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= _ _ 2 6 2 √ _ 2 3 √ 5π, evaluate sec = − _ _ 2 6, evaluate tan(45°). — 5π . _ 6 b. Given sin Solution Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. a. b. tan(45°) = sin(45°) _______ cos(45°) = — 5π sec = _ 6 1 _ 5π cos ___ — Try It... |
Figure 8 12 13 t x Solution We can find the sine using the Pythagorean Identity, cos2 t + sin2 t = 1, and the remaining functions by relating them to sine and cosine. 12 __ 13 2 + sin2 t = 1 sin2 t = 1 − 2 12 __ 13 sin2 t = 1 − 144 ___ 169 sin2 t = 25 ___ 169 sin t = ± √ ____ 25 ___ 169 sin t = ± — 25 √ _ — 16... |
satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days. period of a function The period P of a repeating function f is... |
cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent. If we need to work with degrees and our calculator or software does not have a degree... |
Functions 5.3 SeCTIOn exeRCISeS VeRBAl 1. On an interval of [0, 2π), can the sine and cosine values of a radian measure ever be equal? If so, where? 3. For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle? 5. Tangent and cotangent have a period of π. What doe... |
If sin t = and cot t. — 3 √ and cos t = 1 _ _, find sec t, csc t, tan t, 2 2 42. If sin 40° ≈ 0.643 and cos 40° ≈ 0.766, find sec 40°, csc 40°, tan 40°, and cot 40°. 44. If cos t = 1 _, what is the cos(−t)? 2 43. If sin t =, what is the sin(−t)? — 2 √ _ 2 45. If sec t = 3.1, what is the sec(−t)? 46. If csc t = 0.34, w... |
= 3sin2 x cos x + sec x is even, odd, or neither. 69. Determine whether the function f (x) = csc2 x + sec x f (x) = sin x − 2cos2 x is even, odd, or neither. is even, odd, or neither. For the following exercises, use identities to simplify the expression. sec t ____ 70. csc t tan t csc t 71. ReAl-WORlD APPlICATIOnS 1 ... |
5.4 RIGHT TRIAnGle TRIGOnOMeTRY We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle: cos t = x sin t = y In this section, we will see another way to define trigonometric functions using properties of right tr... |
CahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.” How To… Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle. 1. Find the sine as the ratio of the oppos... |
side to the hypotenuse • cosine as the ratio of the adjacent side to the hypotenuse • tangent as the ratio of the opposite side to the adjacent side • secant as the ratio of the hypotenuse to the adjacent side • cosecant as the ratio of the hypotenuse to the opposite side • cotangent as the ratio of the adjacent side ... |
√ the relation s, s, √ 2s 6 s s 4 s 2s 4 s We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles. Figure 8 Side lengths of special triangles How To… Given trigonometric functions of a special angle, evaluate using side lengths. 1. Use the side lengths shown in Figure 8 fo... |
the side opposite the angle of is also the side 3 π 3 s and 2s. Similarly, cos _ 3 π π π and cos , so sin _ _ _ are exactly the same ratio of the same two sides, √ adjacent to 3 6 6 π and sin _ are also the same ratio using the same two sides, s and 2s. 6 π _ The interrelationship between the sines and ... |
t = sec 2 Table 1 How To… Given the sine and cosine of an angle, find the sine or cosine of its complement. 1. To find the sine of the complementary angle, find the cosine of the original angle. 2. To find the cosine of the complementary angle, find the sine of the original angle. Example 4 Using Cofunction Identiti... |
has one angle of and a hypotenuse of 20. Find the unknown sides and angle of the triangle. 3 Using Right Triangle Trigonometry to Solve 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 heig... |
and the adjacent side is 30 ft long. The opposite side is the unknown height. The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of 57°, letting h be the unknown height. tan θ = opposite _ adjacen... |
= cot(_____) 9. tan 4 For the following exercises, find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse. 10. cos B = 4 __, a = 10 5 11. sin B = 1 __, a = 20 2 13. tan A = 100, b = 100 12. tan A =, b = 6 5 __ 12 14. sin √ GRAPHICAl 15. a = 5, ∡ ... |
