You found and graphed the inverses of relations and functions. (Lesson 1-7)

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2 You found and graphed the inverses of relations and functions. (Lesson 1-7) LEQ: How do we evaluate and graph inverse trigonometric functions & find compositions of trigonometric functions?

3 arcsine function arccosine function arctangent function

4 Evaluate Inverse Sine Functions A. Find the exact value of, if it exists. Find a point on the unit circle on the interval with a y-coordinate of. When t = Therefore,

5 Evaluate Inverse Sine Functions Answer: Check If

6 Evaluate Inverse Sine Functions B. Find the exact value of, if it exists. Find a point on the unit circle on the interval with a y-coordinate of When t =, sin t = Therefore, arcsin

7 Evaluate Inverse Sine Functions Answer: CHECK If arcsin then sin

8 Evaluate Inverse Sine Functions C. Find the exact value of sin 1 ( 2π), if it exists. Because the domain of the inverse sine function is [ 1, 1] and 2π < 1, there is no angle with a sine of 2π. Therefore, the value of sin 1 ( 2π) does not exist. Answer: does not exist

9 Find the exact value of sin 1 0. A. 0 B. C. D. π

10 Evaluate Inverse Cosine Functions A. Find the exact value of cos 1 1, if it exists. Find a point on the unit circle on the interval [0, π] with an x-coordinate of 1. When t = 0, cos t = 1. Therefore, cos 1 1 = 0.

11 Evaluate Inverse Cosine Functions Answer: 0 Check If cos 1 1 = 0, then cos 0 = 1.

12 Evaluate Inverse Cosine Functions B. Find the exact value of, if it exists. Find a point on the unit circle on the interval [0, π] with an x-coordinate of When t = Therefore, arccos

13 Evaluate Inverse Cosine Functions Answer: CHECK If arcos

14 Evaluate Inverse Cosine Functions C. Find the exact value of cos 1 ( 2), if it exists. Since the domain of the inverse cosine function is [ 1, 1] and 2 < 1, there is no angle with a cosine of 2. Therefore, the value of cos 1 ( 2) does not exist. Answer: does not exist

15 Find the exact value of cos 1 ( 1). A. B. C. π D.

16 Evaluate Inverse Tangent Functions A. Find the exact value of, if it exists. Find a point on the unit circle on the interval such that When t =, tan t = Therefore,

17 Evaluate Inverse Tangent Functions Answer: Check If, then tan

18 Evaluate Inverse Tangent Functions B. Find the exact value of arctan 1, if it exists. Find a point on the unit circle in the interval such that When t =, tan t = Therefore, arctan 1 =.

19 Evaluate Inverse Tangent Functions Answer: Check If arctan 1 =, then tan = 1.

20 Find the exact value of arctan. A. B. C. D.

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22 Sketch Graphs of Inverse Trigonometric Functions Sketch the graph of y = arctan By definition, y = arctan and tan y = are equivalent on for < y <, so their graphs are the same. Rewrite tan y = interval as x = 2 tan y and assign values to y on the to make a table to values.

23 Sketch Graphs of Inverse Trigonometric Functions Then plot the points (x, y) and connect them with a smooth curve. Notice that this curve is contained within its asymptotes. Answer:

24 Sketch the graph of y = sin 1 2x. A. C. B. D.

25 Use an Inverse Trigonometric Function A. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground. Write a function modeling the viewing angle θ for a person in the theater whose eye-level when sitting is 6 feet above ground. Draw a diagram to find the measure of the viewing angle. Let θ 1 represent the angle formed from eyelevel to the bottom of the screen, and let θ 2 represent the angle formed from eye-level to the top of the screen.

26 Use an Inverse Trigonometric Function So, the viewing angle is θ = θ 2 θ 1. You can use the tangent function to find θ 1 and θ 2. Because the eye-level of a seated person is 6 feet above the ground, the distance opposite θ 1 is 8 6 feet or 2 feet long.

27 Use an Inverse Trigonometric Function opp = 2 and adj = d Inverse tangent function The distance opposite θ 2 is (32 + 8) 6 feet or 34 feet opp = 34 and adj = d Inverse tangent function

28 Use an Inverse Trigonometric Function So, the viewing angle can be modeled by Answer:

29 Use an Inverse Trigonometric Function B. MOVIES In a movie theater, a 32-foot-tall screen is located 8 feet above ground-level. Determine the distance that corresponds to the maximum viewing angle. The distance at which the maximum viewing angle occurs is the maximum point on the graph. You can use a graphing calculator to find this point.

30 Use an Inverse Trigonometric Function From the graph, you can see that the maximum viewing angle occurs approximately 8.2 feet from the screen. Answer: about 8.2 ft

31 MATH COMPETITION In a classroom, a 4 foot tall screen is located 6 feet above the floor. Write a function modeling the viewing angle θ for a student in the classroom whose eye-level when sitting is 3 feet above the floor. A. C. B. D.

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33 Use Inverse Trigonometric Properties A. Find the exact value of, if it exists. The inverse property applies because interval [ 1, 1]. lies on the Therefore, Answer:

34 Use Inverse Trigonometric Properties B. Find the exact value of, if it exists. Notice that does not lie on the interval [0, π]. However, is coterminal with 2π or which is on the interval [0, π].

35 Use Inverse Trigonometric Properties Therefore,. Answer:

36 Use Inverse Trigonometric Properties C. Find the exact value of, if it exists. Because tan x is not defined when x =, arctan does not exist. Answer: does not exist

37 Find the exact value of arcsin A. B. C. D.

38 Evaluate Compositions of Trigonometric Functions Find the exact value of To simplify the expression, let u = cos 1 so cos u =. Because the cosine function is positive in Quadrants I and IV, and the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I.

39 Evaluate Compositions of Trigonometric Functions Using the Pythagorean Theorem, you can find that the length of the side opposite is 3. Now, solve for sin u. Sine function So, Answer: opp = 3 and hyp = 5

40 Find the exact value of A. B. C. D.

41 Evaluate Compositions of Trigonometric Functions Write cot (arccos x) as an algebraic expression of x that does not involve trigonometric functions. Let u = arcos x, so cos u = x. Because the domain of the inverse cosine function is restricted to Quadrants I and II, u must lie in Quadrant I or II. The solution is similar for each quadrant, so we will solve for Quadrant I.

42 Evaluate Compositions of Trigonometric Functions From the Pythagorean Theorem, you can find that the length of the side opposite to u is for cot u. Cotangent function. Now, solve opp = and adj = x So, cot(arcos x) =.

43 Answer: Evaluate Compositions of Trigonometric Functions

44 Write cos(arctan x) as an algebraic expression of x that does not involve trigonometric functions. A. B. C. D.