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Inverse Trigonometric Ratios

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Inverse Trigonometric Ratios

What if you were told the tangent of \angle Z is 0.6494? How could you find the measure of \angle Z ? After completing this Concept, you'll be able to find angle measures by using the inverse trigonometric ratios.

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CK-12 Foundation: The Inverse Trigonometric Ratios

James Sousa: Introduction to Inverse Trigonometric 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 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} \left (\frac{b}{a} \right ) = m \angle B and  \tan^{-1} \left( \frac{a}{b} \right) = m \angle A .

Inverse Sine: Labeled \sin^{-1} .

\sin^{-1} \left( \frac{b}{c} \right) = m \angle B and  \sin^{-1} \left(\frac{a}{c} \right) = m \angle A .

Inverse Cosine: Labeled \cos^{-1} .

\cos^{-1} \left(\frac{a}{c} \right) = m \angle B and  \cos^{-1} \left( \frac{b}{c} \right) = m \angle 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 [ \text{SIN}^{-1}], [ \text{COS}^{-1}] , and [\text{TAN}^{-1}] . You might also have to hit a shift or 2^{nd} 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.

Example A

Use the sides of the triangle and your calculator to find the value of \angle A . Round your answer to the nearest tenth of a degree.

In reference to \angle A , we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.

\tan A = \frac{20}{25} = \frac{4}{5} . So, \tan^{-1} \frac{4}{5} = m \angle A . Now, use your calculator.

If you are using a TI-83 or 84, the keystrokes would be: [2^{nd}] [TAN] \left( \frac{4}{5} \right) [ENTER] and the screen looks like:

m \angle A \approx 38.7^\circ

Example B

\angle A is an acute angle in a right triangle. Find m \angle 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) m \angle A = \sin^{-1} 0.68 \approx 42.8^\circ

b) m \angle A = \cos^{-1} 0.85 \approx 31.8^\circ

c) m \angle A = \tan^{-1} 0.34 \approx 18.8^\circ

Example C

Solve the right triangle.

To solve this right triangle, we need to find AB, m \angle C and m \angle B . Use only the values you are given.

\underline{AB} : Use the Pythagorean Theorem.

24^2 + AB^2 &= 30^2\\576 + AB^2 &= 900\\AB^2 &= 324\\AB &= \sqrt{324} = 18

\underline{m \angle B} : Use the inverse sine ratio.

\sin B &= \frac{24}{30} = \frac{4}{5}\\\sin^{-1} \left( \frac{4}{5} \right) \approx 53.1^\circ = m \angle B

\underline{m \angle C} : Use the inverse cosine ratio.

\cos C = \frac{24}{30} = \frac{4}{5} \longrightarrow \cos^{-1}\left( \frac{4}{5} \right) \approx  36.9^\circ = m \angle C

CK-12 Foundation: The Inverse Trigonometric Ratios

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 m \angle A .

\underline{AB} : Use the sine ratio.

\sin 62^\circ &= \frac{25}{AB}\\AB &= \frac{25}{\sin 62^\circ}\\AB & \approx 28.31

\underline{BC} : Use the tangent ratio.

\tan 62^\circ &= \frac{25}{BC}\\BC &= \frac{25}{\tan 62^\circ}\\BC & \approx 13.30

\underline{m \angle A} : Use the Triangle Sum Theorem

62^\circ + 90^\circ + m \angle A &= 180^\circ\\m \angle A &= 28^\circ

2. The two acute angles are congruent, making them both 45^\circ . This is a 45-45-90 triangle. You can use the trigonometric ratios or the special right triangle ratios.

Trigonometric Ratios

\tan 45^\circ &= \frac{15}{BC} && \sin 45^\circ = \frac{15}{AC}\\BC &= \frac{15}{\tan 45^\circ} = 15 && \quad \ AC = \frac{15}{\sin 45^\circ} \approx 21.21

45-45-90 Triangle Ratios

BC = AB = 15, AC = 15 \sqrt{2} \approx 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.

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Use your calculator to find m \angle A to the nearest tenth of a degree.

Let \angle A be an acute angle in a right triangle. Find m \angle A to the nearest tenth of a degree.

  1. \sin A = 0.5684
  2. \cos A = 0.1234
  3. \tan A = 2.78
  4. \cos^{-1} 0.9845
  5. \tan^{-1} 15.93
  6. \sin^{-1} 0.7851

Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest tenth.

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