, 267, 280 rational equation 89, 90, 152 rational exponents 37 rational expression 58, 67, 89, 615, 621 rational function 419, 426, 429, 455 rational number 2, 5, 67, 89 Rational Zero Theorem 405, 455 real number line 4, 67 real numbers 4, 67 reciprocal 96, 255, 362 reciprocal function 188, 415 recursive formula 764, 774, 783, 829 reflection 487, 510 regression analysis 554, 557, 560 regression line 326 relation 160, 267 relative 200 remainder 395 Remainder Theorem 403, 455 removable discontinuity 422, 455 restricting the domain 262 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. INDEX C-3 revenue function 588, 674 Richter Scale 493 roots 345, 455 row 625, 674 row matrix 626 row operations 638, 642, 652, 653, 654, 674 row-echelon form 638, 640, 674 trinomial 42, 67 turning point 369, 384, 455 U union of two events 821, 829 upper limit of summation 789, 790, 829 upper triangular form 594 row-equivalent 638, 674 V variable 11, 67 varies directly 448, 455 varies inversely 449, 455 vertex 344, 455, 685, 717, 723 vertex form of a quadratic function 346, 455 vertical asymptote 417, 420, 424, 455 vertical compression 234, 268 vertical line 95, 297, 298, 334 vertical line test 171, 268 vertical reflection 229, 230, 268 vertical shift 222, 223, 268, 293, 483, 506, 544 vertical stretch 234, 268, 292, 508 vertices 685, 687 volume 104, 152 volume of a sphere 361 W whole numbers 2, |
5, 67 X x-axis 74, 152 x-coordinate 75, 152 x-intercept 79, 152, 296 Y y-axis 74, 152 y-coordinate 75, 152 y-intercept 79, 152, 281, 282, 291, 296 Z zero-product property 120, 152 zeros 345, 377, 380, 405, 455 S sample space 819, 829 scalar 628 scalar multiple 628, 674 scalar multiplication 628 scatter plot 322 scientific notation 25, 26, 27, 67 sequence 758, 771, 829 series 789, 829 set-builder notation 142, 183, 184, 268 sigma 789 slope 92, 93, 152, 281, 282, 284, 334 slope-intercept form 281, 334 smooth curve 371, 455 solution set 87, 152, 595, 674 solving systems of linear equations 580, 582 square matrix 626, 663 square root function 188 square root property 123, 152 standard form 95 standard form of a quadratic function 346, 347, 455 stretch 486 substitution method 581, 674 summation notation 789, 790, 829 surface area 436 synthetic division 398, 407, 455 system of equations 637, 638, 640, 641, 655 system of linear equations 315, 578, 580, 581, 675 system of nonlinear equations 605, 675 system of nonlinear inequalities 611, 675 system of three equations in three variables 666 T term 758, 772, 829 term of a polynomial 41, 42, 67 term of a polynomial function 365, 455 transformation 222, 292 translation 721 transverse axis 700, 751 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. introduced. We move on, in Section 2.4, to simplifying trigonometric expressions and proving that a trigonometric equation is an identity. Then, finally, Section 2.5 introduces definitions for circular functions of varying radii, along with applications. 32 The Trigonometric Functions 2.1 Right Triangle Trigonometry Learning Objectives In this section you will: ο· Identify the trigonometric functions. ο· Learn the trigonometric function values for 30 degrees, 45 degrees and 60 degrees. ο· Solve right triangles and related application problems. As we shall see in the sections to come, many applications |
in trigonometry involve finding the measures of the interior angles, and the lengths of the sides, of right triangles. Recall that a right triangle is a triangle containing one right angle and two acute angles. In this section, we will define a new group of functions known as trigonometric functions that will assist us in determining unknown angle measures and/or side lengths for right triangles. Noting that two right triangles are similar if they have one congruent acute angle, we will use properties of similar triangles to establish trigonometric function values for three special angles: 30Β°, 45Β° and 60Β°. Similar Triangles We begin with a definition from geometry. Recall that two triangles are similar if they have the same shape or, more specifically, if their corresponding angles are congruent. Additionally, two triangles are similar if and only if their corresponding sides are proportional. In the following triangles,,, and. Thus, triangle ABC is similar to triangle RST and ARοοοBSοοοCTοοο.ABBCCARSSTTRο½ο½BACSRT 2.1 Right Triangle Trigonometry 33 Trigonometric Functions We consider the generic right triangle below with acute angle ΞΈ. The side with length a is called the side of the triangle adjacent to ΞΈ; the side with length b is called the side of the triangle opposite ΞΈ; and the remaining side c (the side opposite the right angle) is called the hypotenuse. The six commonly used trigonometric functions are defined below. The Trigonometric Functions: Suppose ΞΈ is an acute angle residing in a right triangle. If the length of the side adjacent to ΞΈ is a, the length of the side opposite ΞΈ is b, and the length of the hypotenuse is c, then ο· The sine of ΞΈ, denoted, is defined by. ο· The cosine of ΞΈ, denoted, is defined by ο· The tangent of ΞΈ, denoted, is defined by ο· The cotangent of ΞΈ, denoted, is defined by ο· The secant of ΞΈ, denoted, is defined by.... ο· The cosecant of ΞΈ, denoted, is defined by. ο¨ο©sinο±ο¨ο© |
oppositehypotenusesinbcο±ο½ο½ο¨ο©cosο±ο¨ο©adjacenthypotenusecosacο±ο½ο½ο¨ο©tanο±ο¨ο©oppositeadjacenttanbaο±ο½ο½ο¨ο©cotο±ο¨ο©adjacentoppositecotabο±ο½ο½ο¨ο©secο±ο¨ο©hypotenuseadjacentseccaο±ο½ο½ο¨ο©cscο±ο¨ο©hypotenuseoppositecsccbο±ο½ο½ 34 The Trigonometric Functions The following are important properties of the trigonometric functions. 1. For all right triangles with the same acute angle ΞΈ, because they are similar, the values of the resulting trigonometric functions of ΞΈ will be identical. This is a result of the property of equivalent proportions of corresponding sides within similar triangles. 2. Cosecant, secant and cotangent are reciprocal functions of sine, cosine and tangent, respectively. Thus, if we know the sine, cosine and tangent values for an angle, we can easily determine the remaining three trigonometric functions. In particular, Example 2.1.1. Use the following triangle to evaluate,,,, and. Solution. From the definitions of trigonometric functions, ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©1cscsin1seccos1cottanο±ο±ο±ο±ο±ο±ο½ο½ο½ο¨ο©sinο‘ο¨ο©cosο‘ο¨ο©tanο‘ο¨ο©cscο‘ο¨ο©secο‘ο¨ο©cotο‘ο¨ο©ο¨ο©ο¨ο©oppositehypotenuseadjacenthypotenuseopp |
ositeadjacent4sin53cos54tan3ο‘ο‘ο‘ο½ο½ο½ο½ο½ο½435ο‘ 2.1 Right Triangle Trigonometry 35 The reciprocals of these three function values result in the remaining three trigonometric function values: Example 2.1.2. Verify that the following triangle is similar to the triangle in Example 2.1.1. Then evaluate the trigonometric function values for the angle corresponding to Ξ±. Solution. The side lengths of the second triangle are proportional to the corresponding side lengths of the first triangle by a scale factor of 3: Thus the triangles are similar, with the angle Ξ² being equal in measure to Ξ±. To evaluate the trigonometric function values for Ξ², we save a bit of writing by using the abbreviations opp, adj and hyp in place of opposite, adjacent and hypotenuse, respectively. The trigonometric function values for this similar triangle will be ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©15cscsin415seccos313cottan4ο‘ο‘ο‘ο‘ο‘ο‘ο½ο½ο½ο½ο½ο½912153345ο½ο½ο½.ο¨ο©ο¨ο©ο¨ο©opphypadjhypoppadj12344sin153559333cos1535512344tan9333ο’ο’ο’οο½ο½ο½ο½οοο½ο½ο½ο½οοο½ο½ο½ο½ο12915 ο’ ο§ 36 The Trigonometric Functions Using reciprocal properties, the remaining three values are We note that the trigonometric function values are identical for the two similar triangles in Example 2.1.1 and Example 2.1.2, and observe that trigonometric ratios are not affected by the value of the scale factor. Pythagorean Theorem The Pythagorean Theorem will be useful in our next task: determining trigonometric function values for 30Β°, 45Β° and 60 |
