Solving for a side in a right triangle using the trigonometric ratios

In a right triangle, the trigonometric ratios—sine, cosine, and tangent—are used to relate the angles to the lengths of the sides. Here's a brief overview of each:

1. Sine (sin): This ratio compares the length of the opposite side to the hypotenuse.

   $$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

2. Cosine (cos): This ratio compares the length of the adjacent side to the hypotenuse.

   $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$

3. Tangent (tan): This ratio compares the length of the opposite side to the adjacent side.

   $$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

To solve for a side of a triangle, you can rearrange these formulas based on the known values:

• To find the length of the opposite side:

  $$\text{Opposite} = \sin(\theta) \times \text{Hypotenuse} \quad \text{or} \quad \text{Opposite} = \tan(\theta) \times \text{Adjacent}$$

• To find the length of the adjacent side:

  $$\text{Adjacent} = \cos(\theta) \times \text{Hypotenuse} \quad \text{or} \quad \text{Adjacent} = \frac{\text{Opposite}}{\tan(\theta)}$$

• To find the hypotenuse:

  $$\text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\theta)} \quad \text{or} \quad \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)}$$

By knowing one side length and one angle (other than the right angle), you can calculate the lengths of the other sides using these ratios.