Determine tangent ratios in right triangles

Tangent relates an acute angle in a right triangle to the ratio of the length of the opposite leg to the length of the adjacent leg: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ . The understanding includes identifying these sides relative to a chosen angle, interpreting tangent as a scale-invariant relationship across similar triangles, and using known side lengths or a calculator to determine a ratio or missing measure; hypotenuse-based ratios and tangent beyond acute right-triangle contexts are not included.

Detailed Explanation: 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 $0.75$ .

Learn by doing: Determine tangent ratios in right triangles

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Ratios of Lengths - Length and Ratio to Top Length, Decimal Numbers - Right Angle Line Display

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