and that the angle of depression to the bottom of the tower is 23°. How tall is the tower? From a window in the building, a person determines that the angle of elevation to the top of the tower is 43°, and that the angle of depression to the bottom of the tower is 31°. How tall is the tower? 48. A 200-foot tall monume... |
base of the building. Using this information, find the height of the building. 56. 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 60°, how far from the base of the tree am I? 49 8 CHAPTER 5 trigono... |
square of the cosine of a given angle plus the square of the sine of that angle equals 1 quadrantal angle an angle whose terminal side lies on an axis radian the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle radian measure the ratio of the arc length formed ... |
keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle. • An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a neg... |
useful for determining the sines and cosines of special angles. See Example 3. • Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See Example 4. • The domain of the sine and cosine functions is all real numbers. • The range of both t... |
Pythagorean Identity can be manipulated algebraically to produce new identities. See Example 7. • The trigonometric functions repeat at regular intervals. • The period P of a repeating function f is the smallest interval such that f (x + P) = f (x) for any value of x. • The values of trigonometric functions of special... |
2π in radians that is coterminal with the given angle. 9. − 20π ____ 11 10. 14π ____ 5 For the following exercises, draw the angle provided in standard position on the Cartesian plane. 11. −210° 5π ___ 4 13. 12. 75° 14. − π _ 3 15. Find the linear speed of a point on the equator of the earth if the earth has a radius ... |
cot 3 π _ = sin(_____°) 41. cos 2 42. csc(18°) = sec(_____°) π _ 40. tan 6 For the following exercises, use the given information to find the lengths of the other two sides of the right triangle. 3 __, a = 6 43. cos B = 5 5 __, b = 6 44. tan A = 9 For the following exercises, use Figure 1 to evaluate each trigonom... |
of the sine and cosine functions. π _ 15. Find the exact value of tan. 3 π _ 14. Find the exact value of cot. 4 16. Use reference angles to evaluate csc 7π _. 4 17. Use reference angles to evaluate tan 210°. 18. If csc t = 0.68, what is the csc(−t)? 19. If cos t =, find cos(t − 2π). — 3 √ _ 2 20. Which trigonometric f... |
AnD COSIne FUnCTIOnS Figure 1 light can be separated into colors because of its wavelike properties. (credit: "wonderferret"/ Flickr) White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can ... |
2 2π _ 3 3π _ 4 5π 1 — 3 √ _ 2 As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4. y 1 0 –1 y = cos (x) 2π 3 3π 4 5π 5π 4 3π 2 7π 4 x 2π Figure 4 The cosine function Because we can evaluate the sine and cosine of any real number, both of these functions are defined f... |
even symmetry of the cosine function characteristics of sine and cosine functions The sine and cosine functions have several distinct characteristics: • They are periodic functions with a period of 2π. • The domain of each function is (−∞, ∞) and the range is [−1, 1]. • The graph of y = sin x is symmetric about the or... |
2 π 3π 2 f (x) = sin (2x) Figure 8 period of sinusoidal functions If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms The period is 2π _. ∣ B ∣ y = Asin(Bx) y = Acos(Bx) Identifying the Period of a Sine or Cosine Function Example 1 π _ Determine the period of t... |
–3 –4 Figure 9 amplitude of sinusoidal functions If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms y = Asin(Bx) and y = Acos(Bx) The amplitude is A, and the vertical height from the midline is ∣ A ∣. In addition, notice in the example that 1 ∣ maximum − minim... |
