9.4 Evaluate Inverse
Trigonometric Functions
How are inverse Trigonometric functions used?
How much information must be given about side
lengths in a right triangle in order for you to be able to
find the measures of its acute angles?
Inverse Trig Functions
y
x
Inverse Trig Functions
y
0
x
Inverse Trig Functions
y
x
Evaluate the expression in both radians and degrees.
a. cos–1 √ 3
2
SOLUTION
a. When 0 ≤ θ ≤ π or 0°≤ θ ≤
whose cosine is √ 3
2
π
√3
–1
cos
=
θ =
2
6
180°, the angle
√ 3 30°
=
θ = cos–1
2
y
90°
120°
60°
45°
135°
30°
150°
0°
360°
180°
330°
210°
315°
225°
240°
300°
270°
x
Evaluate the expression in both radians and degrees.
b. sin–1 2
SOLUTION
b. There is no angle whose sine is 2. So, sin–1 2 is
undefined.
Evaluate the expression in both radians and degrees.
c. tan–1 ( – √ 3 )
SOLUTION
c.
When – π < θ < π , or – 90° < θ < 90°, the
2
2
angle whose tangent is – √ 3 is:
θ=
tan–1
π
–
(–√3 ) =
3
θ = tan–1 ( – √ 3 ) = –60°
Evaluate the expression in both radians and degrees.
1.
sin–1 √ 2
2
ANSWER
2.
3.
π , 45°
4
cos–1 1
2
ANSWER
ANSWER – 4π , –45°
4.
π , 60°
3
tan–1 (–1)
sin–1 (– 1 )
2
ANSWER – 6π , –30°
Solve a Trigonometric Equation
5
Solve the equation sin θ = – 8 where 180° < θ < 270°.
SOLUTION
STEP 1
Use a calculator to determine that in the
interval –90° ≤ θ ≤ 90°, the angle whose
5
sine is – 5 is sin–1 –
– 38.7°. This
8
8
angle is in Quadrant IV, as shown.
STEP 2
Find the angle in Quadrant III (where
180° < θ < 270°) that has the same sine
value as the angle in Step 1. The angle is:
θ 180° + 38.7° = 218.7°
CHECK : Use a calculator to check the answer.
5
–

sin 218.7° – 0.625 =
8
Solve the equation for
5. cos θ = 0.4;
ANSWER
6. tan θ = 2.1;
ANSWER
7. sin θ = –0.23;
ANSWER
270° < θ < 360°
about 293.6°
360  66.4  293 .6
180° < θ < 270°
about 244.5°
64.5  180  244 .5
270° < θ < 360°
about 346.7°
360  13.3  346 .7
Solve the equation for
8. tan θ = 4.7;
ANSWER
9. sin θ = 0.62;
ANSWER
180° < θ < 270°
about 258.0°
78  180  258
90° < θ < 180°
about 141.7°
180  38.3  141 .7
10. cos θ = –0.39; 180° < θ < 270°
ANSWER
about 247.0°
360  113  247
SOLUTION
In the right triangle, you are given the lengths of the
side adjacent to θ and the hypotenuse, so use the
inverse cosine function to solve for θ.
cos θ =
6
adj
= 11
hyp
θ = cos
ANSWER The correct answer is C.
–1
6
11
56.9°
Monster Trucks
A monster truck drives off a ramp in order to jump
onto a row of cars. The ramp has a height of 8 feet
and a horizontal length of 20 feet. What is the angle θ
of the ramp?
http://www.youtube.com/watch?v=7SjX7A_FR6g
http://www.youtube.com/watch?v=SrzXaDFZcAo
SOLUTION
STEP 1 Draw: a triangle that represents the ramp.
Write: a trigonometric equation
STEP 2
that involves the ratio of the
ramp’s height and horizontal
length.
8
opp
tan θ =
= 20
adj
STEP 3 Use: a calculator to find the measure of θ.
θ=
tan–1
8
20
21.8°
ANSWER
The angle of the ramp is about 22°.
Find the measure of the angle θ.
11.
SOLUTION
In the right triangle, you are given the lengths of the
side adjacent to θ and the hypotenuse. So, use the
inverse cosine function to solve for θ.
cos θ =
adj
4
= 9
hyp
θ = cos–1 4
9
63.6°
Find the measure of the angle θ.
12.
SOLUTION
In the right triangle, you are given the lengths of the
side opposite to θ and the side adjacent. So, use the
inverse tan function to solve for θ.
tan θ =
opp
10
= 8
adj
θ = tan–1 10
8
51.3°
Find the measure of the angle θ.
13.
SOLUTION
In the right triangle, you are given the lengths of the
side opposite to θ and the hypotenuse. So, use the
inverse sin function to solve for θ.
sin θ =
opp
5
= 12
hyp
θ = sin–1
5
12
24.6°
9.4 Assignment
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