Inverse Trigonometric Ratios
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: November 21, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Inverse Trigonometric Ratios
1
Inverse Trigonometric
Ratios
Here you’ll learn how to apply the three inverse trigonometric ratios, the inverse sine, the inverse cosine, and the
inverse tangent, to find angle measures.
What if you were told the tangent of 6 Z is 0.6494? How could you find the measure of 6 Z? After completing this
Concept, you’ll be able to find angle measures by using the inverse trigonometric ratios.
Watch This
MEDIA
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James Sousa:Introduction toInverseTrigonometric Functions
Guidance
In mathematics, the word inverse means “undo.” For example, addition and subtraction are inverses of each other
because one undoes the other. When we we use the inverse trigonometric ratios, we can find acute angle measures
as long as we are given two sides.
Inverse Tangent: Labeled tan−1 , the “-1” means inverse.
tan−1 ab = m6 B and tan−1 ba = m6 A.
Inverse Sine: Labeled sin−1 .
sin−1 bc = m6 B and sin−1 ac = m6 A.
Inverse Cosine: Labeled cos−1 .
cos−1 ac = m6 B and cos−1 bc = m6 A.
In most problems, to find the measure of the angles you will need to use your calculator. On most scientific and
graphing calculators, the buttons look like [SIN−1 ], [COS−1 ], and [TAN−1 ]. You might also have to hit a shift or 2nd
button to access these functions.
Now that you know both the trig ratios and the inverse trig ratios you can solve a right triangle. To solve a right
triangle, you need to find all sides and angles in it. You will usually use sine, cosine, or tangent; inverse sine, inverse
cosine, or inverse tangent; or the Pythagorean Theorem.
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Example A
Use the sides of the triangle and your calculator to find the value of 6 A. Round your answer to the nearest tenth of a
degree.
In reference to 6 A, we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.
tan A =
20
25
= 45 . So, tan−1 45 = m6 A. Now, use your calculator.
If you are using a TI-83 or 84, the keystrokes would be: [2nd ][TAN]
4
5
[ENTER] and the screen looks like:
m6 A ≈ 38.7◦
Example B
6
A is an acute angle in a right triangle. Find m6 A to the nearest tenth of a degree.
a) sin A = 0.68
b) cos A = 0.85
c) tan A = 0.34
Answers:
a) m6 A = sin−1 0.68 ≈ 42.8◦
b) m6 A = cos−1 0.85 ≈ 31.8◦
c) m6 A = tan−1 0.34 ≈ 18.8◦
Example C
Solve the right triangle.
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Concept 1. Inverse Trigonometric Ratios
To solve this right triangle, we need to find AB, m6 C and m6 B. Use only the values you are given.
AB: Use the Pythagorean Theorem.
242 + AB2 = 302
576 + AB2 = 900
AB2 = 324
√
AB = 324 = 18
m6 B: Use the inverse sine ratio.
sin B =
sin
−1
24 4
=
30 5
4
≈ 53.1◦ = m6 B
5
m6 C: Use the inverse cosine ratio.
24 4
cosC =
= −→ cos−1
30 5
4
≈ 36.9◦ = m6 C
5
Guided Practice
1. Solve the right triangle.
2. Solve the right triangle.
3. When would you use sin and when would you use sin−1 ?
Answers:
1. To solve this right triangle, we need to find AB, BC and m6 A.
AB: Use the sine ratio.
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25
AB
25
AB =
sin 62◦
AB ≈ 28.31
sin 62◦ =
BC: Use the tangent ratio.
25
BC
25
BC =
tan 62◦
BC ≈ 13.30
tan 62◦ =
m6 A: Use the Triangle Sum Theorem
62◦ + 90◦ + m6 A = 180◦
m6 A = 28◦
2. The two acute angles are congruent, making them both 45◦ . This is a 45-45-90 triangle. You can use the
trigonometric ratios or the special right triangle ratios.
Trigonometric Ratios
15
BC
15
BC =
= 15
tan 45◦
tan 45◦ =
15
AC
15
AC =
≈ 21.21
sin 45◦
sin 45◦ =
45-45-90 Triangle Ratios
√
BC = AB = 15, AC = 15 2 ≈ 21.21
3. You would use sin when you are given an angle and you are solving for a missing side. You would use sin−1
when you are given sides and you are solving for a missing angle.
Practice
Use your calculator to find m6 A to the nearest tenth of a degree.
1.
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Concept 1. Inverse Trigonometric Ratios
2.
3.
4.
5.
6.
Let 6 A be an acute angle in a right triangle. Find m6 A to the nearest tenth of a degree.
7.
8.
9.
10.
11.
12.
sin A = 0.5684
cos A = 0.1234
tan A = 2.78
cos−1 0.9845
tan−1 15.93
sin−1 0.7851
Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest
tenth.
13.
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14.
15.
16.
17.
18.
19.
20.
21.
6