�, the more the graph is shifted. Figure 11 shows that the graph of f (x) = sin(x − π) shifts to the right by π units, which π π __ __ is more than we see in the graph of f (x) = sin x − , which shifts to the right by units. 4 4 y 1 f (x) = sin(x) f (x) = sin x − π 4 f (x) = sin(x − π) π 2 π 3π 2 2π 5π 2 x 3π While ... |
− 2. − 2 can be rewritten as f (x) = sin x − − shows a minus sign before C. Therefore f (x) = sin x + 6 6 If the value of C is negative, the shift is to the left. Try It #3 π Determine the direction and magnitude of the phase shift for f (x) = 3cos x − _ . 2 Example 4 Identifying the Vertical Shift of a ... |
513 y 4 3 2 –1 –2 Amplitude: |A| = 3 y = 3 sin (2x) + 1 Midline: y = 1 π 2 π 3π 2 x 2π Period = π Figure 14 Try It #5 – π x 1 __ __ __ Determine the midline, amplitude, period, and phase shift of the function y = . cos 3 3 2 Example 6 Identifying the Equation for a Sinusoidal Function from a Graph Determine the for... |
ifying the Equation for a Sinusoidal Function from a Graph Determine the equation for the sinusoidal function in Figure 17. y 1 –5 –3 –1 1 3 5 7 x –1 –2 –3 –4 –5 Figure 17 Solution With the highest value at 1 and the lowest value at −5, the midline will be halfway between at −2. So D = −2. The distance from the midline... |
0 and D = 0 and work with a simplified form of the equations in the following examples. y = Asin(Bx − C) + D and y = Acos(Bx − C) + D, How To… Given the function y = Asin(Bx), sketch its graph. 1. Identify the amplitude, ∣ A ∣. 2π ___ 2. Identify the period, P =. ∣ B ∣ 3. Start at the origin, with the function increas... |
2 1 –2 –1 1 2 3 4 5 6 x –1 –2 Figure 19 516 CHAPTER 6 periodic Functions Try It #8 Sketch a graph of g(x) = −0.8cos(2x). Determine the midline, amplitude, period, and phase shift. How To… Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. 1. Express the function in the general form ... |
the Properties of a Sinusoidal Function π __ x + π + 3, determine the amplitude, period, phase shift, and horizontal shift. Then graph the Given y = −2cos 2 function. Solution Begin by comparing the equation to the general form and use the steps outlined in Example 9. y = Acos(Bx − C) + D Step 1. The function is a... |
2 x y 3 2 1 –1 –2 –3 Figure 22 518 CHAPTER 6 periodic Functions Analysis Notice that the period of the function is still 2π ; as we travel around the circle, we return to the point (3, 0) for x = 2π, 4π, 6π,... Because the outputs of the graph will now oscillate between –3 and 3, the amplitude of the sine wave is 3. T... |
relative to the board ranges from −1 in. (at time x = 0) to −7 in. (at time x = π) below the board. Assume the position of y is given as a sinusoidal function of x. Sketch a graph of the function, and then find a cosine function that gives the position y in terms of x. y Figure 25 Example 13 Determining a Rider’s Heig... |
520 CHAPTER 6 periodic Functions 6.1 SeCTIOn exeRCISeS VeRBAl 1. Why are the sine and cosine functions called periodic functions? 3. For the equation Acos(Bx + C) + D, what constants affect the range of the function and how do they affect the range? 5. How can the unit circle be used to construct the graph of f(t) = s... |
(t) = 4cos 2 t + 4 1 __ 21. f(t) = −sin t + 2 5π ___ 3 π _ (x − 3) + 7 22. f (x) = 4sin 2 23. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26. 24. Determine the amplitude, period, midline, and an equation involving cosine for the graph ... |
2 –3 –4 –5 21 3 4 5 x –5 –4 –3 –2 5 4 3 2 1 –1–1 –2 –3 –4 –5 21 3 4 5 x Figure 32 Figure 33 522 CHAPTER 6 periodic Functions AlGeBRAIC For the following exercises, let f (x) = sin x. 31. On [0, 2π), solve f (x) = 0. π __ . 33. Evaluate f 2 32. On [0, 2π), solve f (x) = 1 __. 2 34. On [0, 2π), f (x) =. Find all value... |
f (x) = on the window [−5π, 5π] and sin x ____ x explain what the graph shows. ReAl-WORlD APPlICATIOnS 48. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full re... |