Β° angles. The Pythagorean Theorem: The square of the hypotenuse in a right triangle is equal to the sum of the squares of the two shorter sides. In particular, in a right triangle with hypotenuse c and the shorter sides of lengths a and b, Trigonometric Functions of 30Β°, 60Β°, 90Β° Triangles We begin by finding the values of trigonometric functions for 30Β°. We sketch a 30Β°, 60Β°, 90Β° right triangle with hypotenuse of length x and envision this triangle as being half of a 60Β°, 60Β°, 60Β° equilateral triangle with sides of length x. ο¨ο©ο¨ο©ο¨ο©355csc344355sec333333cot344ο’ο’ο’οο½ο½οοο½ο½οοο½ο½ο222cabο½ο«xxx 30 60xl 30 60 2x 2.1 Right Triangle Trigonometry 37 Noting that the altitude of the equilateral triangle bisects its base, it follows that the shortest side of the 30Β°, 60Β°, 90Β° triangle has length. We apply the Pythagorean Theorem to determine the length, l, of the third side of the 30Β°, 60Β°, 90Β° triangle in terms of x. Using the resulting side lengths, along with the definitions of the trigonometric functions, we have Taking the reciprocals of these three function values results in the remaining three trigonometric function values: We note that these trigonometric function values apply to any 30Β° angle. The reader is encouraged to determine the trigonometric function values for 60Β° angles. 2x22222222243432xxxxlxlxllο¦οΆο«ο½ο§ο·ο¨οΈο½οο½ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©30303021sin2323cos221tan332xxxxxxο½ο½ο½ο½ο½ο½ο¨ο©ο¨οΏ½ |
οΏ½ο¨ο©ο¨ο©ο¨ο©ο¨ο©3030303030301csc2sin12seccos31cot3tanο½ο½ο½ο½ο½ο½x 30 60 2x 32x 38 The Trigonometric Functions Trigonometric Functions of 45Β°, 45Β°, 90Β° Triangles To find the values of the trigonometric functions for 45Β°, we sketch a 45Β°, 45Β°, 90Β° right isosceles triangle with hypotenuse h and remaining two sides each of length x. Using the Pythagorean Theorem, the hypotenuse can be written in terms of x as follows. The resulting trigonometric function values for 45Β° are After rationalizing denominators and adding trigonometric functions for 60Β°, we summarize the trigonometric function values for these special cases in the following table. 2222222xxhxhhxο«ο½ο½ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©4545454545454545451sin221cos22tan11csc2sin1sec2cos1cot1tanxxxxxxο½ο½ο½ο½ο½ο½ο½ο½ο½ο½ο½ο½hxx 45 45xx 45 45 2x 2.1 Right Triangle Trigonometry 39 Trigonometric Function Values for 30Β°, 45Β° and 60Β° ΞΈ 30Β° 45Β° 60Β° 2 1 1 2 Solving Right Triangles We will use these values in the next four examples to determine measures of missing angles and sides. This is sometimes referred to as solving right triangles. Example 2.1.3. Find the measure of the missing angle and the lengths of the missing sides of: Solution. The first and easiest task is to find the measure of the missing angle. Since the sum of the angles of a triangle is 180Β°, we know that the missing angle has measure. We now proceed to find the lengths of the remaining two sides of the triangle. Let c denote the length of the hypotenuse of the triangle. From, we |
get ο¨ο©sinο±ο¨ο©cosο±ο¨ο©tanο±ο¨ο©cscο±ο¨ο©secο±ο¨ο©cotο±12323323332222223212323333180309060οοο½ο¨ο©307coscο½ο¨ο©ο¨ο©ο¨ο©3030307cos17cos7sec.cccο½ο½οο½ο 40 The Trigonometric Functions Since, we arrive at the length of the hypotenuse:. At this point, we have two ways to proceed to find the length of the side opposite the 30Β° angle, which weβll denote b. We know the length of the adjacent side is 7 and the length of the hypotenuse is, so we could find the missing side by applying the Pythagorean Theorem and solving the following for b: Alternatively, we could use the definition,, to get. Choosing the latter, we find The triangle with all of its recorded data follows. Example 2.1.4. A right triangle has one angle of 60Β° and a hypotenuse of 20. Find the unknown side lengths and missing angle measure. Solution. Again, we begin with finding the measure of the missing angle. The sum of the angles of a triangle is 180Β°, from which it follows that the missing angle measure is. We assign the missing side lengths as a, for the side adjacent to 60Β°, and b, for the side opposite 60Β°. ο¨ο©3023sec3ο½1433cο½1433ο¨ο©2221437.3bο¦οΆο«ο½ο§ο·ο¨οΈο¨ο©oppositetanadjacentο±ο½ο¨ο©30tan7bο½ο¨ο©307tan37373.3bο½ο½οο½180609030οοο½ 2.1 Right Triangle Trigonometry 41 Since, we find We find length a using the Pythagorean |
Theorem, although the same result could be achieved through solving for a. The triangle with all of its data is recorded below. ο¨ο©60sin20bο½ο¨ο©6020sin3202103.bο½ο½οο½ο¨ο©60cos20aο½ο¨ο©22221032040030010.aaaο«ο½ο½οο½ba20 601020 60 30 103 42 The Trigonometric Functions Solving Applied Problems Right triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. The following example uses trigonometric functions as well as the concept of an βangle of inclinationβ. The angle of inclination, commonly known as the angle of elevation, of an object refers to the angle whose initial side is some kind of horizontal base-line (say, the ground), and whose terminal side is the line-of-sight to an object above the base-line. This is represented schematically below. The angle of inclination (elevation) from the base line to the object is ΞΈ. Example 2.1.5. The angle of inclination, from a point on the ground 30 feet away from the base of a water tower, to the top of the water tower, is 60Β°. Find the height of the water tower to the nearest foot. Solution. We can represent the problem situation using a right triangle as shown. If we let h denote the height of the tower, then we have. From this we get Hence, the water tower is approximately 52 feet tall. Example 2.1.6. In order to determine the height of a California redwood tree, two sightings from the ground, one 200 feet directly behind the other, are made. If the angles of inclination were 45Β° and 30Β°, respectively, how tall is the tree to the nearest foot. ο¨ο©60tan30hο½ο¨ο©6030tan30351.96hο½ο½ο» 2.1 Right Triangle Trigonometry 43 Solution. Sketching the problem situation below, we find ourselves with two unknowns: the height h of the tree |
and the distance x from the base of the tree to the first observation point. Using trigonometric functions, we get a pair of equations: and. Since, the first equation gives, or. Substituting this into the second equation gives Clearing fractions, we get. The result is a linear equation for h, so we proceed to expand the right hand side and gather all the terms involving h to one side. Hence, the tree is approximately 273 feet tall. Example 2.1.7. How long must a ladder be to reach a windowsill 50 feet above the ground if the ladder is resting against the building at an angle of 75Β° with the ground? ο¨ο©45tanhxο½ο¨ο©30tan200hxο½ο«ο¨ο©45tan1ο½1hxο½xhο½ο¨ο©30tan2003or.2003hhhhο½ο«ο½ο«ο¨ο©32003hhο½ο«ο¨ο©ο¨ο©320033320033320033320032003273.2133hhhhhhhhο½ο«ο½ο«ο½οο½ο½ο»οο 44 The Trigonometric Functions Solution. We know that the angle of inclination, or elevation, is 75Β° and that the opposite side is 50 feet in length. The length of the hypotenuse, h, will give us the necessary length for the ladder to reach a height of 50 feet. Using the trigonometric function for sine of 75Β°, we have Noting that trigonometric function values for 75Β° are not included in the table for special cases, use of a calculator is necessary to find an approximate value for. It is good practice to verify that the calculator is set to the correct mode, in this case degrees, before proceeding with calculations. We have found that the height of the ladder is approximately 51.8 feet. This section leads us to the Unit Circle and an alternate definition for trigonometric functions. Through this new definition we will expand the domain for trigonometric functions to include angle measures outside the interval. ο¨ο©ο¨ο©757550sin50sin500.965925851.76hhhhο½οΏ½ |
οΏ½οΏ½ο»ο»ο¨ο©75sinο¨ο©0,90h50 75 2.1 Right Triangle Trigonometry 45 2.1 Exercises 1. For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle. 2. The tangent of an angle compares which sides of the right triangle? 3. What is the relationship between the two acute angles in a right triangle? In Exercises 4 and 5, use the given triangle to evaluate each trigonometric function of angle A. 4. Find,,,, and. ο¨ο©sinAο¨ο©cosAο¨ο©tanAο¨ο©cscAο¨ο©secAο¨ο©cotA 46 The Trigonometric Functions 5. Find,,,, and. In Exercises 6 β 13, find the measurement of the missing angle and the lengths of the missing sides. 6. Find ΞΈ, b, and c. 7. Find ΞΈ, a, and c. ο¨ο©sinAο¨ο©cosAο¨ο©tanAο¨ο©cscAο¨ο©secAο¨ο©cotA 2.1 Right Triangle Trigonometry 47 8. Find Ξ±, a, and b. 9. Find Ξ², a, and c. 10. Find ΞΈ, a, and c. 48 11. Find Ξ±, b, and c. The Trigonometric Functions 12. Find ΞΈ, a, and c. 13. Find Ξ², b, and c. In Exercises 14 β 25, assume that ΞΈ is an acute angle in a right triangle. 14. If and the side adjacent to ΞΈ has length 4, how long is the hypotenuse? 15. If and the hypotenuse has length 5280, how long is the side adjacent to ΞΈ? 16. If and the side opposite ΞΈ has length 117.42, how long is the hypotenuse? 17. If and the hypotenuse has length 10, how long is the side opposite ΞΈ? 12ο±ο½78.123ο±ο½59ο±ο½5ο±ο½ 2.1 Right |
