Right Triangle Trigonometry Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

In a right triangle, the angle

\theta = 30°

, and the hypotenuse is 10. Find the lengths of the opposite and adjacent sides.

1
Step 1: Recall the definitions: $\sin\theta = \frac{\text{opp}}{\text{hyp}}$ and $\cos\theta = \frac{\text{adj}}{\text{hyp}}$ .
2
Step 2: Find the opposite side: $\sin 30° = \frac{\text{opp}}{10}$ . Since $\sin 30° = 0.5$ , we get $\text{opp} = 10 \times 0.5 = 5$ .
3
Step 3: Find the adjacent side: $\cos 30° = \frac{\text{adj}}{10}$ . Since $\cos 30° = \frac{\sqrt{3}}{2} \approx 0.866$ , we get $\text{adj} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66$ .

Answer

Opposite

= 5

, Adjacent

= 5\sqrt{3}

The sine ratio connects the opposite side to the hypotenuse, and the cosine ratio connects the adjacent side to the hypotenuse. For a

30°

angle, these are well-known values:

\sin 30° = \frac{1}{2}

and

\cos 30° = \frac{\sqrt{3}}{2}

. Multiplying each by the hypotenuse length gives the side lengths.

About Right Triangle Trigonometry

The three primary trigonometric ratios—sine, cosine, and tangent—defined as ratios of specific sides in a right triangle.

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