Skill: Determine sine ratios in right triangles

For an acute angle in a right triangle, use the sine ratio:

$$\sin\theta=\frac{\text{length of the opposite side}}{\text{length of the hypotenuse}}$$

• Opposite: the side directly across from the chosen angle
• Hypotenuse: the side across from the right angle; it is always the longest side

Example: In right triangle $(ABC)$, $\angle C=90^\circ$. For $\angle A$, the side opposite is $(BC=6)$ cm, and the hypotenuse is $(AB=10)$ cm. Find $\sin A$.

1. Identify the reference angle: $\angle A$.
2. Identify the opposite side: $(BC=6)$ cm.
3. Identify the hypotenuse: $(AB=10)$ cm.
4. Substitute into the sine formula:

$$\sin A=\frac{\text{opposite}}{\text{hypotenuse}} =\frac{6}{10}$$

5. Simplify:

$$\boxed{\sin A=\frac{3}{5}=0.6}$$

Always choose the sides based on the angle you are using. If you used $\angle B$ instead, the opposite side would be different, so the sine ratio would change.