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Five-Minute Check (over Lesson 4-5) Then/Now New Vocabulary Example 1: Evaluate Inverse Sine Functions Example 2: Evaluate Inverse Cosine Functions Example 3: Evaluate Inverse Tangent Functions Key Concept: Inverse Trigonometric Functions Example 4: Sketch Graphs of Inverse Trigonometric Functions Example 5: Real-World Example: Use an Inverse Trigonometric Function Key Concept: Domain of Compositions of Trigonometric Functions Example 6: Use Inverse Trigonometric Properties Example 7: Evaluate Compositions of Trigonometric Functions Example 8: Evaluate Compositions of Trigonometric Functions Lesson Menu

A. Locate the vertical asymptotes of y = 2 sec x. A. x = nπ, where n is an integer B. , where n is an odd integer C. , where n is an integer D. x = nπ, where n is an odd integer 5–Minute Check 1

B. Sketch the graph of y = 2 sec x. D. 5–Minute Check 1

A. Identify the damping factor f (x) of y = 3x sin x. A. The damping factor is . B. The damping factor is – . C. The damping factor is 3x. D. The damping factor is 3. 5–Minute Check 2

B. Use a graphing calculator to sketch the graphs of y = 3x sin x, the damping factor f (x) of y = 3x sin x, and –f (x) in the same viewing window. A. B. C. D. 5–Minute Check 2

C. Describe the graph of y = 3x sin x. A. The amplitude of the function is increasing as x approaches the origin. B. The amplitude of the function is decreasing as x approaches the origin. C. The amplitude oscillates between f (x) = x2 and f(x) = –x2. D. The amplitude is 2. 5–Minute Check 2

Write an equation for a secant function with a period of 5π , a phase shift of –2π, and a vertical shift of –3. A. B. C. D. 5–Minute Check 3

Evaluate and graph inverse trigonometric functions. You found and graphed the inverses of relations and functions. (Lesson 1-7) Evaluate and graph inverse trigonometric functions. Find compositions of trigonometric functions. Then/Now

arcsine function arccosine function arctangent function Vocabulary

A. Find the exact value of , if it exists. 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, Example 1

Evaluate Inverse Sine Functions Answer: Check If  Example 1

B. Find the exact value of , if it exists. 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 Example 1

CHECK If arcsin then sin  Evaluate Inverse Sine Functions Answer: CHECK If arcsin then sin  Example 1

C. Find the exact value of sin–1 (–2π), if it exists. 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 Example 1

Find the exact value of sin–1 0. B. C. D. π Example 1

A. Find the exact value of cos–11, if it exists. Evaluate Inverse Cosine Functions A. Find the exact value of cos–11, 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–11 = 0. Example 2

Check If cos–1 1 = 0, then cos 0 = 1.  Evaluate Inverse Cosine Functions Answer: 0 Check If cos–1 1 = 0, then cos 0 = 1.  Example 2

B. Find the exact value of , if it exists. 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 Example 2

Evaluate Inverse Cosine Functions Answer: CHECK If arcos  Example 2

C. Find the exact value of cos–1(–2), if it exists. 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 Example 2

Find the exact value of cos–1 (–1). B. C. π D. Example 2

A. Find the exact value of , if it exists. 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, Example 3

Answer: Check If , then tan  Evaluate Inverse Tangent Functions Example 3

B. Find the exact value of arctan 1, if it exists. 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 = . Example 3

Check If arctan 1 = , then tan = 1.  Evaluate Inverse Tangent Functions Answer: Check If arctan 1 = , then tan = 1.  Example 3

Find the exact value of arctan . B. C. D. Example 3

Key Concept 4

Sketch the graph of y = arctan 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 = as x = 2 tan y and assign values to y on the interval to make a table to values. Example 4

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: Example 4

Sketch the graph of y = sin–1 2x. B. C. D. Example 4

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 θ1 represent the angle formed from eye-level to the bottom of the screen, and let θ2 represent the angle formed from eye-level to the top of the screen. Example 5

Use an Inverse Trigonometric Function So, the viewing angle is θ = θ2 – θ1. You can use the tangent function to find θ1 and θ2. Because the eye-level of a seated person is 6 feet above the ground, the distance opposite θ1 is 8 – 6 feet or 2 feet long. Example 5

Inverse tangent function Use an Inverse Trigonometric Function opp = 2 and adj = d Inverse tangent function The distance opposite θ2 is (32 + 8) – 6 feet or 34 feet opp = 34 and adj = d Inverse tangent function Example 5

So, the viewing angle can be modeled by Use an Inverse Trigonometric Function So, the viewing angle can be modeled by Answer: Example 5

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. Example 5

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 Example 5

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. B. C. D. Example 5

Key Concept 6

A. Find the exact value of , if it exists. Use Inverse Trigonometric Properties A. Find the exact value of , if it exists. The inverse property applies because lies on the interval [–1, 1]. Therefore, Answer: Example 6

B. Find the exact value of , if it exists. 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]. Example 6

Use Inverse Trigonometric Properties Therefore, . Answer: Example 6

C. Find the exact value of , if it exists. 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 Example 6

Find the exact value of arcsin B. C. D. Example 6

To simplify the expression, let u = cos–1 so cos u = . 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. Example 7

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 opp = 3 and hyp = 5 So, Answer: Example 7

Find the exact value of A. B. C. D. Example 7

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. Example 8

Evaluate Compositions of Trigonometric Functions From the Pythagorean Theorem, you can find that the length of the side opposite to u is . Now, solve for cot u. Cotangent function opp = and adj = x So, cot(arcos x) = . Example 8

Evaluate Compositions of Trigonometric Functions Answer: Example 8

Write cos(arctan x) as an algebraic expression of x that does not involve trigonometric functions. Example 8

End of the Lesson