Skill: Select an appropriate trigonometric ratio

To choose a trigonometric ratio, first focus on the given acute angle. Then name the sides from the angle’s point of view:

• Opposite: the side directly across from the angle
• Adjacent: the side next to the angle that is not the hypotenuse
• Hypotenuse: the longest side, always opposite the right angle

Use the ratio that includes the sides you know and the side you want:

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

Example

A right triangle has an angle of $40^\circ$. The side next to the $40^\circ$ angle has length $x$, and the hypotenuse has length $15$. Find $x$.

Step 1: Identify the sides relative to the $40^\circ$ angle.

• Adjacent side: $x$
• Hypotenuse: $15$

Step 2: Choose the ratio.

The available sides are adjacent and hypotenuse, so use cosine:

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

Step 3: Substitute the known values.

$$\cos(40^\circ)=\frac{x}{15}$$

Step 4: Solve for $x$.

Multiply both sides by $15$:

$$x=15\cos(40^\circ)$$

Using a calculator:

$$x\approx 11.5$$

So, the missing side is approximately

$$\boxed{11.5}$$

Always identify the sides in relation to the chosen angle before selecting sine, cosine, or tangent.