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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 − 6x) + 3. nearest thousandth. 1... |
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 − 10) + log(3) = log(51) If there is no solution, write no solution. 28. The formula for measuring sound intensity in decibels D is... |
P(t) = __ 1 + 25e−0.75t, 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 de... |
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... |
d. −4.3 × 106 e. ≈ 1.24 × 1015 13. Number of cells: 3 × 1013; length of a cell: 8 × 10−6 m; total length: 2.4 × 108 m or 240,000,000 m 10. a. $1.52 × 105 16h10 _ 49 1 _ c20d12 9. a. e. — 4. Section 1.3 1. a. 15 b. 3 c. 4 d. 17 2. 5∣ x ∣ ∣ y ∣ √ 2yz Notice the absolute value signs around x and y? That's because their v... |
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 −2 y = 1 __ (−2) + 2 = 1 (−2, 1) 2 (−1) + 2 = 3 −1 y = 1 ( −1, 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 − 7) __ (x + 5)(x − 3) y... |
= 0 + 2i √ 1. √ 3. (3 − 4i) − (2 + 5i) = 1 − 9i 6 2. 5 _ − i 4. 2 5. 18 + i 6. −3 − 4i 7. −1 –5 –4 –3 –2 i 5 4 3 2 1 –1–1 –2 –3 –4 –5 21 3 4 5 r Section 2.5 1. (x − 6)(x + 1) = 0; x = 6, x = −1 3. (x + 5)(x − 5) = 0; x = −5, x = 5 x = 7, x = −3 2 1 _ _ 4. (3x + 2)(4x + 1) = 0; x = − 5. x = 0, x = −10,, x = − 4 3 2 22 ... |
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) = √ 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... |
) (−1, 28) 40 20 –4 –3 –2 –1 –20 –40 –60 –80 –100 –120 Section 3.4 1. a. (fg)(x) = f (x)g(x) = (x − 1)(x 2 − 1f − g)(x) = f (x) − g(x) = (x − 1) − (x2 − 1) = x − 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 gravitationa... |
7. x −2 0 g(x) −5 −10 −15 −20 4 2 x −2 h(x) 15 0 10 2 4 5 unknown 7. y f (x) = x2 h(x) = f (− x)= (− x)2 Notice: h(x) = f (−x) looks the same as f (x). –10 –8 –5 –4 –3 –2 5 4 3 2 1 0 –1–1 –2 –3 –4 –5 8. Even 21 3 4 5 9. x x g(x) 6 4 2 9 12 15 8 0 10. g(x) = 3x − 2 g(x) = −f (x) = −x2 1 __ 11. g(x) = f ( x ) so using t... |
, 7), (−6, 9), or (−9, 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) = − 1 __ x 2 x + 6 10. y = − 1 __ 3 42 6 8 10 x –10 –8 –6 –4 10 8 6 4 2 –2 –2 –4 –6 –8 –10 y 10 8 6 4 2 –6 –4 –2 –2 –4 –6 –8 –10 y = 2x Section 4.2 1. C(x) = 0.25x + 25,000; The y-intercept is (0, 25,000). ... |
as x → ±∞, f (x) → −∞ because of the negative coefficient. The leading coefficient is −1. 4. As x → ∞, f (x) → −∞; as x → −∞, f (x) → −∞. 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... |
. 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 = –3; horizontal asymptote at y = 4. squared function, we find the rational form. 7. For the transformed reciprocal f (x) = 1 ______ (x − 3)2... |
). 3 x-intercepts at (2, 0) and (–2, 0). (–2, 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 − 8x + 15 − 2. 3x 3 − 3x 2 + 21x − 150 + 78 _ 4x + 5 1,090 _ x + 7 3. 3x 2 − 4x + 1 Section 5.5 1. f (−3) =... |
129(1.3526)t million people. — 2 )x ; Answers may vary due to 5. f (x) = 2(1.5)x 6. f (x) = √ 2 ( √ round-off error. the answer should be very close to 1.4142(1.4142)x. 7. y ≈ 12 · 1.85x 8. About $3,644,675.88 10. e−0.5 ≈ 0.60653 11. $3,659,823.44 12. 3.77E-26(This is calculator notation for the number written as 3.77... |
y = 0 x The domain is ( −∞, ∞); the range is (0, ∞); the horizontal asymptote is y = 0. 1 __ 6. f(x) = − ex − 2; the domain is ( −∞, ∞); the range is 3 (− ∞, 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 ... |
