MAT1793-ABED_1: Using the Tangent Ratio

Constructing Knowledge

The ratio you have been using in much of this lesson is called the tangent ratio. This ratio can be written as \(\text{tangent of angle}\,\theta=\frac{\text{length opposite}\,\theta}{\text{length adjacent to}\,\theta}\) or more simply as \(\text{tan}\,\theta=\frac{\text{opp}}{\text{adj}}\).

Tangent Ratio

The ratio of the side length opposite an angle to the side length adjacent to that angle in a right triangle

Multimedia

A video explaining the relationship between similar triangles and the tangent ratio is provided below.

When solving for an unknown side length or angle measure in a right triangle using the tangent ratio, follow these steps:

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.

Step 2: State the appropriate ratio.

Step 3: Substitute known values, and calculate the unknown value.

EXAMPLE 1

Determine the value of tan θ.

Solution

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.

Step 2: State the appropriate ratio

\(\text{tan}\,\theta=\frac{\text{opp}\,\theta}{\text{adj}\,\theta}\)

Step 3: Substitute known values, and calculate the unknown value.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}\,\theta}{\text{adj}\,\theta} \\ \\ &=\frac{6}{10} \\ \\ &=0.6 \\ \end{align}\)

Notice that the tangent ratio always uses the two sides of the triangle that touch the right angle. The side opposite the right angle is called the hypotenuse. The hypotenuse is not used with the tangent ratio.

Hypotenuse

The longest side of a right triangle and the side opposite the right angle in a right triangle