Skill: Determine tangent ratios in right triangles

For an acute angle in a right triangle, tangent compares the two legs:

$$\tan\theta=\frac{\text{opposite leg}}{\text{adjacent leg}}$$

• Opposite: the leg directly across from the chosen angle.
• Adjacent: the leg next to the chosen angle that is not the hypotenuse.
• Do not use the hypotenuse in a tangent ratio.

Example

In right triangle $ABC$, $\angle C$ is the right angle. Relative to $\angle A$:

• $BC=6$ is the opposite leg.
• $AC=8$ is the adjacent leg.

Find $\tan A$.

Step 1: Identify the two sides relative to $\angle A$.

The opposite side is $BC=6$, and the adjacent side is $AC=8$.

Step 2: Substitute into the tangent formula.

$$\tan A=\frac{\text{opposite}}{\text{adjacent}} =\frac{6}{8}$$

Step 3: Simplify the ratio.

$$\tan A=\frac{3}{4}=0.75$$

So, $\boxed{\tan A=0.75}$.

The size of the triangle does not change this ratio. Any right triangle with the same acute angle has similar side proportions, so its opposite-to-adjacent ratio will also be $0.75$.