A right triangle has legs that measure 2 units and

### Inverse of Trigonometric Functions

In the previous concept we used the trigonometric functions sine, cosine and tangent to find the ratio of particular sides in a right triangle given an angle. In this concept we will use the inverses of these functions,

#### Find the inverse of the following problems

Find the measure of angle

sinA=0.8336 tanA=1.3527 cosA=0.2785

Using the calculator we get the following:

sin−1(0.8336)≈56∘ tan−1(1.3527)≈54∘ cos−1(0.2785)≈74∘

Find the measures of the unknown angles in the triangle shown. Round answers to the nearest degree.

We can solve for either

Recall that in a right triangle, the acute angles are always complementary, so

Solve the right triangle shown below. Round all answers to the nearest tenth.

We can solve for either angle

Now we can find

Method 1: We can using trigonometry and the cosine ratio:

Method 2: We can subtract

Either method is valid, but be careful with Method 2 because a miscalculation of angle

### Examples

#### Example 1

Earlier, you were asked what are the measures of the triangle's acute angles.

First, let's find the hypotenuse, then we can solve for either angle.

One of the acute angles will have a sine of

Now we can find

Find the measure of angle

#### Example 2

#### Example 3

#### Example 4

\begin{align*}\cos A=0.8911\end{align*}

\begin{align*}\cos^{-1} (0.8911) \approx 27^\circ\end{align*}

#### Example 5

Find the measures of the unknown angles in the triangle shown. Round answers to the nearest degree.

\begin{align*}x=\cos^{-1} \left(\frac{13}{20} \right) \approx 49^\circ; \quad y=\sin^{-1} \left(\frac{13}{20} \right) \approx 41^\circ\end{align*}

#### Example 4

Solve the triangle. Round side lengths to the nearest tenth and angles to the nearest degree.

\begin{align*}m \angle A=\cos^{-1} \left(\frac{17}{38} \right) \approx 63^\circ; \quad m \angle B=\sin^{-1} \left(\frac{17}{38} \right) \approx 27^\circ; \quad a=\sqrt{38^2-17^2} \approx 34.0\end{align*}

### Review

Use your calculator to find the measure of angle \begin{align*}B\end{align*}. Round answers to the nearest degree.

- \begin{align*}\tan B=0.9523\end{align*}
- \begin{align*}\sin B=0.8659\end{align*}
- \begin{align*}\cos B=0.1568\end{align*}
- \begin{align*}\sin B=0.2234\end{align*}
- \begin{align*}\cos B=0.4855\end{align*}
- \begin{align*}\tan B=0.3649\end{align*}

Find the measures of the unknown acute angles. Round measures to the nearest degree.

Solve the following right triangles. Round angle measures to the nearest degree and side lengths to the nearest tenth.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 13.3.