You found and graphed the inverses of relations and ... · Evaluate Inverse Sine Functions C. Find the exact value of sin–1 (–2π), if it exists. Because the domain of the inverse

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?

• 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 . When t =

Therefore,


Answer:

Check If


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


with a y-coordinate of When t = , sin t =

Therefore, arcsin


Answer:

CHECK If arcsin then sin


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.


Answer: 0

Check If cos–1 1 = 0, then cos 0 = 1.



Find a point on the unit circle on the interval [0, π] with

an x-coordinate of When t =

Therefore, arccos


CHECK If arcos

Answer:


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.


Find the exact value of cos–1 (–1).

A.

B.

C. π

D.

Evaluate Inverse Tangent Functions



such that When t = ,

tan t = Therefore,


Answer:

Check If , then tan


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 = .


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 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.

Answer:

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.

Sketch the graph of y = sin–1 2x.

A.

B.

C.

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 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.


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.


Function

opp = 2 and adj = d

Inverse tangent function

The distance opposite θ2 is (32 + 8) – 6 feet or 34 feet

Inverse tangent function

opp = 34 and adj = d


Function

So, the viewing angle can be modeled by

Answer:


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.

Answer: about 8.2 ft


Function

From the graph, you can see that the maximum viewing angle occurs approximately 8.2 feet from the screen.

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.

Use Inverse Trigonometric Properties


Therefore,

The inverse property applies because lies on the interval [–1, 1].

Answer:



Notice that does not lie on the interval [0, π].

However, is coterminal with – 2π or which

is on the interval [0, π].


Therefore, .

Answer:



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

Because tan x is not defined when x = ,

arctan does not exist.

Find the exact value of arcsin

A.

B.

C.

D.

Evaluate Compositions of Trigonometric

Functions

Find the exact value of

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.

To simplify the expression, let u = cos–1

so cos u = .


Functions

Using the Pythagorean Theorem, you can find that the length of the side opposite is 3. Now, solve for sin u.

opp = 3 and hyp = 5

Sine function

So,

Answer:

Find the exact value of

A.

B.

C.

D.

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.


Functions


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

So, cot(arcos x) = .

opp = and adj = x


Functions

Answer:

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

A.

B.

C.

D.

Documents

You found and graphed the inverses of relations and ... · Evaluate Inverse Sine Functions C. Find the exact value of sin–1 (–2π), if it exists. Because the domain of the inverse