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# Inverse Trigonometric Ratios

## Solving for an angle given a trigonometric ratio.

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Inverse Trigonometric Ratios

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

### Watch This

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 tan1\begin{align*}\tan^{-1}\end{align*}, the “-1” means inverse.

tan1(ba)=mB\begin{align*}\tan^{-1} \left (\frac{b}{a} \right ) = m \angle B\end{align*} and tan1(ab)=mA\begin{align*} \tan^{-1} \left( \frac{a}{b} \right) = m \angle A\end{align*}.

Inverse Sine: Labeled sin1\begin{align*}\sin^{-1}\end{align*}.

sin1(bc)=mB\begin{align*}\sin^{-1} \left( \frac{b}{c} \right) = m \angle B\end{align*} and sin1(ac)=mA\begin{align*} \sin^{-1} \left(\frac{a}{c} \right) = m \angle A\end{align*}.

Inverse Cosine: Labeled cos1\begin{align*}\cos^{-1}\end{align*}.

cos1(ac)=mB\begin{align*}\cos^{-1} \left(\frac{a}{c} \right) = m \angle B\end{align*} and cos1(bc)=mA\begin{align*} \cos^{-1} \left( \frac{b}{c} \right) = m \angle A\end{align*}.

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 [SIN1],[COS1]\begin{align*}[ \text{SIN}^{-1}], [ \text{COS}^{-1}]\end{align*}, and [TAN1]\begin{align*}[\text{TAN}^{-1}]\end{align*}. You might also have to hit a shift or 2nd\begin{align*}2^{nd}\end{align*} 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 A\begin{align*}\angle A\end{align*}. Round your answer to the nearest tenth of a degree.

In reference to A\begin{align*}\angle A\end{align*}, we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.

tanA=2025=45\begin{align*}\tan A = \frac{20}{25} = \frac{4}{5}\end{align*}. So, tan145=mA\begin{align*}\tan^{-1} \frac{4}{5} = m \angle A\end{align*}. Now, use your calculator.

If you are using a TI-83 or 84, the keystrokes would be: [2nd]\begin{align*}[2^{nd}]\end{align*}[TAN](45)\begin{align*}\left( \frac{4}{5} \right)\end{align*} [ENTER] and the screen looks like:

mA38.7

#### Example B

\begin{align*}\angle A\end{align*} is an acute angle in a right triangle. Find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

a) \begin{align*}\sin A = 0.68\end{align*}

b) \begin{align*}\cos A = 0.85\end{align*}

c) \begin{align*}\tan A = 0.34\end{align*}

a) \begin{align*}m \angle A = \sin^{-1} 0.68 \approx 42.8^\circ\end{align*}

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

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

#### Example C

Solve the right triangle.

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

\begin{align*}\underline{AB}\end{align*}: Use the Pythagorean Theorem.

\begin{align*}\underline{m \angle B}\end{align*}: Use the inverse sine ratio.

\begin{align*}\underline{m \angle C}\end{align*}: Use the inverse cosine ratio.

CK-12 Foundation: The Inverse Trigonometric Ratios

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### Guided Practice

1. Solve the right triangle.

2. Solve the right triangle.

3. When would you use \begin{align*}\sin\end{align*} and when would you use \begin{align*}\sin^{-1}\end{align*}?

1. To solve this right triangle, we need to find \begin{align*}AB, BC\end{align*} and \begin{align*}m \angle A\end{align*}.

\begin{align*}\underline{AB}\end{align*}: Use the sine ratio.

\begin{align*}\underline{BC}\end{align*}: Use the tangent ratio.

\begin{align*}\underline{m \angle A}\end{align*}: Use the Triangle Sum Theorem

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

Trigonometric Ratios

45-45-90 Triangle Ratios

3. You would use \begin{align*}\sin\end{align*} when you are given an angle and you are solving for a missing side. You would use \begin{align*}\sin^{-1}\end{align*} when you are given sides and you are solving for a missing angle.

### Explore More

Use your calculator to find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

Let \begin{align*}\angle A\end{align*} be an acute angle in a right triangle. Find \begin{align*}m \angle A\end{align*} to the nearest tenth of a degree.

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

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.10.

### Vocabulary Language: English

Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Conic

Conic

Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
cosine

cosine

The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse.
Hypotenuse

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.
sine

sine

The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.
Slope

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Tangent

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.