Page 814 - Algebra 1

Inverse trigonometric functions can be used to find missing angle measures. On a calculator, these are sin-1, cos-1, and tan-1. Because they are inverse functions, sin-1(sin A) = A, and the same principle follows for cosine and tangent.
Using Trigonometry to Find Missing Angle Measures
Math Reasoning
Generalize When do you use the sine function and when do you use the inverse sine (sin-1) function?
Example
4
a. Find the measure of ∠A. Round to the nearest hundredth of a degree.
SOLUTION
Use the cosine ratio since you know the adjacent leg and the hypotenuse.
c o s A = _6 11
_6 cos-1(cos A) = cos-1 (11 )
∠A ≈ 56.94°
b. Find the measures of ∠A and ∠B. Round to the
B
11
nearest hundredth of a degree.
SOLUTION
Use the tangent ratio since you know the lengths of the legs.
t a n A = _2 t a n B = _5 52
C 6 A
B
2 A 5 C
Hint
You can also find the measure of the second acute angle of a right triangle by subtracting the first angle from 90°. For example, if
m ∠A = 21.80°,
then
m ∠B = 90° - 21.80° = 68.20°.
_2 tan-1(tan A) = tan-1(5 )
∠A ≈ 21.80°
Application: Indirect Measurement
Use the tangent ratio since the problem involves both legs.
tan 35° = _10,000 _x
x·tan35°= 10,000 ·x x
-1
tan (tan B) = tan
-1 _5 (2 )
∠B ≈ 68.20°
Example
5
If an airplane takes off at a 35° angle with the ground, how far has the plane traveled horizontally when it reaches an altitude
of 10,000 feet?
SOLUTION
10,000 feet
x
35°
x · tan 35° = 1_0,000
x = 10,000 ≈ 14,281
tan 35°
The plane has traveled about 14,281 feet horizontally.
Lesson 117
799