functions. Recall that tan x = sin x ____ cos x The period of the tangent function is π because the graph repeats itself on intervals of kπ where k is a constant. If we to π __ graph the tangent function on − π __, we can see the behavior of the graph on one complete cycle. If we look at any 2 2 larger interval, we wi... |
At these values, the tangent function is undefined, = 0 and cos for which cos x = 0. For example, cos 2 2 π 3π _ _ so the graph of y = tan x has discontinuities at x =. At these values, the graph of the tangent has vertical and 2 2 π _ asymptotes. Figure 1 represents the graph of y = tan x. The tangent is positi... |
the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we π P P wish. Our limited domain is then the interval − P _ __ __ __ and the graph has vertical asymptotes at ± where On − π _ __ __ , the graph will come up from the left asymptote at x = −, cross ... |
0.5) = 0.5tan 0.5π ____ 2 π __ = 0.5tan 4 = 0.5 This means the curve must pass through the points (0.5, 0.5), (0, 0), and (−0.5, −0.5). The only inflection point is at the origin. Figure 2 shows the graph of one period of the function. 526 CHAPTER 6 periodic Functions πx y = 0.5 tan 2 y 4 2 (0.5, 0.5) x (–0.5, ... |
� B ∣ C __. B 4. Identify C and determine the phase shift, C __ 5. Draw the graph of y = Atan(Bx) shifted to the right by B + C __ 6. Sketch the vertical asymptotes, which occur at x = B π ____ 2 ∣ B ∣ and up by D. k, where k is an odd integer. 7. Plot any three reference points and draw the graph through these points.... |
Stretched Tangent Find a formula for the function graphed in Figure 4. x = 4 x = 12 x = −12 x = −4 y 6 4 2 –10 –6 –2 2 6 10 x –2 –4 –6 Figure 4 A stretched tangent function 528 CHAPTER 6 periodic Functions Solution The graph has the shape of a tangent function. Step 1. One cycle extends from –4 to 4, so the period is ... |
below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant. Note that, because cosine is an even function, secant is also an even function. That is, sec(−x) = sec x. x = − 3π 2 y x = 3π 2 y = cos (x) –2π y = sec (x) x 2π 8 4 –4 –8 Figure 6 Gr... |
of x where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines. Note that, since sine is an odd function, the cosecant function is also an odd function. That is, csc(−x) = −cscx. The graph of cosecant, which is shown in Figure 7, is similar to the graph of secant. x = −2π y x... |
D • The stretching factor is ∣ A ∣. • The period is 2π ___. ∣ B ∣ C __ • The domain is x ≠ B + π ____ 2 ∣ B ∣ k, where k is an odd integer. • The range is (−∞, − ∣ A ∣ + D] ∪ [ ∣ A ∣ + D, ∞). π C ____ __ • The vertical asymptotes occur at x = 2 ∣ B ∣ B • There is no amplitude. • y = Asec(Bx) is an even function becaus... |
so the stretching factor is 2.5. Step 3. B = 0.4 so P = = 5π. The period is 5π units. Step 4. Sketch the graph of the function g(x) = 2.5cos(0.4x). 2π ___ 0.4 Step 5. Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function. SECTION 6.2 graphs oF the other trigonometric Function... |
� __ __ Step 1. Express the function given in the form y = 4sec + 1. x − 2 3 Step 2. The stretching/compressing factor is ∣ A ∣ = 4. Step 3. The period is Step 4. The phase shift is 2π ___ ∣ B ∣ = = 2π _ π _ 3 3 2π __ ___ · 1 π = 6 π __ 2 _ = π __ 3 π 3 __ __ = · 2 π = 1.5 C __ B 532 CHAPTER 6 periodic Functions C ... |
hing a Variation of the Cosecant Function Graph one period of f (x) = −3csc(4x). Solution Step 1. The given function is already written in the general form, y = Acsc(Bx). Step 2. ∣ A ∣ = ∣ −3 ∣ = 3, so the stretching factor is 3. π π __ __ = Step 3. B = 4, so P = units.. The period is 2 2 Step 4. Sketch the graph of th... |
a graph of y = 2csc 2 Solution π __ Step 1. Express the function given in the form y = 2csc x + 1. 2 Step 2. Identify the stretching/compressing factor, ∣ A ∣ = 2. 2π _ π __ 2 = 0. Step 4. The phase shift is Step 3. The period is 2 _ π = 4. · 2π ___ ∣ B ∣ 2π _ 1 = = 0 _ π __ 2 Step 5. Draw the graph of y = Acsc(... |
of the tangent function because these two functions are reciprocals of one another. See Figure 13. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases. The cotangent graph has ... |