Triangle Trigonometry 49 18. If and the hypotenuse has length 10, how long is the side adjacent to ΞΈ? 19. If and the side opposite ΞΈ has length 306, how long is the side adjacent to ΞΈ? 20. If and the side opposite ΞΈ has length 4, how long is the side adjacent to ΞΈ? 21. If and the hypotenuse has length 10, how long is the side opposite ΞΈ? 22. If and the side adjacent to ΞΈ has length 2, how long is the side opposite ΞΈ? 23. If 24. If and the side opposite ΞΈ has length 14, how long is the hypotenuse? and the hypotenuse has length 3.98, how long is the side adjacent to ΞΈ? 25. If and the side adjacent to ΞΈ has length 31, how long is the side opposite ΞΈ? 26. Find x. 27. Find x. 5ο±ο½37.5ο±ο½30ο±ο½15ο±ο½87ο±ο½38.2ο±ο½2.05ο±ο½42ο±ο½ The Trigonometric Functions 50 28. Find x. 29. Find x. 30. A tree standing vertically on level ground casts a 120 foot long shadow. The angle of elevation from the end of the shadow to the top of the tree is 21.4Β°. Find the height of the tree to the nearest foot. With the help of your classmates, research the term umbra versa and see what it has to do with the shadow in this problem. 31. The broadcast tower for radio station WSAZ (Home of βAlgebra in the Morning with Carl and Jeffβ) has two enormous flashing red lights on it: one at the very top and one a few feet below the top. From a point 5000 feet away from the base of the tower on level ground the angle of elevation to the top light is 7.970Β° and to the second light is 7.125Β°. Find the distance between the lights to the nearest foot. 32. In this section, we defined the angle of inclination (also known as the angle of elevation) and in this exercise we introduce a related angle β the angle of depression (also known as the angle of declination). The angle of depression of an object refers to the angle whose initial side is a |
horizontal line above the object and whose terminal side is the line-of-sight to the object below the horizontal. This is represented schematically below. 2.1 Right Triangle Trigonometry 51 The angle of depression from the horizontal to the object is ΞΈ. (a) Show that if the horizontal is above and parallel to level ground then the angle of depression (from observer to object) and the angle of inclination (from object to observer) will be congruent because they are alternate interior angles. (b) From a fire tower 200 feet above level ground in the Sasquatch National Forest, a ranger spots a fire off in the distance. The angle of depression to the fire is 2.5Β°. How far away from the base of the tower is the fire? (c) The ranger in part (b) sees a Sasquatch running directly from the fire towards the fire tower. The ranger takes two sightings. At the first sighting, the angle of depression from the tower to the Sasquatch is 6Β°. The second sighting, taken just 10 seconds later, gives the angle of depression as 6.5Β°. How far did the Sasquatch travel in those 10 seconds? Round your answer to the nearest foot. How fast is it running in miles per hour? Round your answer to the nearest mile per hour. If the Sasquatch keeps up this pace, how long will it take for the Sasquatch to reach the fire tower from his location at the second sighting? Round your answer to the nearest minute. 33. When I stand 30 feet away from a tree at home, the angle of elevation to the top of the tree is 50Β° and the angle of depression to the base of the tree is 10Β°. What is the height of the tree? Round your answer to the nearest foot. 34. From the observation deck of the lighthouse at Sasquatch Point 50 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the lighthouse. The first sighting had an angle of depression of 8.2Β° and the second sighting had an angle of depression of 25.9Β°. How far had the boat traveled between the sightings? 35. A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it makes a 43Β° angle with the ground. How tall is the tower? How far away from the base of the tower does the wire hit the ground? 52 The Trigonometric Functions 36 |
. A 33 foot ladder leans against a building, so that the angle between the ground and the ladder is 80Β°. How high does the ladder reach up the side of the building? 37. A 23 foot ladder leans against a building so that the angle between the ground and the ladder is 80Β°. How high does the ladder reach up the side of the building? 38. The angle of elevation to the top of a building in New York City is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. 39. The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building. 40. Assuming that a 370 foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60Β°, how far from the base of the tree am I? 41. Let Ξ± and Ξ² be the two acute angles of a right triangle. (Thus Ξ± and Ξ² are complementary angles.) Show that and. The fact that co-functions of complementary angles are equal in this case is not an accident and a more general result will be given in Chapter 4. 42. Let Ξ± and Ξ² be the two acute angles of a right triangle. (Thus Ξ± and Ξ² are complementary angles.) Show that and. ο¨ο©ο¨ο©sincosο‘ο’ο½ο¨ο©ο¨ο©sincosο’ο‘ο½ο¨ο©ο¨ο©seccscο‘ο’ο½ο¨ο©ο¨ο©tancotο‘ο’ο½ 2.2 Determining Cosine and Sine Values from the Unit Circle 53 2.2 Determining Cosine and Sine Values from the Unit Circle Learning Objectives In this section you will: ο· Sketch oriented arcs on the Unit Circle. ο· Determine the cosine and sine values of an angle from a point on the Unit Circle. ο· Learn and apply the Pythagorean identity. ο· Apply the Reference Angle Theorem. ο· Learn the cosine |
and sine values for the common angles: 0Β°, 30Β°, 45Β°, 60Β° and 90Β°, or for their equivalent radian measures. ο· Learn the signs of the cosine and sine functions in each quadrant. We have already defined the Trigonometric Functions as functions of acute angles within right triangles. In this section, we will expand upon that definition by redefining the cosine and sine functions using the Unit Circle. Thus, the new designation βCircular Functionsβ will often be used in place of βTrigonometric Functionsβ as we adopt this new definition. The Unit Circle Consider the Unit Circle,, as shown, with the angle ΞΈ in standard position and the corresponding arc measuring s units in length. By the definition established in Section 1.2, and the fact that the Unit Circle has radius 1, the radian measure of ΞΈ is so that, once again blurring the distinction between an angle and its measure, we have. In order to identify real numbers with oriented angles, we make good use of this fact by essentially βwrappingβ the real number line around the Unit Circle and associating to each real number t an oriented arc on the Unit Circle with initial point. 221xyο«ο½1ssrsο½ο½sο±ο½ο¨ο©1,0xy11 ο± s 54 The Trigonometric Functions Given a real number and the vertical line containing the (vertical) interval, we βwrapβ the interval around the Unit Circle in a counter-clockwise fashion. The resulting arc has a length of t units. Therefore, the corresponding angle has radian measure equal to t. If, we wrap the interval clockwise around the Unit Circle. Since we have defined clockwise rotation as having negative radian measure, the angle determined by this arc has the negative radian measure equal to t. Note that if, we are at the point on the x-axis which corresponds to an angle with radian measure 0. Thus, we identify each real number t with the corresponding angle having radian measure of t. Example 2.2.1. Sketch the oriented arc on the Unit Circle corresponding to each of the following real numbers. 1. 2. 3. 4. Solution. 1. The arc associated with is the arc on the Unit Circle which subtends the angle in radian measure. Since is of a |
