Section 7.7 Notes
To solve a right triangle means to find the
measures of all of its sides and angles.
You can solve a right triangle if you know either of
the following:
1. two side lengths
2.
one side length and the measure of an acute
angle.
Inverse Trigonometric Functions
Let A be an acute angle.
Inverse Tangent:
If tan A = x, then tan−1x = mA.
 BC 
tan 
  mA
 AC 
1
Inverse Sine:
If sin A = x, then sin−1x = mA.
 BC 
sin 
  mA
 AB 
1
Inverse Cosine:
If cos A = x, then cos−1x = mA.
 AC 
cos 
  mA
 AB 
1
Example 1
Use a calculator to approximate the measure of
Q to the nearest tenth of a degree.
12
tan Q 
8
1  12 
tan    mQ
8
mQ  56.3
Example 2
Use a calculator to approximate the measure of
A to the nearest tenth of a degree.
16
tan A 
20
1  16 
tan    mA
 20 
mA  38.7
Example 3
Let C be an acute angle in a right triangle. Use
a calculator to approximate the measure of C to
the nearest tenth of a degree.
a.
sin C = 0.24
1
mC  13.9
sin  0.24   mC
b.
cos C = 0.37
1
cos  0.37   mC
mC  68.3
Example 4
Let A and B be acute angles in two right
triangles. Use a calculator to approximate the
measures of A and B to the nearest tenth of a
degree.
1
a.
sin A = 0.76 sin  0.76   mA
mA  49.5
b.
cos B = 0.17 cos 1  0.17   mB
mB  80.2
Example 5
Solve the right triangle formed by the water slide
shown in the figure. Round decimal answers to
the nearest tenth.
mY = 90°
mZ = (90 – 42)° = 48°
ZY
sin 42 
50
ZY  50sin 42
ZY  33.5 ft.
XY
cos 42 
50
XY  50cos 42
XY  37.2 ft.
Example 6
Solve the right triangle. Round decimal answers
to the nearest tenth.
mC = 90°
mB = (90 – 23)° = 67°
BC
sin 23 
40
BC  40sin 23
BC  15.6 ft.
AC
cos 23 
40
AC  40cos 23
AC  36.8 ft.
Example 7
You are building a track for a model train. You
want the track to incline from the first level to
the second level, 4 inches higher, in 96 inches.
Is the angle of elevation less than 3°?
4
tan x 
96
1  4 
x  tan  
 96 
x  2.4
Yes, the angle of elevation
is less than 3°
Example 8
A road rises 10 feet in a horizontal distance of
200 feet. What is the angle of inclination to the
nearest tenth?
10 ft.
10
x°
tan x 
200 ft.
200
1  10 
x  tan 

 200 
x  2.9
The angle of inclination is about 2.9°.