Solving for an angle in a right triangle using the trigonometric ratios

In a right triangle, the trigonometric ratios can be used to solve for angles. The primary ratios are:

1. Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

2. Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

3. Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

To find an angle (θ), you rearrange these formulas and apply the inverse trigonometric functions:

• $\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$
• $\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$
• $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

By knowing two sides of the triangle, you can use these ratios to calculate the unknown angle.