∞), 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, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0. 7. 8 10 –10 –8 –6 –4 –2–1 –2 –3 –4 –5 y 4 3 2 1 –10 –8... |
) (2x + 3)4 ; this answer could also be written log ( 5(x − 1)3 √ ___________ (7x − 1) x12(x + 5)4 ________ 11. log 12. The pH increases by about 0.301. 13. ln(8) ____ ln(0.5) 14. ln(100) _____ ≈ ln(5) 4.6051 _____ 1.6094 = 2.861 4 x3(x + 5) ) _______. (2x + 3) Section 6.6 1. x = −2 2. x = −1 2 4. The equation has no ... |
a. The logistic regression model that fits these data is y = 25.65665979 _____________________ 1 + 6.113686306e−0.3852149008x. 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 f... |
−√ _________ 71. √ — — x+√ ___________ 73. — √ ____ Section 1.4 9. 5. 11. x +x+ 15. b− b+b−b+ 1. Th Th 3. 7. 13. w +w+ 17. x −x− 21. v −v+ 25. y −y+ 29. y −y+ 35. −m+ 41. y −y −y+ 45. a−b 47. t −tu+u 49. t +x +t−tx−x 51. r +rd−d 55. t −t +t+ 37. q − 39. t +t −t −t+ 27. p+p+ 31. c − 57. a +ac−ac −c 23. n−n+n− 43. p−p −... |
x− 13. 21. 29. 37. 43. 51. 25. p+ p+ x+y xy 27. d+ d+ 35. a− a−a− 33. 39. z +z+ z −z− 41. x+xy+y x+xy+y+ 45. 53. +ab b x+ x− 47. a−b 49. c +c− c +c+ 55. y+ 57. ANSWERS 31. 33. − x y − – – – – – – – – – – 35. x − y 37. x − y Chapter 1 Review Exercises 1. − 3. 5. y= 7. m 11. 13. 15. a 17. 19. a 9. x y 27. 25. √ — 33. − ... |
Th Section 2.2 1. 5. x 7. x= 9. x= 11. x= 13. x= 17. x≠−x=− 19. x≠x= 21. x≠x=− ) :(, = − ) 31. :(,) :(, m= 15. x=− ; ; 23. 29. 33. ; 27. :(, 35. = = 39. =; :(, x+ ___ : 43. y=− y=− :(,) :(,) ) 51. y=− 49. ) x+ y= 25. m=− =; = ; = − 41. :; ___ y=− x+ 45. ; 37. 47. 53. x+y= Download the Ope... |
±√ — −±√ ± x≈−x≈ 45. = 53. ± 25. x=−x= — ±√ 31. x= 37. — ±√ ± 41. x= 43. x= 47. = 49. = 51. x≈x≈ 55. ax +bx+c= b x=− c x + a a c b b =− x+ + x+ a a a ) ( x+ b a x+b a =± √ = b a b− ac a b− ac a — −b±√b ac a − x= xx+= Th 57. 61. x−x−x+=300. Thx x= 63. 59. 1. 3. ii Section 2.6 5. −+i 7. +i 11. i 13. 9. − + i −−−− −− ... |
> −6 and x > −2 x > −2, (−2, +∞) x < − 3 or x ≥ 1 (−∞, −3) ∪ [1, ∞) All real numbers (−∞, ∞) 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 (−∞, ∞) (−∞, −4] ∪ [8, +∞) 1 ) 27. [2,5] 29. No solution [−11, −3] y 45. 65. −∞+∞ 67 69. x=− 71. − 73.... |
30. 34. 40. 43. [−10, 12] 42. [6, 12] (−∞, −1) ∪ (3, ∞) y 12 10 8 6 4 2 –10 –8 –6 –4 –2–2 –4 –6 –8 –10 –12 2 4 6 8 10 x 32. ( −∞, − _ ) ∪ (4, ∞) 36. No solution 38. (−5, 11) 49. Where the blue line is above the red line; point of intersection is x = −3. (−∞, −3) y 12 10 8 6 4 2 –10 –8 –6 –4 –2–2 –4 –6 –8 –10 –12 53. 5... |
∣ a+∣ fa+h=∣ a+h−∣−∣ a+h+∣ 33. gx−ga _ x−a 37. a. f = b. x=− =x + a+ x ≠ a 35. a. f −= b. x= __ 39. a. r=− t 43. 49. 41. b. f−= c. t= 45. 47. 53. a. f = b. f x=−x=− 51. 55. 57. 59. 65. 69. f−=f−=f=f=f= 71. f−=f−=f=f=f= 67. f x=x= 61. 63. 85. Th − 87. Th − – 89. a.g=b.Th f 100 s 91. a. fts 200 ft. b. Thfts 350 ft. Secti... |
�∞ 55 Th Th 57. 59. fx= — x− √ 61. a. Th b. Th c. Th Section 3.3 8. − 6. x+h 1. 3. Th 10. _________ 14. +h x+h− 31. x+a 33. 25. 29. + −∞−∪∞ h+h+ ≈− b+ − + 23. 20. 18. 12. 26. 16. 35. 27. − 37. −∞∪ ∪∞ − 40. 42. −− 44. a. − b. − 45. −−∞∞ 47. −−−− −∞ −−−−∞ −−−−∞ − 49. 51. 57. 53. 55. b= ≈− ≈ Section 3.4 1. g f g... |
+ — — x + √ ; gfx= x x+ √ 15. fgx= x x __ x≠gfx=x−x≠ 17. fgx= x++ — −x √ __ ) b. ( −∞ 21. a.g∘f x=− 19. fghx= 23. a. ∪∞x=−b.∞c.∞ 25. ∞ 31. fx= 27. fx=xgx=x− x ; gx=x+ 29. fx= x ; gx= √ x− x− x+ x ; gx=x+ 35. fx=√ 33. fx= x ; gx= √ — — — Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANS... |
