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• How do I use inverse sine, cosine, and tangent functions to find the angles of a triangle?
5.4 Solve Right Triangles
Inverse Trigonometric RatiosLet be an acute angle.A
A
B
C
Tangent Inverse then,A tan If xA.tan 1 mx A
AC
BCtan 1 m
Sine Inverse then,Asin If yA.sin 1 my
AAB
BCsin 1 m
Cosine Inverse then,A cos If zA.cos 1 mz
AAB
ACcos 1 m
5.4 Solve Right Triangles
Example 1 Use an inverse tangent to find an angle measureUse an inverse tangent to find an angle measure
Use a calculator to approximate the measure of to the nearest tenth of a degree.
A
A
C
B 20
16
Because tan A = _____
= _____
= _____,20
16
5
48.0
A.___tan 1 m8.0 Using a calculator,
._______________tan 1 8.0 . . . 65980825.38
So, the measure of is approximately _______.A o38.7
5.4 Solve Right TrianglesCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
1. In Example 1, use a calculator and an inverse tangent to approximate to the nearest tenth of a degree.
A
C
B 20
16C
tan C 20
1625.1
25.1tan 1 o3.51
5.4 Solve Right Triangles
Example 2 Use an inverse sine and an inverse cosineUse an inverse sine and an inverse cosine
Let and be acute angles in two right triangles. Use a calculator to approximate the measure of and to the nearest tenth of a degree.
A BA B
a. sin A = 0.76
b. cos B = 0.17
Solution
________ a. Am 76.0sin 1
_____ o5.49
________ b. Bm 17.0cos 1
_____ o2.80
5.4 Solve Right Triangles
Example 3 Solve a right triangleSolve a right triangle
Solve the right triangle. Round your answer to the nearest tenth.
ft 40
A
B
Co23
SolutionStep 1 Find by using the Triangle Sum Theorem.Bm
Bm oo 2390____o180Bm____o67
o67
5.4 Solve Right Triangles
Example 3 Solve a right triangleSolve a right triangle
Solve the right triangle. Round your answer to the nearest tenth.
ft 40
A
B
Co23
SolutionStep 2 Approximate BC using a ______ ratio.
40
BC_______ o23sin
o67
sine
Write ratio for _________.o23sin
Multiply each side by ___.40BC___________ o23sin40Approximate ________.
o23sinBC___________ 3907.040Simplify and round answer.BC____ 6.15
ft 5.61
5.4 Solve Right Triangles
Example 3 Solve a right triangleSolve a right triangle
Solve the right triangle. Round your answer to the nearest tenth.
ft 40
A
B
Co23
SolutionStep 3 Approximate AC using a ______ ratio.
40
AC_______ o23cos
o67
cosine
Write ratio for _________.o23cos
Multiply each side by ___.40AC___________ o23cos40Approximate ________.
o23cosAC___________ 9205.040Simplify and round answer.AC____ 8.36
ft 5.61
ft 6.83
5.4 Solve Right Triangles
Example 3 Solve a right triangleSolve a right triangle
Solve the right triangle. Round your answer to the nearest tenth.
ft 40
A
B
Co23
Solution
o67ft 5.61
ft 6.83
The angle measures are _____, _____, and _____. o23 o67 o90
The side lengths are _____ feet, about _______ feet, and about ______ feet.
40 6.158.36
5.4 Solve Right TrianglesCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
2. Find to the nearest tenth of a degree if cos T = 0.64.
Tm
Tm 64.0cos 1 o2.503. Find to the nearest tenth of a degree if sin
D = 0.48.Dm
Dm 48.0sin 1 o7.28
5.4 Solve Right TrianglesCheckpoint. Complete the following exercises.Checkpoint. Complete the following exercises.
4. Solve a right triangle that has a 50o angle and a 15 inch hypotenuse.
Bm 5090180 o40Bm
AC
B
in 51
o50
o40
o50sin15
a15 15
1550sin o a in 5.11
in 1.51
o50cos15
b15 15
1550cos o b in 6.9
in .69
5.4 Solve Right Triangles
Pg. 187, 5.4 #1-30