functions (cosine and sine, respectively) k, where k is an integer. π ___ ∣ B ∣ + How To… Given a modified cotangent function of the form f (x) = Acot(Bx), graph one period. 1. Express the function in the form f (x) = Acot(Bx). 2. Identify the stretching factor, ∣ A ∣. 3. Identify the period, P = π ___. ∣ B ∣ 4. Draw ... |
C __. B C __ 5. Draw the graph of y = Atan(Bx) shifted to the right by B 4. Identify the phase shift, C __ 6. Sketch the asymptotes x = B π ___ ∣ B ∣ 7. Plot any three reference points and draw the graph through these points. k, where k is an integer. + and up by D. Graphing a Modified Cotangent Example 9 π π __ __ ... |
do we determine the distance? We can use the tangent function. SECTION 6.2 graphs oF the other trigonometric Functions 537 Example 10 Using Trigonometric Functions to Solve Real-World Scenarios π __ Suppose the function y = 5tan t marks the distance in the movement of a light beam from the top of a police car 4 ac... |
On exeRCISeS VeRBAl 1. Explain how the graph of the sine function can be used to graph y = csc x. 2. How can the graph of y = cos x be used to construct the graph of y = sec x? 3. Explain why the period of tan x is equal to π. 4. Why are there no intercepts on the graph of 5. How does the period of y = csc x compare wi... |
x) 4 31. f (x) = 7sec(5x) π __ − 2 34. f (x) = −sec x − 3 π __ (x + 1) 20. h(x) = 2sec 4 π __ 23. p(x) = tan x − 2 26. f (x) = π tan(πx − π) − π π __ x + π 21. m(x) = 6csc 3 24. f (x) = 4tan(x) 27. f (x) = 2csc(x) 29. f (x) = 4sec(3x) 30. f (x) = −3cot(2x) csc(πx) 32. f (x) = 9 __ 10 π 35. f (x) = 7 __ ... |
x) x = −2π x = π 10 6 2 – 3π 8 π 8 –6 –10 x 5π 8 10 6 2 –6 –10 x = −π x = 2π x x = π 4 x = − 5π 8 x = − π 8 44. f (x) 45. 10 6 2 –2π –π π 2π x −0.01 –10 x = −0.005 TeCHnOlOGY x = 3π 8 f (x) 4 3 2 1 –1 –2 –3 –4 x 0.01 x = 0.005 For the following exercises, use a graphing calculator to graph two periods of the given func... |
the right. (See Figure 19.) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance d(x), in kilometers, from the fisherman to the boat is given by the function d(x) = 1.5sec(x). a. What is a reasonable domain for d(x)? b. Graph d(x) on this domain. c. Find and discuss the mea... |
functions. • Use a calculator to evaluate inverse trigonometric functions. • Find exact values of composite functions with inverse trigonometric functions. 6.3 InVeRSe TRIGOnOMeTRIC FUnCTIOnS For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are... |
function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each fun... |
x = cos y. The inverse cosine function is sometimes called the arccosine function, and notated arccos x. y = cos−1 x has domain [−1, 1] and range [0, π] • The inverse tangent function y = tan−1 x means x = tan y. The inverse tangent function is sometimes called the arctangent function, and notated arctan x. π y = tan−... |
−1 x = y., 2 2 Example 1 Writing a Relation for an Inverse Function 5π __ 12 ≈ 0.96593, write a relation involving the inverse sine. Given sin Solution Use the relation for the inverse sine. If sin y = x, then sin−1 x = y. In this problem, x = 0.96593, and y = 5π __. 12 sin−1 (0.96593) ≈ 5π ___ 12 Try It #1 Given c... |
are multiple values that would satisfy this relationship, such 2 π π 5π, but we know we need the angle in the interval − π 1 _ _ _ _ _ = , so the answer will be sin− and as 6 Remember that the inverse is a function, so for each input, we will get exactly one output. 2 b. To evaluate sin−1 − √ , we know that _ ... |
solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places. In these examples and exercises, the answers will be interpreted as angles and we will use θ as the independent variable. The value displayed on the calculator may be in degrees or radian... |
triangle in Figure 9 for the angle θ. 10 6 θ Figure 9 546 CHAPTER 6 periodic Functions Finding exact Values of Composite Functions with Inverse Trigonometric Functions There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact valu... |