– – – — — fx= ∣ x−∣− fx= √ x+− fx= x− fx= ∣ x+∣− — fx= − √ x fx= −x++ fx= √ 31. 33. 35. 37. 39. 41. 43. 45. 49. g x Thg f f −x + 47. 51. Th 53. 55. Thg Thg f f y Thg 59. 57. f 63. gx= x+− hift 67. hift y gx= ∣ −x ∣ 61. gx= x−+ 65. 69. hift hift – – – – – – – – – – – 71. y Th 73. hift – – – – – – – – – Download the Ope... |
xgfx=x — 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 – f –– – – –– – 69. f −x=+ – – – – – 57. 59. 61. − 63. x 65. 67. x f −1(x) 71. 73. f −x = x − d _ t= td= Th x 49. – – – – 53. y – – – – – – y Chapter 3 Review Exercises 1. 3. 5. f−... |
21. −∞−∞ 23. − 27. fx={ |x| x ≤ x > 29. x = 31. x − 33. f −x=− 19. f−x 25. f= −x_ CHAPTER 4 Section 4.1 3. dt = −t 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. − 31. y = x− 7. 17. 23. 13. 15. 21. 29. _ x − 19. 25. 33. y = − x+ 37. 39. 41. y... |
b. c. d. e. Pt=+t f. 47. a. Cx=x + b. The fl c. 49. a. Pt=t + 51. a. Rt= −t + b. b. c. 53. 55. 57. 31. y =−t + 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=−x+r=− 23. 25. y=x+r= 27. y=−x+ r=− 31. y=x−r= −(−− f... |
∞ 21. −∞∞∞ −∞ 27. fx=x+x+ 29. fx=x−x+ 31. fx=− __ __ 35. −) x= +√ 33. fx=x−x+ 37. ( ) − x= — +√ )( ( 25. −∞∞−∞ − √ __ x+ x − − − − − −) 39. ( x= +√ )( ( −# √ ) — — y −−−−− − − − − − x 41. fx=x+x+ 43. fx=−x−x− 45. fx=− __ x− x+ 47. fx=x+x+ 49. fx=−x+x 51. Th h. Th 53. Thhift 55. Th 57. −∞∞ −∞ 59. −∞∞ ∞ 61. fx=x+ 63. fx=... |
→−∞ x→∞fx→∞ 57. y− x− x→−∞fx→∞ x→∞fx→∞ y y −−−− − − − − − − x −−−−−− − − − − − x x y Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-13 59. yx − x→−∞fx→−∞ x→∞fx→∞ 61. fx=x− 63. f x=x−x+ x 65. f x=x+ 67. Vm=m+ m+ m+ 69. Vx=x−x+ x y −−−−− − − − − − Section 5.3 x 9. − 7. −− 5. Th 3... |
−−− 69. : −− 71. − 73. f x=x−x+ 77. f x=x−x+ x+ π x+ x+ x+ 79. f x= 75. f x=x−x+ x Section 5.4 1. Th 3. x+ + x+ x− 5. x+ x+ 9. x−+ x− 7. x− x− x+ 11. x−+ x− x+ 13. x −x+ x−x+ 15. x+ x+ + x− 17. x−x+ − x+ 19. x−x+ − 23. x−x+ − 27. x+ x+ x− x+ x+ 21. x+ x+ 25. x−x+ 29. x−x+ 31. x−x+ 41. 35. x−x+ x− 39. x−x− 43. 33. x−x+... |
15 8, ± ± − (+) ✓ ( ( ))( (+)) ◆ 3 1, 3 2, ± 3 4, ± Download the OpenStax text for free at http://cnx.org/content/col11759/latest. B-14 ANSWERS 44. f x y= − −−− 38. f x 40. f x −− − − − − − − x −−−−−− − − − − − x −− − 42. f x x − − − − − − −− − − − − 46. f x −−−−−− − − − − − x −−−−−− − − − − − x 47. 49. 53. ... |
) ( hx 41. x= y= sx x= −− − − −−− y= x 45. x=− y= − ax − −−− y= x x=− −− − − − − 49. x=− y= −( − ) wx x= x= − y=x+ x= x − −−−− y= x − − − − −− − − − − −− − − − − Download the OpenStax text for free at http://cnx.org/content/col11759/latest. ANSWERS B-15 51. fx= 55. fx= ⋅ x− x− ________ x− x− x+ x+ 53. fx= x+x− x+x+ 57... |
= 17. f−x= √ x− −x ______ ∞ 11. f−x= ± √ √ 15. f−x= x− — −x 19. f−x= 21. f−x=−x 27. f−x= x− ______ x+ 23. f−x= 29. f−x=√ x+ ______ x — x+− x−+ __________ 25. f−x= x+ ______ −x x x x 41. −∪∞ f x −−−−− − − − − − 45. −∞−ċ− f x − − − −− − − − − − 49. −−− y −−− − – − − − − − x−b a −h x x x 53. f−x= √ 57. th = √ x−b 51. f−x=... |
. y= — x √ xz w 21. y= 41. y= x y −−−− − − − − − − __ 45. y= x y − −−−− − − − − − 39. y= __ 43. y= √ — x y − − − −− − − − − − x 47. ≈ 49. ≈ 51. 53. 55. x 1. f x→∞ x→∞f x→∞ − 5. f x=x− 7. 9. − 13. { −− } 11. x+x+x++ x− 15. −− 17. f x=− x−x−x+ 19. 21. −( ) x=−y= y 23. f−x=x−+x≥ x+ x− 25. f−x= −−−− − − − − − x = − − 27. y... |