revolution, we have an arc which begins at the point and proceeds counter-clockwise up to midway through Quadrant II. 2. Since one revolution is radians, and is negative, we graph the arc which begins at and proceeds clockwise for one full revolution. 0tοΎ1xο½οο0,tοο0,t0tοΌοο,0t0tο½ο¨ο©1,034tο°ο½2tο°ο½ο2tο½ο117tο½34tο°ο½34ο°34ο°38ο¨ο©1,02ο°2tο°ο½οο¨ο©1,0xy11 t t 0tοΎxy11 t t 0tοΌxy11 34tο°ο½ 2.2 Determining Cosine and Sine Values from the Unit Circle 55 3. Like, is negative, so we begin our arc at and proceed clockwise around the Unit Circle. With and, we find rotating 2 radians clockwise from the point lands us in Quadrant III between and. To more accurately place the endpoint, we proceed as we did in Example 1.1.3, successively halving the angle measure until we find, which tells us our arc extends, clockwise, almost a quarter of the way into Quadrant III. 4. Since 117 is positive, the arc corresponding to begins at and proceeds counter-clockwise. As 117 is much greater than, we wrap around the Unit Circle several times before finally reaching our endpoint. We approximate as 18.62 which tells us we complete 18 revolutions counter-clockwise with 0.62, or just shy of of a revolution, remaining. In other words, the terminal side of the angle which measures 117 radians in standard position is just short of being midway through Quadrant III. The Cosine and Sine as Circular Functions This leads to our new definition for the cosine and sine functions. Consider an angle ΞΈ in standard position and let P denote the point where the terminal side of ΞΈ intersects the Unit Circle. 2tο°ο½ο2tο½οο¨ο© |
1,01.572ο°ο»3.14ο°ο»ο¨ο©1,02ο°οο°ο51.968ο°οο»ο117tο½ο¨ο©1,02ο°1172ο°58xy11 2tο°ο½οxy11 2tο½οxy11 117tο½ 56 The Trigonometric Functions By associating the point P with the angle ΞΈ, we are assigning a position on the Unit Circle to the angle ΞΈ. The x-coordinate of P is called the cosine of ΞΈ, written, while the y-coordinate of P is called the sine of ΞΈ, written.15 The reader is encouraged to verify that these rules that match an angle with its cosine and sine satisfy the definition of a function: for each angle ΞΈ, there is only one associated value of and only one associated value of. It is important to note that any angle that is not labeled as being in degrees is, by default, assumed to be in radians. In the following example, the angles in number 2 and number 4 are radian measures: radians and radians, respectively. Example 2.2.2. Find the cosine and sine of the following angles. 1. 2. 3. 4. 5. Solution. 1. To find and, we plot the angle in standard position and find the point on the terminal side of ΞΈ which lies on the Unit Circle. Since 270Β° represents of a counter-clockwise revolution, the terminal side of ΞΈ lies along the negative y-axis. Hence, the point we seek is so that and. 15 The etymology of the name βsineβ is quite colorful, and the interested reader is invited to research it; the βcoβ in βcosineβ is explained in Section 4.2. ο¨ο©cosο±ο¨ο©sinο±ο¨ο©cosο±ο¨ο©sinο±ο±ο°ο½ο6ο°ο±ο½270ο±ο½ο±ο°ο½ο45οΏ½ |
οΏ½οΏ½ο½6ο°ο±ο½60ο±ο½ο¨ο©270cosο¨ο©270sin270ο±ο½34ο¨ο©0,1οο¨ο©270cos0ο½ο¨ο©270sin1ο½ο 2.2 Determining Cosine and Sine Values from the Unit Circle 57 2. The angle represents one half of a clockwise revolution so its terminal side lies on the negative x-axis. The point on the Unit Circle that lies on the negative x-axis is, from which and. 3. When we sketch in standard position, we see that its terminal side does not lie along any of the coordinate axes. We let denote the point on the terminal side of ΞΈ which lies on the Unit Circle. By definition, and. If we drop a perpendicular line segment from P to the x-axis, we obtain a 45Β°, 45Β°, 90Β° right isosceles triangle whose legs have lengths x and y units. From the properties of isosceles triangles, it follows that. ο±ο°ο½οο¨ο©1,0οο¨ο©cos1ο°οο½οο¨ο©sin0ο°οο½45ο±ο½ο¨ο©,Pxyο¨ο©45cosxο½ο¨ο©45sinyο½yxο½ 58 The Trigonometric Functions lies on the Unit Circle, so. Substituting into this equation yields, or Now, lies in the first quadrant where, so. Since, we can also conclude that. Finally, we have and. 4. As before, the terminal side does not lie on any of the coordinate axes so we proceed using a triangle approach. Letting denote the point on the terminal side of ΞΈ which lies on the Unit Circle,we drop a perpendicular line segment from P to the x-axis to form a 30Β°, 60Β°, 90Β° right triangle. ο¨ο©,Pxy221xyο«ο½yxο½221xο½12.22xο½ο± |
ο½ο±ο¨ο©,Pxy0xοΎ22xο½yxο½22yο½ο¨ο©2cos452ο½ο¨ο©452sin2ο½ο¨ο©,Pxy 2.2 Determining Cosine and Sine Values from the Unit Circle 59 Noting that we have half of an equilateral triangle with sides of length 1, we find, so. Since lies on the Unit Circle, we substitute into to get, or. In the first quadrant, so. 5. Plotting in standard position, we find ΞΈ is not a quadrantal angle and set about using a triangle approach. Once again, we get a 30Β°, 60Β°, 90Β° right triangle and, after computations similar to part 4 of this example, we find and. It is not by accident that the last three angles in Example 2.2.2 are 30Β°, 45Β° and 60Β° (or, and, respectively). In Section 2.1 we used right triangles to obtain these same cosine and sine values for 30Β°, 45Β° and 60Β°. In this section, the Unit Circle approach to calculating trigonometric function values allows us to expand the domain imposed by acute angles within a right triangle to include negative angles, and other angles outside the interval. Knowing which quadrant an angle ΞΈ terminates in will help us determine whether and are positive or negative, as indicated below. 12yο½ο¨ο©1sin62ο°ο½ο¨ο©,Pxy12yο½221xyο«ο½234xο½32xο½ο±0xοΎο¨ο©3cos62xο°ο½ο½60ο±ο½ο¨ο©601cos2xο½ο½ο¨ο©603sin2yο½ο½6ο°4ο°3ο°ο¨ο©0,90ο¨ο©cosο±ο¨ο©sinο± 60 The Trigonometric Functions Sign of Cosine and Sine in Each Quadrant The Pythagorean Identity In Example 2.2.2, it was quite easy |
to find the cosine and sine of the quadrantal angles, but for non- quadrantal angles, the task is more involved. In these latter cases, we made good use of the fact that the point lies on the Unit Circle,. If we substitute and into, we get the identity. An unfortunate convention, from a function notation perspective, is to write as and as. We will follow this convention. Thus, our identity results in the following theorem, one of the most important results in trigonometry. Theorem 2.1. The Pythagorean Identity: For any angle ΞΈ,. The moniker βPythagoreanβ brings to mind the Pythagorean Theorem, from which both the distance formula and the equation for a circle are ultimately derived. The word βidentityβ reminds us that, regardless of the angle ΞΈ, the equation in Theorem 2.1 is always true. If one of or is known, Theorem 2.1 can be used to determine the other, up to a sign. If, in addition, we know where the terminal side of ΞΈ lies when in standard position, we can remove the ambiguity of the sign and completely determine the missing value as the next example illustrates. ο¨ο©ο¨ο©ο¨ο©ο¨ο©,cos,sinPxyο±ο±ο½221xyο«ο½ο¨ο©cosxο±ο½ο¨ο©sinyο±ο½221xyο«ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©ο¨ο©2cosο±ο¨ο©2cosο±ο¨ο©ο¨ο©2sinο±ο¨ο©2sinο±ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©cosο±ο¨ο©sinο±ο¨ο©ο±xy I II III IV ο¨ο©ο¨οΏ½ |
οΏ½οΏ½cos0sin0ο±ο±οΎοΎ ο¨ο©ο¨ο©cos0sin0ο±ο±οΌοΎ ο¨ο©ο¨ο©cos0sin0ο±ο±οΌοΌ ο¨ο©ο¨ο©cos0sin0ο±ο±οΎοΌ 2.2 Determining Cosine and Sine Values from the Unit Circle 61 Example 2.2.3. Using the given information about ΞΈ, find the indicated value. 1. If ΞΈ is a Quadrant II angle with, find. with, find., find. 2. If 3. If Solution. 1. When we substitute into the Pythagorean identity,, we obtain. Solving, we find. Since ΞΈ is a Quadrant II angle, its terminal side, when plotted in standard position, lies in Quadrant II. In Quadrant II, the x-coordinates are negative. Hence,. 2. Substituting into gives. Since we are given that, we know ΞΈ is a Quadrant III angle. Since x and y are negative in Quadrant III, both sine and cosine are negative in Quadrant III. Hence, we conclude. 3. When we substitute into, we find. ο¨ο©3sin5ο±ο½ο¨ο©cosο±32ο°ο°ο±οΌοΌο¨ο©5cos5ο±ο½οο¨ο©sinο±ο¨ο©sin1ο±ο½ο¨ο©cosο±ο¨ο©3sin5ο±ο½ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©29cos125ο±ο«ο½ο¨ο©4cos5ο±ο½ο±ο¨ο©4cos5ο±ο½οο¨ο©5cos5ο±ο½οο¨οΏ½ |
οΏ½ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©225sin55ο±ο½ο±ο½ο±32ο°ο°ο±οΌοΌο¨ο©25sin5ο±ο½οο¨ο©sin1ο±ο½ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©cos0ο±ο½xy1 35 ο¨ο©3cos,5ο±ο¦οΆο§ο·ο¨οΈ The Trigonometric Functions 62 Symmetry Another tool which helps immensely in determining cosines and sines of angles is the symmetry inherent in the Unit Circle. Suppose, for instance, we wish to know the cosine and sine of. We plot ΞΈ in standard position and, as usual, let denote the point on the terminal side of ΞΈ which lies on the Unit Circle. Note that the terminal side of ΞΈ lies radians short of one half revolution. In Example 2.2.2, we determined that and. This means that the point on the terminal side of the angle, when plotted in standard position, is. From the figure, it is clear that the point can be obtained by reflecting the point about the y-axis. Hence, and. Reference Angles In the above scenario, angle is called the reference angle for the angle. 56ο°ο±ο½ο¨ο©,Pxy6ο°ο¨ο©3cos62ο°ο½ο¨ο©1sin62ο°ο½6ο°31,22ο¦οΆο§ο·ο¨οΈο¨ο©,Pxy31,22ο¦οΆο§ο·ο¨οΈο¨ο©53cos62ο°ο½οο¨ο©51sin62ο°ο½6ο°56ο° 2.2 Determining Cosine and Sine Values from the |
