Skill: Determine missing angles using inverse trigonometry

To find a missing acute angle in a right triangle:

1. Identify the sides relative to the unknown angle:

   • Opposite: across from the angle
   • Adjacent: next to the angle, but not the hypotenuse
   • Hypotenuse: opposite the right angle and the longest side

2. Choose the ratio that uses the known sides:

$$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

3. Use the inverse function on your calculator to find the angle:

$$\theta=\sin^{-1}(\text{ratio}),\quad \theta=\cos^{-1}(\text{ratio}),\quad\text{or}\quad \theta=\tan^{-1}(\text{ratio})$$

Make sure your calculator is in degree mode.

Example

A right triangle has an opposite side of $7$ cm and an adjacent side of $24$ cm. Find the angle $\theta$.

The known sides are the opposite and adjacent sides, so use tangent:

$$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$$

Substitute the side lengths:

$$\tan(\theta)=\frac{7}{24}$$

To undo the tangent, use the inverse tangent:

$$\theta=\tan^{-1}\left(\frac{7}{24}\right)$$

Enter this into a calculator in degree mode:

$$\theta\approx 16.3^\circ$$

Therefore, the missing angle is:

$$\boxed{16.3^\circ}$$

The inverse trigonometric function gives the angle that produces the known side ratio.