In=( + n ) 59. ff x=a⋅( ) x b b> 1. Thn> fx=a⋅( ) x b 61. 67. x =ae−nx=ae−nx 63. =ab−x=aen− r ) − − 65. Section 6.2 1. x. Th 3. gx=−xy 5. gx) =−x+y 7. gx=( ) y x, ✓ ◆ : ;:= :(,);:= 13. :(,);:= 15. 17. : 19. 21. 0, 1 1024 ◆ ✓ ;:= 9. 11. 23. y 25. y −− − − − − − − − − fx= x x − − −fx= − x −− − − − − − − −f x=x... |
. 61. ≈ − ≈ 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. ( −∞, )−∞∞ 9. ( − ∞)−∞∞ 11. ∞x= ∞#) x=− 13. ( − 15. −∞x=− 17. ( ∞#) x= x→( ) 19. −∞x=−... |
) — y √ = b )= () ( ()+ 9. xy () 25. 31. 27. 23. 29. 33. = ( ≈ a−b b 39. a b− ≈− 37. ≈ x= x+−x−=( x = x+ _____ )= x− () 35. 41. 42. = = = x+ _____ x− x+ _____ − x− x− x+ _______ _____ − x− x− x+−x+ _____________ x− x− _____ x− = x= +− −= −x= bnn 1. Th bn= nn nb = nb Section 6.6 1. 3. Th. Th x= 7. 9. 5. x= ... |
://cnx.org/content/col11759/latest. B-20 ANSWERS 27. y=bx bb≠ 1. Th __ __ y=bx y=xb ey=exb y=exb S ) ( 25. M= S S M=( ) S S M =( ) S SM =S 29. A=e−tA≈ 33. f t=e−t; 35. r≈− 37. f t=etftP ≈ 39. f t=et 41. ≈ 31. 43. ≈ 45. Section 6.8 1. 3. t fi. Th 5. y 7. 13. p≈ 19. 21. 11. P = 15. y 17. 9. y x y 31. x 23. 25. 27. f x=x ... |
� 55. f t=tf ≈g 57. x 59. 61. y x 63. y =x; y =e–x 65. 67. y =−x y Chapter 6 Practice Test x 1. 5. y 3. y f−x= −x fx= −x 7. a= 9. x = 11. ≈− 13. x< x = x→−f x→−∞ x→−∞f x→∞ −−−−− − x 15. t 17. y+z+ x− 19. x = _______ + __ ≈ 23. 25. x = 21. a= + ________ — 27. x =± √ ____ 29. f t=e−t; 31. Tt=e−t+T≈°F Download the OpenSta... |
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 21... |
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 7... |
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 imagi... |
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... |
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 func... |
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 scient... |
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 u... |
oppositehypotenusesinbccosadjacenthypotenusecosactanoppositeadjacenttanbacotadjacentoppositecotabsechypotenuseadjacentseccacschypotenuseoppositecsccb 34 The Trigonometric Functions The following are important properties of the trigonometric functions. 1. For all right triangle... |
ositeadjacent4sin53cos54tan3435 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... |
° 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 30°, 60°, 90° Triangles We begin by finding the value... |
�3030303030301csc2sin12seccos31cot3tanx 30 60 2x 32x 38 The Trigonometric Functions Trigonometric Functions of 45°, 45°, 90° Triangles To find the values of the trigonometric functions for 45°, we sketch a 45°, 45°, 90° right isosceles triangle with hypotenuse h and remaining two sides each of length x. U... |
get sincostancscseccot12323323332222223212323333180309060307cosc3030307cos17cos7sec.ccc 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 30° angle, which we’ll deno... |
Theorem, although the same result could be achieved through solving for a. The triangle with all of its data is recorded below. 60sin20b6020sin3202103.b60cos20a22221032040030010.aaaba20 601020 60 30 103 42 The Trigonometric Functions Solving Applied Problems Right triangle trigonometry has many prac... |
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 t... |
��75sin0,90h50 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 r... |