Unit Circle 63 In general, for a non-quadrantal angle ΞΈ, the reference angle for ΞΈ (usually denoted Ξ±) is the acute angle made between the terminal side of ΞΈ and the x-axis. ο· ο· If ΞΈ is a Quadrant I or IV angle, Ξ± is the angle between the terminal side of ΞΈ and the positive x- axis. If ΞΈ is a Quadrant II or III angle, Ξ± is the angle between the terminal side of ΞΈ and the negative x-axis. If we let P denote the point, then P lies on the Unit Circle. Since the Unit Circle possesses symmetry with respect to the x-axis, y-axis and origin, regardless of where the terminal side of ΞΈ lies, there is a point Q symmetric with P which determines ΞΈβs reference angle, Ξ±, as seen in the following illustration. Reference angle Ξ± for a Quadrant I angle Reference angle Ξ± for a Quadrant II angle Reference angle Ξ± for a Quadrant III angle Reference angle Ξ± for a Quadrant IV angle ο¨ο©ο¨ο©ο¨ο©cos,sinο±ο± 64 The Trigonometric Functions We have just outlined the proof of the following theorem. Theorem 2.2. Reference Angle Theorem: Suppose Ξ± is the reference angle for ΞΈ. Then and, where the sign, + or β, is determined by the quadrant in which the terminal side of ΞΈ lies. In light of Theorem 2.2, it pays to know the cosine and sine values for certain common angles. In the following table, we summarize the values which we consider essential. Cosine and Sine Values of Common Angles ΞΈ degrees ΞΈ radians 0Β° 30Β° 45Β° 60Β° 90Β° 0 1 0 0 1 Example 2.2.4. Find the cosine and sine of the following angles. 1. 2. 3. 4. ο¨ο©ο¨ο©coscosο±ο‘ο½ο±ο¨ο©ο¨ο©sinsinο±ο‘ο½ο±ο¨ο©cosο±ο¨ο©sinο±6ο°32124ο°22223ο°12322ο°225ο±οΏ½ |
οΏ½οΏ½116ο°ο±ο½54ο°ο±ο½ο73ο°ο±ο½xy ο¨ο©1,0 13,22ο¦οΆο§ο·ο§ο·ο¨οΈ 22,22ο¦οΆο§ο·ο§ο·ο¨οΈ 31,22ο¦οΆο§ο·ο§ο·ο¨οΈ ο¨ο©0,1 2.2 Determining Cosine and Sine Values from the Unit Circle 65 Solution. 1. We begin by plotting in standard position and find its terminal side overshoots the negative x-axis to land in Quadrant III. Hence, we obtain a reference angle Ξ± by subtracting: Since ΞΈ is a Quadrant III angle, and. The Reference Angle Theorem yields: and. 2. The terminal side of, when plotted in standard position, lies in Quadrant IV, just shy of the positive x- axis. To find the reference angle Ξ±, we subtract: Since ΞΈ is a Quadrant IV angle, and, so the Reference Angle Theorem gives: and. 3. To plot, we rotate clockwise an angle of from the positive x-axis. The terminal side of ΞΈ, therefore, lies in Quadrant II making an angle of radians with respect to the negative x- 225ο±ο½18022518045.ο‘ο±ο½οο½οο½ο¨ο©cos0ο±οΌο¨ο©sin0ο±οΌο¨ο©ο¨ο©225452coscos2ο½οο½οο¨ο©ο¨ο©225452sinsin2ο½οο½ο116ο°ο±ο½116622.ο°ο°ο‘ο°ο±ο°ο½οο½οο½ο¨ο©cos0ο±οΎο¨ο©sin |
0ο±οΌο¨ο©ο¨ο©113coscos662ο°ο°ο½ο½ο¨ο©ο¨ο©111sinsin662ο°ο°ο½οο½ο54ο°ο±ο½ο54ο°544ο°ο°ο‘ο°ο½οο½ 66 The Trigonometric Functions axis. Since ΞΈ is a Quadrant II angle, the Reference Angle Theorem gives: and. 4. Since the angle measures more than, we find the terminal side of ΞΈ by rotating one full revolution followed by an additional radians. Since ΞΈ and Ξ± are coterminal, and. The reader may have noticed that, when expressed in radian measure, the reference angle for a non- quadrantal angle is easy to spot. Reduced fraction multiples of Ο with a denominator of 6 have as a reference angle. Those with a denominator of 4 have as their reference angle, and those with a denominator of 3 have as their reference angle. ο¨ο©ο¨ο©52coscos442ο°ο°οο½οο½οο¨ο©ο¨ο©52sinsin442ο°ο°οο½ο½73ο°ο±ο½623ο°ο°ο½7233ο°ο°ο‘ο°ο½οο½ο¨ο©ο¨ο©71coscos332ο°ο°ο½ο½ο¨ο©ο¨ο©73sinsin332ο°ο°ο½ο½6ο°4ο°3ο° 2.2 Determining Cosine and Sine Values from the Unit Circle 67 The next example summarizes all of the important ideas discussed in this section. Example 2.2.5. Suppose Ξ± is an acute angle with. 1. Find and use this to plot Ξ± in standard position. 2. Find the sine and cosine of the following angles. (a) |
(b) (c) (d) ο¨ο©5cos13ο‘ο½ο¨ο©sinο‘ο±ο°ο‘ο½ο«2ο±ο°ο‘ο½ο3ο±ο°ο‘ο½ο2ο°ο±ο‘ο½ο« 68 Solution. The Trigonometric Functions 1. Proceeding as in Example 2.2.3, we substitute into and find. Since Ξ± is an acute (and therefore Quadrant I) angle, is positive. Hence,. To plot Ξ± in standard position, we begin our rotation from the positive x-axis to the ray which contains the point. 2. (a) To find the cosine and sine of, we first plot ΞΈ in standard position. We can imagine the sum of the angles as a sequence of two rotation: a rotation of Ο radians followed by a rotation of Ξ± radians.16 We see that Ξ± is the reference angle for ΞΈ, so by the Reference Angle Theorem, and. Since the terminal side of ΞΈ falls in Quadrant III, both and are negative. Hence, and. (b) Rewriting as, we can plot ΞΈ by visualizing one complete revolution counter-clockwise followed by a clockwise revolution, or βbacking upβ of Ξ± radians. We see that Ξ± is ΞΈβs reference angle, and since ΞΈ is a Quadrant IV angle, the Reference Angle Theorem gives: and. 16 Since Ο + Ξ± = Ξ± + Ο, ΞΈ may be plotted by reversing the order of rotations given here. Try it! ο¨ο©5cos13ο‘ο½ο¨ο©ο¨ο©22cossin1ο‘ο‘ο«ο½ο¨ο©12sin13ο‘ο½ο±ο¨ο©sinο‘ο¨ο©12sin13ο‘ο½ο¨ο©ο¨ο©ο¨ο©512cos,sin,1313ο‘ο‘ο¦οΆο½ο§ο·ο¨ |
οΈο±ο°ο‘ο½ο«ο°ο‘ο«ο¨ο©ο¨ο©5coscos13ο±ο‘ο½ο±ο½ο±ο¨ο©ο¨ο©12sinsin13ο±ο‘ο½ο±ο½ο±ο¨ο©cosο±ο¨ο©sinο±ο¨ο©5cos13ο±ο½οο¨ο©12sin13ο±ο½ο2ο±ο°ο‘ο½οο¨ο©2ο±ο°ο‘ο½ο«οο¨ο©5cos13ο±ο½ο¨ο©12sin13ο±ο½ο 2.2 Determining Cosine and Sine Values from the Unit Circle 69 (c) Taking a cue from the previous problem, we rewrite as. The angle represents one and a half revolutions counter-clockwise, so that when we βback upβ Ξ± radians, we end up in Quadrant II. Using the Reference Angle Theorem, we get and. (d) To plot, we first rotate radians and follow up with Ξ± radians. The reference angle here is not Ξ±, so the Reference Angle Theorem is not immediately applicable. (Itβs important that you see why this is the case. Take a moment to think about this before reading on.) Let be the point on the terminal side of ΞΈ which lies on the Unit Circle so that and. Once we graph Ξ± in standard position, we use the fact that equal angles subtend equal chords to show that the dotted lines in the following figure are equal. Hence,. Similarly, we find. 3ο±ο°ο‘ο½οο¨ο©3ο±ο°ο‘ο½ο«ο3ο°ο¨ο©5cos13ο±ο½οο¨ο©12sin13ο±ο½2ο° |
ο±ο‘ο½ο«2ο°ο¨ο©,Qxyο¨ο©cosxο±ο½ο¨ο©sinyο±ο½ο¨ο©12cos13xο±ο½ο½οο¨ο©5sin13yο±ο½ο½ 70 The Trigonometric Functions We close this section by noting that we can easily extend the functions cosine and sine to real numbers by identifying a real number t with the angle radians. Using this identification, we define and. In practice this means expressions like and can be found by regarding the inputs as angles in radian measure or real numbers; the choice is the readerβs. If we trace the identification of real numbers t with angles ΞΈ in radian measure to its roots, as explained at the beginning of this section, we can spell out this correspondence more precisely. For each real number t, we associate an oriented arc t units in length with initial point and endpoint. tο±ο½ο¨ο©ο¨ο©coscostο±ο½ο¨ο©ο¨ο©sinsintο±ο½ο¨ο©cosο°ο¨ο©2sinο¨ο©1,0ο¨ο©ο¨ο©ο¨ο©cos,sinPtt 2.2 Determining Cosine and Sine Values from the Unit Circle 71 2.2 Exercises In Exercises 1 β 5, sketch the oriented arc on the Unit Circle which corresponds to the given real number. 1. 2. 3. 4. 5. In Exercises 6 β 9, use the given sign of the cosine and sine functions to find the quadrant in which the terminal point determined by t lies. 6. 8. and and 7. 9. and and 10. Use the numbers 0, 1, 2, 3 and 4 to complete the following table of cosine and sine values for common angles. (This exercise serves as a memory tool for remembering these values.) Ξ 0 In Exercises 11 β 30, find the exact value of the cosine and sine of the given angle. 11. 15. 12 |
. 16. 13. 17. 14. 18. 56tο°ο½tο°ο½ο6tο½2tο½ο12tο½ο¨ο©cos0tοΌο¨ο©sin0tοΌο¨ο©cos0tοΎο¨ο©sin0tοΎο¨ο©cos0tοΌο¨ο©sin0tοΎο¨ο©cos0tοΎο¨ο©sin0tοΌο¨ο©cosο±ο¨ο©sinο±226ο°224ο°223ο°222ο°220ο±ο½4ο°ο±ο½3ο°ο±ο½2ο°ο±ο½23ο°ο±ο½34ο°ο±ο½ο±ο°ο½76ο°ο±ο½ 72 19. 23. 27. The Trigonometric Functions 20. 24. 28. 21. 25. 29. 22. 26. 30. In Exercises 31 β 40, use the results developed throughout the section to find the requested value. 31. If with ΞΈ in Quadrant IV, what is? 32. If with ΞΈ in Quadrant I, what is? 33. If with ΞΈ in Quadrant II, what is? 34. If with ΞΈ in Quadrant III, what is? 35. If with ΞΈ in Quadrant III, what is 36. If with ΞΈ in Quadrant IV, what is 37. If and, what is? 38. If and, what is 39. If and, what is???? 40. If and, what is? 54ο°ο±ο½43ο°ο±ο½32ο°ο±ο½53ο°ο±ο½74ο°ο±ο½236ο°ο±ο½132ο°ο±ο½ο436ο°ο±ο½ο34οΏ½ |
