Inverses of Trigonometric Functions

1
Inverses of Trigonometric Functions
13-4
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Objectives
Evaluate inverse trigonometric functions. Use
trigonometric equations and inverse trigonometric
functions to solve problems.
3
Vocabulary
inverse sine functions inverse cosine
function inverse tangent function
4
You have evaluated trigonometric functions for a
given angle. You can also find the measure of
angles given the value of a trigonometric
function by using an inverse trigonometric
relation.
5
(No Transcript)
6
The inverses of the trigonometric functions are
not functions themselves because there are many
values of ? for a particular value of a. For
example, suppose that you want to find cos-1 .
Based on the unit circle, angles that measure
and radians have a cosine of . So do all
angles that are coterminal with these angles.
7
Example 1 Finding Trigonometric Inverses
Find all possible values of cos-1 .
Use the x-coordinates of points on the unit
circle.
8
Example 1 Continued
Find all possible values of cos-1 .
Add integer multiples of 2? radians, where n is
an integer
9
Check It Out! Example 1
Find all possible values of tan-11.
10
Because more than one value of ? produces the
same output value for a given trigonometric
function, it is necessary to restrict the domain
of each trigonometric function in order to define
the inverse trigonometric functions.
11
Trigonometric functions with restricted domains
are indicated with a capital letter. The domains
of the Sine, Cosine, and Tangent functions are
restricted as follows.
? is restricted to Quadrants I and IV.
? is restricted to Quadrants I and II.
? is restricted to Quadrants I and IV.
12
These functions can be used to define the inverse
trigonometric functions. For each value of a in
the domain of the inverse trigonometric
functions, there is only one value of ?.
Therefore, even though tan-1 has many values,
Tan-11 has only one value.
13
(No Transcript)
14
(No Transcript)
15
Example 2A Evaluating Inverse Trigonometric
Functions
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
Use x-coordinates of points on the unit circle.
16
Example 2B Evaluating Inverse Trigonometric
Functions
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
17
Check It Out! Example 2a
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
18
Check It Out! Example 2b
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
19
Example 3 Safety Application
A painter needs to lean a 30 ft ladder against a
wall. Safety guidelines recommend that the
distance between the base of the ladder and the
wall should be of the length of the ladder. To
the nearest degree, what acute angle should the
ladder make with the ground?
20
Example 3 Continued
21
Example 3 Continued
Step 2 Find the value of ?.
Use the cosine ratio.
Substitute 7.5 for adj. and 30 for hyp. Then
simplify.
The angle between the ladder and the ground
should be about 76
22
Check It Out! Example 3
A group of hikers wants to walk form a lake to an
unusual rock formation.
The formation is 1 mile east and 0.75 mile north
of the lake. To the nearest degree, in what
direction should the hikers head from the lake to
reach the rock formation?
23
Example 4A Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use the
given restrictions.
sin ? 0.4, for 90 ? 90
The restrictions on ? are the same as those for
the inverse sine function.
Use the inverse sine function on your calculator.
?? Sin-1(0.4) 23.6
24
Example 4B Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use the
given restrictions.
sin ? 0.4, for 90 ? 270
The terminal side of ? is restricted to Quadrants
ll and lll. Since sin ? gt 0, find the angle in
Quadrant ll that has the same sine value as 23.6.
? has a reference angle of 23.6, and 90 lt ? lt
180.
? 180 23.6 156.4
25
Check It Out! Example 4a
Solve each equations to the nearest tenth. Use
the given restrictions.
tan ? 2, for 90 lt ? lt 90
26
Check It Out! Example 4b
Solve each equations to the nearest tenth. Use
the given restrictions.
tan ? 2, for 90 lt ? lt 180