−, 2 2 π 3π __ __ =. 4 4 How To… Given an expression of the form f −1(f(θ)) where f(θ) = sin θ, cos θ, or tan θ, evaluate. 1. If θ is in the restricted domain of f, then f −1(f(θ)) = θ. 2. If not, then find an angle ϕ within the restricted domain of f such that f(ϕ) = f(θ). Then f −1(f(θ)) = ϕ. Example 5 Using ... |
Try It #5 π __ and tan−1 tan Evaluate tan−1 tan 8 11π ___ 9 . Evaluating Compositions of the Form f −1(g(x )) Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We ... |
x) = − x., 2 2 2 π , then find another angle y in − π π 4. If x is not in − π __ __ __ __ such that sin y = sin x.,, 2 2 2 2 π __ cos−1 (sin x) = − y 2 Example 6 Evaluating the Composition of an Inverse Sine with a Cosine Evaluate sin−1 cos a. by direct evaluation. 13π ___ 6 b. by the method described p... |
. 5 4 , which means cos θ = 4 __ __ Solution Beginning with the inside, we can say there is some angle such that θ = cos−1 , 5 5 and we are looking for sin θ. We can use the Pythagorean identity to do this. sin2 θ + cos2 θ = 1 Use our known value for cosine. 4 __ sin2 θ + 5 2 = 1 Solve for sine. sin2 θ = 1 − 16 _... |
−1 4 7 _ = — 65 √ = — 65 7 √ ______ 65 Try It #8 7 __ . Evaluate cos sin−1 9 Finding the Cosine of the Inverse Sine of an Algebraic Expression Example 9 x _ Find a simplified expression for cos sin−1 for −3 ≤ x ≤ 3. 3 x _ Solution We know there is an angle θ such that sin θ =. 3 sin2 θ + cos2 θ = 1 x _... |
� 2 11. cos−1 − √ ____ 2 14. tan−1 (−1) — 2. Since the functions y = cos x and y = cos−1 x are π ___ inverse functions, why is cos−1 cos − not equal 6 π __ to −? 6 4. Most calculators do not have a key to evaluate sec−1(2). Explain how this can be done using the cosine function or the inverse cosine functio... |
3 π _ 26. cos−1 sin 3 4π _ 29. tan−1 sin 3 4 _ 32. cos sin−1 5 35. cos tan−1 12 __ 5 SECTION 6.3 section exercises 551 For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle. 37. tan(sin−1 (x − 1)) 38. sin(cos−1 (1 − x))... |
For what value of x does sin x = sin−1 x? Use a graphing calculator to approximate the answer. 53. Suppose a 13-foot ladder is leaning against a 54. Suppose you drive 0.6 miles on a road so that the building, reaching to the bottom of a second-floor window 12 feet above the ground. What angle, in radians, does the lad... |
arccosine another name for the inverse cosine; arccos x = cos−1 x arcsine another name for the inverse sine; arcsin x = sin−1 x arctangent another name for the inverse tangent; arctan x = tan−1 x inverse cosine function the function cos−1 x, which is the inverse of the cosine function and the angle that has a cosine e... |
general shape as a sine or cosine function. • In the general formula for a sinusoidal function, the period is P = See Example 1. 2π _ ∣B∣ • In the general formula for a sinusoidal function, ∣A∣ represents amplitude. If ∣A∣ > 1, the function is stretched, whereas if ∣A∣ < 1, the function is compressed. See Example 2. •... |
Bx − C) + D is a cotangent with vertical and/or horizontal stretch/compression and shift. See Example, ± 3π __ 2,... 8 and Example 9. • Real-world scenarios can be solved using graphs of trigonometric functions. See Example 10. 6.3 Inverse Trigonometric Functions • An inverse function is one that “undoes” another funct... |
RCISeS GRAPHS OF THe SIne AnD COSIne FUnCTIOnS For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes. 1. f (x) = −3cos x + 3 2π _ 4. f (x) = −2sin . f (x) = 6sin 3x − 6 1 _ 2. f (x) = sin x 4 π _ 5. f (x) = 3sin x... |
city. 19. What is the largest and smallest population the city 20. Graph the function on the domain of [0, 40]. may have? 21. What are the amplitude, period, and phase shift for 22. Over this domain, when does the population reach the function? 18,000? 13,000? 23. What is the predicted population in 2007? 2010? For th... |