οΏ½ο±ο½ο6ο°ο±ο½ο103ο°ο±ο½117ο±ο°ο½ο¨ο©7sin25ο±ο½οο¨ο©cosο±ο¨ο©4cos9ο±ο½ο¨ο©sinο±ο¨ο©5sin13ο±ο½ο¨ο©cosο±ο¨ο©2cos11ο±ο½οο¨ο©sinο±ο¨ο©2sin3ο±ο½οο¨ο©cosο±ο¨ο©28cos53ο±ο½ο¨ο©sinο±ο¨ο©25sin5ο±ο½2ο°ο±ο°οΌοΌο¨ο©cosο±ο¨ο©10cos10ο±ο½522ο°ο°ο±οΌοΌο¨ο©sinο±ο¨ο©sin0.42ο±ο½ο32ο°ο°ο±οΌοΌο¨ο©cosο±ο¨ο©cos0.98ο±ο½ο2ο°ο±ο°οΌοΌο¨ο©sinο± 2.3 The Six Circular Functions 73 2.3 The Six Circular Functions Learning Objectives In this section you will: ο· Determine the values of the six circular functions from the coordinates of a point on the Unit Circle. ο· Learn and apply the reciprocal and quotient identities. ο· Learn and apply the Generalized Reference Angle Theorem. ο· Find angles that satisfy circular function equations. In this section, we extend the definition of cosine and sine as points on the Unit Circle to include the remaining four circular functions: tangent, cotangent, secant and cosecant. The Circular Functions The Circ |
ular Functions: Suppose ΞΈ is an angle plotted in standard position and is the point on the terminal side of ΞΈ which lies on the Unit Circle. The circular functions are defined as follows. ο· The sine of ΞΈ, denoted, is defined by ο· The cosine of ΞΈ, denoted, is defined by.. ο· The tangent of ΞΈ, denoted, is defined by, provided ο· The cosecant of ΞΈ, denoted, is defined by, provided.. ο· The secant of ΞΈ, denoted, is defined by, provided. ο· The cotangent of ΞΈ, denoted, is defined by, provided. In Section 2.2, we defined and for angles ΞΈ using the coordinate values of points on the Unit Circle. As such, these functions earn the moniker circular functions.17 17 In Section 2.1 we also showed cosine and sine to be functions of an angle residing in a right triangle so we could just as easily call them trigonometric functions. You will find that we do indeed use the phrase βtrigonometric functionβ interchangeably with βcircular functionβ. ο¨ο©,Pxyο¨ο©sinο±ο¨ο©sinyο±ο½ο¨ο©cosο±ο¨ο©cosxο±ο½ο¨ο©tanο±ο¨ο©tanyxο±ο½0xοΉο¨ο©cscο±ο¨ο©1cscyο±ο½0yοΉο¨ο©secο±ο¨ο©1secxο±ο½0xοΉο¨ο©cotο±ο¨ο©cotxyο±ο½0yοΉο¨ο©cosο±ο¨ο©sinο± 74 The Trigonometric Functions Historical Note: While we left the history of the name βsineβ as an interesting research project in Section 2.2, the names βtangentβ and βsecantβ can be explained using the diagram below. |
Consider the acute angle ΞΈ in standard position. Let denote, as usual, the point on the terminal side of ΞΈ which lies on the Unit Circle and let denote the point on the terminal side of ΞΈ which lies on the vertical line. The word βtangentβ comes from the Latin meaning βto touchβ. The line is a tangent line to the Unit Circle since it intersects, or touches, the circle at only one point, namely. Dropping perpendiculars from P and Q creates the pair of similar triangles βOPA and βOQB. Thus which gives, where this last equality comes from the definition of the tangent of ΞΈ. We have just shown that for acute angles ΞΈ, is the y-coordinate of the point on the terminal side of ΞΈ which lies on the tangent line. The word βsecantβ means βto cutβ. A secant line is any line that cuts through a circle at two points. The line containing the terminal side of ΞΈ is a secant line that intersects the Unit Circle in Quadrants I and III. With the point P lying on the Unit Circle, the length of the hypotenuse of βOPA is 1. If we let h denote the length of the hypotenuse of βOQB, we have, from similar triangles, that or. Hence, for an acute angle ΞΈ, is the length of the line segment which lies on the secant line determined by the terminal side of ΞΈ and βcuts offβ the tangent line. Not only do these observations help explain the names of these functions, they serve as the basis for a fundamental inequality needed for calculus which weβll explore in the Exercises. Of the six circular functions, only cosine and sine are defined for all angles. Since and in their definitions as circular functions, it is customary to rephrase the remaining four ο¨ο©,Pxyο¨ο©1,'Qy1xο½1xο½ο¨ο©1,0'1yyxο½ο¨ο©'tanyyxο±ο½ο½ο¨ο©tanο±1xο½11hxο½ο¨ο©1sechxο±ο½ο½ο¨ |
ο©secο±1xο½ο¨ο©cosxο±ο½ο¨ο©sinyο±ο½ 2.3 The Six Circular Functions 75 circular functions in terms of cosine and sine. The following theorem is a result of simply replacing x with and y with in the definitions presented at the beginning of this section. Reciprocal and Quotient Identities Theorem 2.3. Reciprocal and Quotient Identities: ο· ο· ο· ο·, provided ; if then is undefined., provided, provided, provided ; if ; if ; if then is undefined. then is undefined. then is undefined. Before using Theorem 2.3 in an example, the following mnemonic may help with remembering the signs of the trigonometric functions in each quadrant. We assign the first letter of each word in the phrase βAll Students Take Calculusβ to Quadrants I, II, III and IV, respectively. Note that cosine, sine and tangent are All positive in Quadrant I, the Sine alone is positive in Quadrant II, then Tangent alone is positive in Quadrant III and the Cosine alone is positive in Quadrant IV. It is high time for an example. Example 2.3.1. Find the indicated value, if it exists. 1. 2. 3. ο¨ο©cosο±ο¨ο©sinο±ο¨ο©ο¨ο©ο¨ο©sintancosyxο±ο±ο±ο½ο½ο¨ο©cos0ο±οΉο¨ο©cos0ο±ο½ο¨ο©tanο±ο¨ο©ο¨ο©ο¨ο©coscotsinxyο±ο±ο±ο½ο½ο¨ο©sin0ο±οΉο¨ο©sin0ο±ο½ο¨ο©cotο±ο¨ο©ο¨ο©11seccosxο±ο±ο½ο½ο¨ο©cos0οΏ½ |
οΏ½οΏ½οΉο¨ο©cos0ο±ο½ο¨ο©secο±ο¨ο©ο¨ο©11cscsinyο±ο±ο½ο½ο¨ο©sin0ο±οΉο¨ο©sin0ο±ο½ο¨ο©cscο±ο¨ο©60secο¨ο©7csc4ο°ο¨ο©3cotxyASTC The Trigonometric Functions 76 4. 5. 6. Solution., where ΞΈ is any angle coterminal with., where and ΞΈ is a Quadrant IV angle., where and. 1. From Theorem 2.3, the reciprocal identity for secant will help us out here. 2. We apply the reciprocal identity for cosecant and note that. 3. Since radians is not one of the common angles from Section 2.2, we resort to the calculator for a decimal approximation. We use the quotient identity for cotangent and check that our calculator is in radian mode. Noting that, this problem could also be solved as follows. ο¨ο©tanο±32ο°ο¨ο©cosο±ο¨ο©csc5ο±ο½οο¨ο©sinο±ο¨ο©tan3ο±ο½32ο°ο°ο±οΌοΌο¨ο©ο¨ο©ο¨ο©1260601seccos12ο½ο½ο½ο¨ο©72sin42ο°ο½οο¨ο©ο¨ο©71csc74sin41222ο°ο°ο½ο½ο¦οΆοο§ο·ο¨οΈο½ο3ο±ο½ο¨ο©ο¨ο©ο¨ο©333coscotsin7.015ο½ο»οο¨οΏ½ |
οΏ½ο¨ο©1cottanο±ο±ο½ 2.3 The Six Circular Functions 77 4. If ΞΈ is coterminal with, then and. Attempting to compute results in, so is undefined. 5. We are given that. From, it follows that. Then we have the following. 6. It is given that. From the quotient identity for tangent, we know. Be careful! We can NOT assume any values for and. We CAN assume that. ο¨ο©ο¨ο©331cottan7.015ο½ο»ο32ο°ο¨ο©ο¨ο©3coscos02ο±ο°ο½ο½ο¨ο©ο¨ο©3sinsin12ο±ο°ο½ο½οο¨ο©ο¨ο©ο¨ο©sintancosο±ο±ο±ο½10οο¨ο©tanο±ο¨ο©csc5ο±ο½οο¨ο©ο¨ο©1cscsinο±ο±ο½ο¨ο©1sin5ο±ο½οο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©2221cos154cos52cos52cos525cos5ο±ο±ο±ο±ο±ο¦οΆο«οο½ο§ο·ο¨οΈο½ο½ο±ο½ο½ Pythagorean identity cosΞΈ>0 in Quadrant IVο¨ο©tan3ο±ο½ο¨ο©ο¨ο©sin3cosο±ο±ο½ο¨ο©sinο±ο¨ο©cosο±ο¨ο©ο¨ο©sin3cosο±ο±ο½ 78 The |
Trigonometric Functions While the reciprocal and quotient identities presented in Theorem 2.3 allow us to always convert problems involving tangent, cotangent, secant and cosecant to problems involving cosine and sine, it is not always convenient to do so.18 The tangent and cotangent values of the common angles are summarized in the following chart. 18 As we shall see shortly, when solving equations involving secant and cosecant, we usually convert back to cosines and sines. However, when solving for tangent or cotangent, we usually stick with what weβre dealt. ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©222222sin3cos1sincos31sinsin131sinsin1910sin199sin103sin103sin10310sin10ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο½ο½ο¦οΆο«ο½ο§ο·ο¨οΈο«ο½ο½ο½ο½ο±ο½οο½οPythagorean identitysinΞΈ<0 in Quadrant III 2.3 The Six Circular Functions 79 Tangent and Cotangent Values of Common Angles ΞΈ degrees ΞΈ radians 0Β° 30Β° 45Β° 60Β° 90Β° 0 1 0 0 undefined 1 1 0 1 undefined 0 Finding Angles that Satisfy Cosine and Sine Equations Our next example asks us to solve some very basic trigonometric equations.19 Example 2.3.2. Find all of the angles which satisfy the given equation. 1. 2. 3. Solution. Since there is no context in the problem to indicate whether to use degrees or radians, we will default to using radian measure in each of these problems. This choice will be justified later in the text when we study what is known |
