LESSON 4 6 Inverse Trigonometric Functions FiveMinute Check

LESSON 4– 6 Inverse Trigonometric Functions

Five-Minute Check (over Lesson 4 -5) TEKS 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

Over Lesson 4 -5 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

Over Lesson 4 -5 B. Sketch the graph of y = 2 sec x. A. C. B. D.

Over Lesson 4 -5 A. Identify the damping factor f (x) of y = 3 x sin x. A. The damping factor is B. The damping factor is – C. The damping factor is 3 x. D. The damping factor is 3. . .

Over Lesson 4 -5 B. Use a graphing calculator to sketch the graphs of y = 3 x sin x, the damping factor f (x) of y = 3 x sin x, and –f (x) in the same viewing window. A. C. B. D.

Over Lesson 4 -5 C. Describe the graph of y = 3 x 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) = x 2 and f(x) = –x 2. D. The amplitude is 2.

Over Lesson 4 -5 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.

Targeted TEKS P. 2(F) Graph exponential logarithmic rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions. P. 2(H) Graph arcsin x and arcos x and describe the limitations on the domain. Also addresses P. 2(I) and P. 4(F). Mathematical Processes P. 1(B), P. 1(D)

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

• arcsine function • arccosine function • arctangent function

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 Therefore, . When t =

Evaluate Inverse Sine Functions Answer: Check If

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 Therefore, arcsin When t = , sin t =

Evaluate Inverse Sine Functions Answer: CHECK If arcsin then sin

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

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

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.

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

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

Evaluate Inverse Cosine Functions Answer: CHECK If arcos

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

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

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 tan t = When t = Therefore, ,

Evaluate Inverse Tangent Functions Answer: Check If , then tan

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 = Therefore, arctan 1 = . , tan t =

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

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

Sketch Graphs of Inverse Trigonometric Functions Sketch the graph of y = arctan By definition, y = arctan for <y< and tan y = are equivalent on , 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.

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:

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

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 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.

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.

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

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

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.

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

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.

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

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

Use Inverse Trigonometric Properties Therefore, Answer: .

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

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

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.

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:

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

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.

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 = So, cot(arcos x) = and adj = x .

Evaluate Compositions of Trigonometric Functions Answer:

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

LESSON 4– 6 Inverse Trigonometric Functions