Skill: Determine missing sides using trigonometry

In a right triangle, choose the given acute angle first. Then name the sides relative to that angle:

• Opposite: across from the angle
• Adjacent: next to the angle, but not the hypotenuse
• Hypotenuse: across from the right angle; it is always the longest side

Use SOHCAHTOA to choose a trigonometric ratio:

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

Example

Find the missing side length $x$.

Suppose a right triangle has:

• an acute angle of $35^\circ$,
• an adjacent side of length $8$,
• an opposite side of length $x$.

Since the known side and unknown side are the adjacent and opposite sides, use tangent:

$$\tan(35^\circ)=\frac{x}{8}$$

Multiply both sides by $8$ to isolate $x$:

$$x=8\tan(35^\circ)$$

Make sure your calculator is in degree mode, then calculate:

$$x\approx 8(0.7002)$$ $$x\approx 5.6$$

Therefore, the missing side is approximately

$$\boxed{5.6\text{ units}}$$

The answer makes sense because the $35^\circ$ angle is less than $45^\circ$, so the opposite side should be shorter than the adjacent side. Indeed, $5.6<8$.