Triangle Trigonometry 49 18. If and the hypotenuse has length 10, how long is the side adjacent to θ? 19. If and the side opposite θ has length 306, how long is the side adjacent to θ? 20. If and the side opposite θ has length 4, how long is the side adjacent to θ? 21. If and the hypotenuse has length 10, how long is ... |
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 θ. (a) Show that if the horizontal is above and parallel to level gr... |
. A 33 foot ladder leans against a building, so that the angle between the ground and the ladder is 80°. 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 80°. How high does the ladder reach up the side of the... |
and sine values for the common angles: 0°, 30°, 45°, 60° and 90°, or for their equivalent radian measures. 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 t... |
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. 0t1x0,t0,t0t,0t0t1,034t2t2t117t34t3434381... |
1,01.5723.141,0251.968117t1,02117258xy11 2txy11 2txy11 117t 56 The Trigonometric Functions By associating the point P with the angle θ, we are assigning a position on the Unit Circle to the angle θ. The x-coordinate of P is called the cosine of θ, written, while the y-coordinate of P is called ... |
��660270cos270sin270340,1270cos0270sin1 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. W... |
,Pxy0x22xyx22y2cos452452sin2,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 positi... |
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, ... |
��cos0sin0 cos0sin0 cos0sin0 cos0sin0 2.2 Determining Cosine and Sine Values from the Unit Circle 61 Example 2.2.3. Using the given information about θ, find the indicated value. 1. If θ is a Quadrant II angle with, find. with, find., find. 2. If 3. If Solution. 1. When we substitute into th... |
�22cossin1225sin553225sin5sin122cossin1cos0xy1 35 3cos,5 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 an... |
Unit Circle 63 In general, for a non-quadrantal angle θ, the reference angle for θ (usually denoted α) is the acute angle made between the terminal side of θ and the x-axis. If θ is a Quadrant I or IV angle, α is the angle between the terminal side of θ and the positive x- axis. If θ is a Quadrant II or III angle,... |
��1165473xy 1,0 13,22 22,22 31,22 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 α by sub... |
0113coscos662111sinsin6625454544 66 The Trigonometric Functions axis. Since θ is a Quadrant II angle, the Reference Angle Theorem gives: and. 4. Since the angle measures more than, we find the terminal side of θ by rotating one full revolution followed by an additional radians. Since θ a... |
(b) (c) (d) 5cos13sin232 68 Solution. The Trigonometric Functions 1. Proceeding as in Example 2.2.3, we substitute into and find. Since α is an acute (and therefore Quadrant I) angle, is positive. Hence,. To plot α in standard position, we begin our rotation from the positive x-axis to the r... |
5coscos1312sinsin13cossin5cos1312sin13225cos1312sin13 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 wh... |
2,Qxycosxsiny12cos13x5sin13y 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 expression... |
. 16. 13. 17. 14. 18. 56tt6t2t12tcos0tsin0tcos0tsin0tcos0tsin0tcos0tsin0tcossin226224223222220432233476 72 19. 23. 27. The Trigonometric Functions 20. 24. 28. 21. 25. 29. 22. 26. 30. In Exercises 31 – 40, use the results developed throughout the section ... |
�61031177sin25cos4cos9sin5sin13cos2cos11sin2sin3cos28cos53sin25sin52cos10cos10522sinsin0.4232coscos0.982sin 2.3 The Six Circular Functions 73 2.3 The Six Circular Functions Learning Objectives In this section you... |