as Analytic Trigonometry. In those sections to come, radian measure will be the only appropriate angle measure so it is worth the time to become fluent in radians now. 1. If, then the terminal side of ΞΈ, when plotted in standard position, intersects the Unit Circle at. This means ΞΈ is a Quadrant I or IV angle with reference angle. 19 We will study trigonometric equations more formally in Chapter 6. Enjoy these relatively straightforward exercises while they last! ο¨ο©cosο±ο¨ο©sinο±ο¨ο©ο¨ο©ο¨ο©sincostanο±ο±ο±ο½ο¨ο©ο¨ο©ο¨ο©cossincotο±ο±ο±ο½6ο°32121333ο½34ο°22223ο°123231333ο½2ο°ο¨ο©1cos2ο±ο½ο¨ο©1sin2ο±ο½οο¨ο©cos0ο±ο½ο¨ο©1cos2ο±ο½12xο½3ο° 80 The Trigonometric Functions One solution in Quadrant I is, and since all other Quadrant I solutions must be coterminal with, we find for integers k.20 Proceeding similarly for the Quadrant IV case, we find the solution to is, so our answer in this quadrant is for integers k. 2. If, then when ΞΈ is plotted in standard position, its terminal side intersects the Unit Circle at. From this, we determine ΞΈ is a Quadrant III or Quadrant IV angle with reference angle. 20 Recall in Section 1.2, two angles in radian measure are coterminal if and only if they differ by an integer multiple of 2Ο. Hence to describe all angles coterminal with a given angle, we add 2Οk for integers k = 0, Β±1, Β±2, Β·Β·Β·. 3ο°ο±ο½3ο°23kο°ο±ο°ο½ο«ο¨ο©1cos2οΏ½ |
οΏ½ο½53ο°523kο°ο±ο°ο½ο«ο¨ο©1sin2ο±ο½ο12yο½ο6ο° 2.3 The Six Circular Functions 81 In Quadrant III, one solution is, so we capture all Quadrant III solutions by adding integer multiples of 2Ο:. In Quadrant IV, one solution is so all the solutions here are of the form for integers k. 3. The angles with are quadrantal angles whose terminal sides, when plotted in standard position, lie along the y-axis. While, technically speaking, isnβt a reference angle, we can nonetheless use it to find our answers. If we follow the procedure set forth in the previous examples, we find and for integers k. While this solution is correct, it can be shortened to for integers k. (Can you see why this works from the diagram?) One of the key items to take from Example 2.3.2 is that, in general, solutions to trigonometric equations consist of infinitely many answers. To get a feel for these answers, the reader is encouraged to follow our mantra βWhen in doubt, write it out!β This is especially important when checking answers to the Exercises. For example, in number 2, another Quadrant IV solution to is. Hence, the family of Quadrant IV answers to number 2 above could have been written for integers k. While on the surface this family may look different than the stated solution of for integers k, we leave it to the reader to show they represent the same list of angles. 76ο°726kο°ο±ο°ο½ο«116ο°1126kο°ο±ο°ο½ο«ο¨ο©cos0ο±ο½2ο°22kο°ο±ο°ο½ο«322kο°ο±ο°ο½ο«2kο°ο±ο°ο½ο«ο¨ο©1sin2ο±ο½ο6ο°ο±ο½ο26kο°ο±ο°ο½ο |
ο«1126kο°ο±ο°ο½ο« 82 The Trigonometric Functions Finding Angles that Satisfy Other Circular Function Equations Before determining angles in equations of the other four circular functions, we introduce the Generalized Reference Angle Theorem. This theorem results from coupling the reciprocal and quotient identities, Theorem 2.3, with the Reference Angle Theorem, Theorem 2.2. Theorem 2.4. Generalized Reference Angle Theorem: The values of the circular functions of an angle, if they exist, are the same, up to a sign, as the corresponding circular functions of the reference angle. More specifically, if Ξ± is the reference angle for ΞΈ, then,,,, and. The sign, + or β, is determined by the quadrant in which the terminal side of ΞΈ lies. We put Theorem 2.4 to good use in the following example. Example 2.3.3. Find all angles which satisfy the given equation. 1. Solution. 2. 3. 1. To solve, we convert to a cosine and get, or. This is the same equation we solved in Example 2.3.2, number 1, so we know the answer is or for integers k. 2. Noting that, we see. According to Theorem 2.4, we know the solutions to must, therefore, have a reference angle of. Our next task is to determine in which quadrants the solutions to this equation lie. ο¨ο©ο¨ο©sinsinο±ο‘ο½ο±ο¨ο©ο¨ο©coscosο±ο‘ο½ο±ο¨ο©ο¨ο©tantanο±ο‘ο½ο±ο¨ο©ο¨ο©csccscο±ο‘ο½ο±ο¨ο©ο¨ο©secsecο±ο‘ο½ο±ο¨ο©ο¨ο©cotcotο±ο‘ο½ο±ο¨ο©sec2ο±ο½ο¨ο©tan3ο±ο½ο¨ο©cot1 |
ο±ο½οο¨ο©sec2ο±ο½ο¨ο©12cosο±ο½ο¨ο©1cos2ο±ο½23kο°ο±ο°ο½ο«523kο°ο±ο°ο½ο«ο¨ο©ο¨ο©ο¨ο©sin3tan3cos3ο°ο°ο°ο½ο¨ο©tan33ο°ο½ο¨ο©tan3ο±ο½3ο° 2.3 The Six Circular Functions 83 Since tangent is defined as the ratio of points on the Unit Circle with, tangent is positive when x and y have the same sign (i.e., when they are both positive or both negative.) This happens in Quadrants I and III. In Quadrant I we get the solutions for integers k, and for Quadrant III we get for integers k. While these descriptions of the solutions are correct, they can be combined as for integers k. The latter form of the solution is best understood looking at the geometry of the situation in the following diagram.21 3. We see that has a cotangent of 1, which means the solutions to have a reference angle of. 21 See Example 2.3.2, number 3, for another example of this kind of simplification of the solution. yxο¨ο©,xy0xοΉ23kο°ο±ο°ο½ο«423kο°ο±ο°ο½ο«3kο°ο±ο°ο½ο«4ο°ο¨ο©cot1ο±ο½ο4ο° 84 The Trigonometric Functions To find the quadrants in which our solutions lie, we note that for a point on the Unit Circle where. If is negative, then x and y must have different signs (i.e., one positive and one negative.) Hence, our solutions lie in Quadrants II and IV. Our Quadrant II solution is, and for Quadrant IV we get, for integers k. Can these be combined? Indeed they can! One |
such way to capture all the solutions is for integers k. Suppose we are asked to solve an equation such as. As we have already mentioned, the distinction between t as a real number and as an angle radians is often blurred. Indeed, we solve in the exact same manner22 as we did in Example 2.3.2 number 2. Our solution is only cosmetically different in that the variable used is t rather than ΞΈ: or for integers k. As we progress in our study of the trigonometric functions, keep in mind that any properties developed which regard them as functions of angles in radian measure apply equally well if the inputs are regarded as real numbers. 22 Well, to be pedantic, we would be technically using reference numbers or reference arcs instead of reference angles, but the idea is the same. ο¨ο©cotxyο±ο½ο¨ο©,xy0yοΉο¨ο©cotο±324kο°ο±ο°ο½ο«724kο°ο±ο°ο½ο«34kο°ο±ο°ο½ο«ο¨ο©1sin2tο½οtο±ο½ο¨ο©1sin2tο½ο726tkο°ο°ο½ο«1126tkο°ο°ο½ο« 85 2.3 Exercises In Exercises 1 β 20, find the exact value or state that it is undefined. The Trigonometric Functions 1. 5. 9. 13. 17. 2. 6. 10. 14. 18. 3. 7. 11. 15. 19. 4. 8. 12. 16. 20. In Exercises 21 β 44, find all angles which satisfy the given equation. 21. 24. 27. 30. 33. 36. 39. 42. 22. 25. 28. 3 1. 3 4. 37. 40. 43. 23. 26. 29. 3 2.csc( ΞΈ) 35. 38. 41. 44. ο¨ο©tan4ο°ο¨ο©sec6ο°ο¨ο©5csc6ο°ο¨οΏ½ |
οΏ½4cot3ο°ο¨ο©11tan6ο°οο¨ο©3sec2ο°οο¨ο©csc3ο°οο¨ο©13cot2ο°ο¨ο©tan117ο°ο¨ο©5sec3ο°οο¨ο©csc3ο°ο¨ο©cot5ο°οο¨ο©31tan2ο°ο¨ο©sec4ο°ο¨ο©7csc4ο°οο¨ο©7cot6ο°ο¨ο©2tan3ο°ο¨ο©sec7ο°οο¨ο©csc2ο°ο¨ο©3cot4ο°ο¨ο©1sin2ο±ο½ο¨ο©3cos2ο±ο½οο¨ο©sin0ο±ο½ο¨ο©2cos2ο±ο½ο¨ο©3sin2ο±ο½ο¨ο©cos1ο±ο½οο¨ο©sin1ο±ο½οο¨ο©3cos2ο±ο½ο¨ο©cos1.001ο±ο½οο¨ο©tan3ο±ο½sec()2ο±ο½1ο½οο¨ο©3cot3ο±ο½ο¨ο©tan0ο±ο½ο¨ο©sec1ο±ο½ο¨ο©csc2ο±ο½ο¨ο©cot0ο±ο½ο¨ο©tan1ο±ο½οο¨ο©sec0ο±ο½ο¨ο©12cscο±ο½οΏ½ |
οΏ½οΏ½ο¨ο©sec1ο±ο½οο¨ο©tan3ο±ο½οο¨ο©csc2ο±ο½οο¨ο©cot1ο±ο½ο 86 The Trigonometric Functions In Exercises 45 β 61, solve the equation for t. Give exact values. (See the comments following Example 2.3.3.) 45. 48. 51. 54. 57. 60. 46. 49. 52. 55. 58. 61. 47. 50. 53. 56. 59. 62. Explain why the fact that does not necessarily mean and. (See the solution to number 6 in Example 2.3.1.) ο¨ο©cos0tο½ο¨ο©2sin2tο½οο¨ο©cos3tο½ο¨ο©1sin2tο½οο¨ο©1cos2tο½ο¨ο©sin2tο½οο¨ο©cos1tο½ο¨ο©sin1tο½ο¨ο©2cos2tο½οο¨ο©cot1tο½ο¨ο©3tan3tο½ο¨ο©23sec3tο½οο¨ο©csc0tο½ο¨ο©cot3tο½οο¨ο©3tan3tο½οο¨ο©23sec3tο½ο¨ο©23csc3tο½ο¨ο©31tan3ο±ο½ο½ο¨ο©sin3ο±ο½ο¨ο©cos1ο±ο½ 2.4 Verifying Trigonometric Identities 87 2.4 Verifying Trigonometric Identities Learning Objectives In this section you will: ο· Learn and apply the Pythagorean identities and conjugates. ο· Simplify trigonometric expressions. ο· Pro |
