Inverses of Trigonometric
Functions

10-4
Inverses of Trigonometric
Functions
°
°
°
°
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
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.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Reading Math
The expression sin -1 is read as “the inverse sine.”
In this notation, -1 indicates the inverse of the sine function, NOT the reciprocal of the sine function.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
The inverses of the trigonometric functions are not functions themselves because there are many values of θ 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.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 1: Finding Trigonometric Inverses
Find all possible values of cos -1 .
Step 1 Find the values between
0 and 2  radians for which cos θ is equal to .
Use the x-coordinates of points on the unit circle.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 1 Continued
Find all possible values of cos -1 .
Step 2 Find the angles that are coterminal with angles measuring and radians.
Add integer multiples of
2
 radians, where n is an integer
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Because more than one value of θ 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.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Trigonometric functions with restricted domains are indicated with a capital letter. The domains of the Sine, Cosine, and Tangent functions are restricted as follows.
Sinθ = sinθ for {θ| }
θ is restricted to Quadrants I and IV.
Cosθ = cosθ for {θ| }
θ is restricted to Quadrants I and II.
Tanθ = tanθ for {θ| }
θ is restricted to Quadrants I and IV.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
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 θ. Therefore, even though tan -1 has many values, Tan -1 1 has only one value.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Reading Math
The inverse trigonometric functions are also called the arcsine, arccosine, and arctangent functions.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 2A: Evaluating Inverse Trigonometric
Functions
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.

Find value of θ for
Cosine or whose
.
Use x-coordinates of points on the unit circle.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Check It Out!
Example 2b
Evaluate each inverse trigonometric function.
Give your answer in both radians and degrees.
0 = Cos θ
(0, 1)

Find value of θ for or whose
Cosine is 0.
Use x-coordinates of points on the unit circle.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
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?
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 3 Continued
Step 1 Draw a diagram. The base of the ladder should be (30) = 7.5 ft from the wall. The angle between the ladder and the ground θ is the measure of an acute angle of a right triangle.
Holt McDougal Algebra 2
7.5
θ

10-4
Inverses of Trigonometric
Functions
Example 3 Continued
Step 2 Find the value of θ.
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 °
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
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?
Step 1 Draw a diagram. The base of the triangle should be 1 mile. The angle North from that point to the rock is 0.75 miles.
θ is the measure of an acute angle of a right triangle.
Lake
Holt McDougal Algebra 2
θ
1 mi
Rock
0.75 mi

10-4
Inverses of Trigonometric
Functions
Check It Out!
Example 3 Continued
Step 2 Find the value of θ
Use the tangent ratio.
Substitute 0.75 for opp. and 1 for adj. Then simplify.
The angle the hikers should take is about 37 ° north of east.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 4A: Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use the given restrictions.
sin θ = 0.4, for – 90 ° ≤ θ ≤ 90 °
The restrictions on θ are the same as those for the inverse sine function.
 = Sin -1 (0.4) ≈ 23.6
°
Use the inverse sine function on your calculator.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Example 4B: Solving Trigonometric Equations
Solve each equation to the nearest tenth. Use the given restrictions.
sin θ = 0.4, for 90 ° ≤ θ ≤ 270 °
The terminal side of θ is restricted to Quadrants ll and lll. Since sin θ > 0, find the angle in Quadrant ll that has the same sine value as 23.6
° .
θ ≈ 180 ° – 23.6
° ≈ 156.4
°
θ has a reference angle of 23.6
° , and
90 ° < θ < 180 ° .
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Check It Out!
Example 4a
Solve each equations to the nearest tenth.
Use the given restrictions.
tan θ = – 2, for – 90 ° < θ < 90 °
The restrictions on θ are the same for those of the inverse tangent function.
θ = Tan -1 – 2 ≈ – 63.4
° Use the inverse tangent function on your calculator.
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Check It Out!
Example 4b
Solve each equations to the nearest tenth.
Use the given restrictions.
tan θ = – 2, for 90 ° < θ < 180 °
The terminal side of θ is restricted to Quadrant II.
Since tan θ < 0, find the angle in Quadrant II that has the same value as – 63.4
°.
116.6
°
– 63.4
°
Holt McDougal Algebra 2
θ ≈ 180 ° – 63.4
° ≈ 116.6
°

10-4
Inverses of Trigonometric
Functions
Lesson Quiz: Part I
1. Find all possible values of cos -1 ( – 1).
2. Evaluate Sin -1 Give your answer in both radians and degrees.
3. A road has a 5% grade, which means that there is a
5 ft rise for 100 ft of horizontal distance. At what angle does the road rise from the horizontal? Round to the nearest tenth of a degree.
2.9
°
Holt McDougal Algebra 2

10-4
Inverses of Trigonometric
Functions
Lesson Quiz: Part II
Solve each equation to the nearest tenth.
Use the given restrictions.
4. cos θ = 0.3, for 0 ° ≤ θ ≤ 180 ° θ ≈ 72.5
°
5. cos θ = 0.3, for 270 ° < θ < 360 ° θ ≈ 287.5
°
Holt McDougal Algebra 2