ular Functions: Suppose θ is an angle plotted in standard position and is the point on the terminal side of θ which lies on the Unit Circle. The circular functions are defined as follows. The sine of θ, denoted, is defined by The cosine of θ, denoted, is defined by.. The tangent of θ, denoted, is defined by, prov... |
Consider the acute angle θ in standard position. Let denote, as usual, the point on the terminal side of θ which lies on the Unit Circle and let denote the point on the terminal side of θ which lies on the vertical line. The word ‘tangent’ comes from the Latin meaning ‘to touch’. The line is a tangent line to the Unit... |
sec1xcosxsiny 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 Identi... |
��cos0sec11cscsinysin0sin0csc60sec7csc43cotxyASTC The Trigonometric Functions 76 4. 5. 6. Solution., where θ is any angle coterminal with., where and θ 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 rec... |
�1cottan 2.3 The Six Circular Functions 77 4. If θ 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... |
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 su... |
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 θ, when plotted in standard position, intersects the Unit Circle at. This means θ is a Quadrant I or IV angle with ref... |
�53523k1sin212y6 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 2π:. 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 termin... |
1126k 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... |
sec212cos1cos223k523ksin3tan3cos3tan33tan33 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 Qua... |
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... |
�4cot311tan63sec2csc313cot2tan1175sec3csc3cot531tan2sec47csc47cot62tan3sec7csc23cot41sin23cos2sin02cos23sin2cos1sin13cos2cos1.001tan3sec()213cot3tan0sec1csc2cot0tan1sec012csc� |
��sec1tan3csc2cot1 86 The Trigonometric Functions In Exercises 45 – 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 s... |
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 remai... |
�sin022cossin12sin 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 Comm... |
��221sincos221cossin221tanseccos022sectan122sec1tan221cotcscsin022csccot122csc1cot1sincsctansinsecsectansectan1sec11tancossin336sectan1sin1sin� |
�sin1cos1cossin 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 ... |
�� 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... |
�22231sin31sin331sin1sin1sin1sin33sin33sin1sin6sin1sin6sincos1sin6coscos6sectan Pythagorean identityreciprocal and quotient identities1cos1cos1cos 2.4 Verifying Trigonometric Identities 91 In the preceding example, number 6, we see that mult... |
��221cossinsin1cos1cos1cossin1cos1cossin1cossinsin1cossinsin1cossinbinomial multiplicationPyth agorean identity1cos1cos12121cos1cos1cos1cos221cos1cos1cossin1sin1sin221sin1sin1sincossec1... |
��22sec1sec1sec1tansectansectan22sectansectansectan1csc1csc122csc1csc1csc1cotcsccotcsccot22csccotcsccotcsccot1 92 The Trigonometric Functions Verifying trigonometric identities requires a health... |
the situation) follows and ample practice is provided for you in the Exercises. Strategies for Verifying Identities Try working on the more complicated side of the identity. 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 oth... |
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