csc x and explain any observations. x _ 41. Graph the function f (x) = − 1 x3 _ 3! x5 _ 5! x7 _ 7! + − on the interval [−1, 1] and compare the graph to the graph of f (x) = sin x on the same interval. Describe any observations. 556 CHAPTER 6 periodic Functions CHAPTeR 6 PRACTICe TeST For the following exercises, sketc... |
− 3 17. y = sin 6 18. y = 8sin 7π _ 6 x + 7π _ + 6 2 19. The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t is the number of ho... |
the interval [0, 2π]? 30. What is the smallest possible value for f (x)? For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift. π _ 32. Sine curve with amplitude 3, period, and phase 3 π _, 2 shift (h, k) = 4 π _ 33. Cosine curve with ampl... |
ne 7.1 Solving Trigonometric equations with Identities 7.2 Sum and Difference Identities 7.3 Double-Angle, Half-Angle, and Reduction Formulas 7.4 Sum-to-Product and Product-to-Sum Formulas 7.5 Solving Trigonometric equations 7.6 Modeling with Trigonometric equations Introduction Math is everywhere, even in places we mi... |
all of the trigonometric functions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity can be written in many ways. To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression u... |
θ cos2 θ + sin2 θ ___________ cos2 θ + = = = 1 _____ cos2 θ = sec2 θ The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity ... |
π __ __ = cos 4 4 ≈ 0.707 See Figure 3. y – π, 4 0.707 2 π, 4 0.707 –2π –π –2 x π 2π f(x) = cos x For all θ in the domain of the sine and cosine functions, respectively, we can state the following: Figure 3 Graph of y = cos θ • Since sin(−θ) = −sin θ, sine is an odd function. • Since, cos(−θ) = cos θ, cosine is ... |
�) 1 ____ cos θ To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities. The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciproc... |
cos θ cos θ ____ sin θ Example 1 Graphing the Equations of an Identity Graph both sides of the identity cot θ =. In other words, on the graphing calculator, graph y = cot θ and y = 1 ____ tan θ 1 ____. tan θ Solution See Figure 4. y = cot θ = 1 tan θ y 10 6 2 – 5π 2 – 3π 2 π 2– π 2 3π 2 5π 2 θ –6 –10 Figure 4 564 CHAP... |
igonometric Identity Involving sec2 θ = 1 − sin2 x = cos2 x Since sin(−x) = −sin x Difference of squares cos2 x = 1 − sin2 x Verify the identity sec2 θ − 1 ________ sec2 θ = sin2 θ Solution As the left side is more complicated, let’s begin there. sec2 θ − 1 ________ = sec2 θ (tan2 θ + 1) − 1 _____________ sec2 θ sec2 θ... |
� 1 ____ cos θ Thus, = 2 sin θ _____ cos2 θ = 2 sin θ ________ 1 − sin2 θ 2tan θ sec θ = 2 sin θ ________ 1 − sin2 θ Substitute 1 − sin2 θ for cos2 θ Example 6 Verifying an Identity Using Algebra and Even/Odd Identities Verify the identity: sin2(−θ) − cos2(−θ) _________________ sin(−θ) − cos(−θ) = cos θ − sin θ Solut... |
sin2 x + cos2 x ___________ sin2 x + = (1 − cos2 x) 1 ____ = (sin2 x) sin2 x = 1 Find the common denominator. Using Algebra to Simplify Trigonometric expressions We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expression... |
. This is the difference of squares. Thus, 4 cos2 θ − 1 = (2 cos θ)2 − 1 = (2 cos θ − 1)(2 cos θ + 1) Analysis If this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and l... |
) = sec x on the interval [−π, π]. How can we tell whether the function is even or odd by only observing the graph of f (x) = sec x? 3. After examining the reciprocal identity for sec t, explain why the function is undefined at certain points. 4. All of the Pythagorean identities are related. Describe how to manipulate... |
− sin x ; sec x and tan x 24. tan x; sec x 26. sec x; sin x 28. cot x; csc x 25. sec x; cot x 27. cot x; sin x For the following exercises, verify the identity. 29. cos x − cos3 x = cos x sin2 x 30. cos x(tan x − sec(−x)) = sin x − 1 31. 1 + sin2 x ________ = cos2 x 1 _____ cos2 x + sin2 x _____ cos2 x = 1 + 2 tan2 x ... |
functions. Use sum and difference formulas to verify identities. 7. 2 SUM AnD DIFFeRenCe IDenTITIeS Figure 1 Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr) How can the height of a mountain be mea... |
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