ve that a trigonometric equation is an identity. We have already seen the importance of identities in trigonometry. Our next task is to use the reciprocal and quotient identities found in Theorem 2.3, coupled with the Pythagorean identity found in Theorem 2.1, to derive the new Pythagorean-like identities for the remaining four circular functions. The Pythagorean Identities Theorem 2.1 states that, for any angle ΞΈ,. Through manipulating this identity, we will obtain two alternate versions relating secant and tangent, followed by cosecant and cotangent. To obtain an identity relating secant and tangent, we start with and, assuming, divide both sides of the equation by. The result is the Pythagorean identity. We next look for an identity relating cosecant and cotangent. We assume that and divide both sides of by. ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©cos0ο±οΉο¨ο©2cosο±ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22222222222cossin1cossin1coscoscossin11coscos1tansecο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο«ο½ο«ο½ο¦οΆο¦οΆο«ο½ο§ο·ο§ο·ο¨οΈο¨οΈο«ο½ from quotient & reciprocal identities ο¨ο©ο¨ο©221tansecο±ο±ο«ο½οΏ½ |
οΏ½ο©sin0ο±οΉο¨ο©ο¨ο©22cossin1ο±ο±ο«ο½ο¨ο©2sinο± 88 The Trigonometric Functions Thus, we have, our third Pythagorean Identity,. The three Pythagorean Identities, along with some of their other common forms, are summarized in the following theorem. Theorem 2.5. The Pythagorean Identities: 1. 2. 3. Common Alternate Forms:, provided Common Alternate Forms:, provided Common Alternate Forms: and and and Trigonometric identities play an important role, both in trigonometry and calculus. Weβll use them in this book to find the values of the circular functions of an angle and to solve equations. In calculus, they are needed to rewrite expressions in a format that enables or simplifies integration. In this next example, we make good use of Theorem 2.3 and Theorem 2.5. Example 2.4.1. Verify the following identities. Assume that all quantities are defined. 1. 3. 5. 2. 4. 6. ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22222222222cossin1cossin1sinsinsincos11sinsincot1cscο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο«ο½ο«ο½ο¦οΆο¦οΆο«ο½ο§ο·ο§ο·ο¨οΈο¨οΈο«ο½from quotient & reciprocal ident itiesο¨ο©ο¨ο©22cot1cscο±ο±ο«ο½ο¨ο©ο¨ο©22cossin1ο±οΏ½ |
οΏ½οΏ½ο«ο½ο¨ο©ο¨ο©221sincosο±ο±οο½ο¨ο©ο¨ο©221cossinο±ο±οο½ο¨ο©ο¨ο©221tansecο±ο±ο«ο½ο¨ο©cos0ο±οΉο¨ο©ο¨ο©22sectan1ο±ο±οο½ο¨ο©ο¨ο©22sec1tanο±ο±οο½ο¨ο©ο¨ο©221cotcscο±ο±ο«ο½ο¨ο©sin0ο±οΉο¨ο©ο¨ο©22csccot1ο±ο±οο½ο¨ο©ο¨ο©22csc1cotο±ο±οο½ο¨ο©ο¨ο©1sincscο±ο±ο½ο¨ο©ο¨ο©ο¨ο©tansinsecο±ο±ο±ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©sectansectan1ο±ο±ο±ο±οο«ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©sec11tancossinο±ο±ο±ο±ο½οοο¨ο©ο¨ο©ο¨ο©ο¨ο©336sectan1sin1sinο±ο±ο±ο±ο½οοο«ο¨ο©ο¨ο©ο¨ο©ο¨οΏ½ |
οΏ½sin1cos1cossinο±ο±ο±ο±ο«ο½ο 2.4 Verifying Trigonometric Identities 89 Solution. In verifying identities, we typically start with the more complicated side of the equation and use known identities to transform it into the other side of the equation. 1. To verify we start with the left side, using the reciprocal identity for cosecant. 2. We start with the right hand side of. 3. We begin with the left hand side of the equation. 4. While both sides of the equation contain fractions, the left side affords us more opportunities to use our identities. ο¨ο©ο¨ο©1sincscο±ο±ο½ο¨ο©ο¨ο©ο¨ο©111cscsinsinο±ο±ο±ο½ο¦οΆο§ο·ο¨οΈο½ο¨ο©ο¨ο©ο¨ο©tansinsecο±ο±ο±ο½ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©tan1sinsecsincossincosο±ο±ο±ο±ο±ο±ο±ο½ο½ο½ from reciprocal identity for secantfrom quotient identity for tangentο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©2222sectansectansecsectantansectansectan1ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±οο«ο½ο«οοο½οοΏ½ |
οΏ½οΏ½ binomial multiplicationPythagorean iden tity 90 The Trigonometric Functions 5. The right hand side of the equation seems to hold promise. We find a common denominator. 6. It is debatable which side of the equation is more complicated. One thing which stands out is that the denominator on the left hand side is, while the numerator on the right hand side is. This suggests the strategy of starting with the left hand side and multiplying the numerator and denominator by the quantity. ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©1seccossin1tan1cos1coscossincos1coscoscossincoscoscos1cossinο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο¦οΆο§ο·ο¨οΈο½οο¦οΆοο§ο·ο¨οΈο¦οΆο§ο·ο¨οΈο½οο¦οΆοο§ο·ο¨οΈο¦οΆο§ο·ο¨οΈο½ο¦οΆοο§ο·ο¨οΈο½οο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨οΏ½ |
οΏ½ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22231sin31sin331sin1sin1sin1sin33sin33sin1sin6sin1sin6sincos1sin6coscos6sectanο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο«οοοο½οο«οο«ο«οο«ο½οο½οο½ο¦οΆο¦οΆο½ο§ο·ο§ο·ο¨οΈο¨οΈο½ Pythagorean identityreciprocal and quotient identitiesο¨ο©1cosο±οο¨ο©1cosο±ο«ο¨ο©1cosο±ο« 2.4 Verifying Trigonometric Identities 91 In the preceding example, number 6, we see that multiplying by produces a difference of squares that can be simplified to one term using Theorem 2.5. This is exactly the same kind of phenomenon that occurs when we multiply expressions such as by. For this reason, the quantities and are called βPythagorean conjugatesβ. The following list includes other Pythagorean conjugates. Pythagorean Conjugates ο· ο· ο· ο· ο· ο· : : : and and and and and : and : : ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©οΏ½ |
οΏ½οΏ½ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©221cossinsin1cos1cos1cossin1cos1cossin1cossinsin1cossinsin1cossinο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο±ο«ο½οοοο«ο«ο½οο«ο½ο«ο½ο«ο½binomial multiplicationPyth agorean identityο¨ο©1cosο±οο¨ο©1cosο±ο«12ο12ο«ο¨ο©ο¨ο©1cosο±οο¨ο©ο¨ο©1cosο±ο«ο¨ο©1cosο±οο¨ο©1cosο±ο«ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©221cos1cos1cossinο±ο±ο±ο±οο«ο½οο½ο¨ο©1sinο±οο¨ο©1sinο±ο«ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©221sin1sin1sincosο±ο±ο±ο±οο«ο½οο½ο¨ο©sec1ο±οο¨ο©sec1ο±ο«ο¨ο©ο¨ο©ο¨οΏ½ |
οΏ½οΏ½ο¨ο©ο¨ο©ο¨ο©22sec1sec1sec1tanο±ο±ο±ο±οο«ο½οο½ο¨ο©ο¨ο©sectanο±ο±οο¨ο©ο¨ο©sectanο±ο±ο«ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22sectansectansectan1ο±ο±ο±ο±ο±ο±οο«ο½οο½ο¨ο©csc1ο±οο¨ο©csc1ο±ο«ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22csc1csc1csc1cotο±ο±ο±ο±οο«ο½οο½ο¨ο©ο¨ο©csccotο±ο±οο¨ο©ο¨ο©csccotο±ο±ο«ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©ο¨ο©22csccotcsccotcsccot1ο±ο±ο±ο±ο±ο±οο«ο½οο½ 92 The Trigonometric Functions Verifying trigonometric identities requires a healthy mix of tenacity and inspiration. You will need to spend many hours struggling just to become proficient in the basics. Like many things in life, there is no short-cut here. There is no complete algorithm for verifying identities. Nevertheless, a summary of some strategies which may be helpful (depending on |
the situation) follows and ample practice is provided for you in the Exercises. Strategies for Verifying Identities ο· Try working on the more complicated side of the identity. ο· Use the Reciprocal and Quotient Identities in Theorem 2.3 to write functions on one side of the identity in terms of the functions on the other side of the identity. Simplify any resulting complex fractions. ο· Add rational expressions with unlike denominators by obtaining common denominators. ο· Use the Pythagorean identities in Theorem 2.5 to exchange sines and cosines, secants and tangents, cosecants and cotangents, and simplify sums or differences of squares to one term. ο· Multiply numerator and denominator by Pythagorean conjugates in order to take advantage of the Pythagorean identities in Theorem 2.5. ο· If you find yourself stuck working with one side of the identity, try starting with the other side of the identity and see if you can find a way to bridge the two parts of your work. Most importantly, keep in mind that we are not solving equations. To verify identities, we choose one side of the identity and work with that side until it matches the other side. Verifying identities is an important skill and we will work with identities again in Chapter 4, as more tools become available. Time spent now in developing some proficiency will be useful throughout the course. 2.4 Verifying Trigonometric Identities 93 2.4 Exercises In Exercises 1 β 47, verify the identity. Assume that all quantities are defined. 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. ο¨ο©ο¨ο©cossec1ο±ο±ο½ο¨ο©ο¨ο©ο¨ο©tancossinο±ο±ο±ο½ο¨ο©ο¨ο©sincsc1ο±ο±ο½ο¨ο©ο¨ο©tancot1ο±ο±ο½οΏ½ |
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