Tangent Ratio

Comparing Opposite and Adjacent Sides

The tangent ratio, written as $\tan$ , compares the side opposite an angle with the adjacent side in a right triangle. Both sides are measured from the same acute angle. Because only the ratio is used, a larger triangle with the same angle gives the same tangent value.

Tangent

Tangent is the ratio between the length of the opposite side (the side opposite to the known angle) and the adjacent side (the side adjacent to the angle) in a right-angled triangle.

\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}

Visualization of Tangent in a Triangle

The

\tan

ratio compares the opposite side to the adjacent side.

For angle $\theta$ in a right triangle:

The opposite side is the side facing the angle $\theta$ .
The adjacent side is the side next to the angle $\theta$ other than the hypotenuse.
Tangent is calculated by dividing the length of the opposite side by the adjacent side

Examples of Tangent Values

Tangent values for special angles can be written exactly, with decimals shown only as approximations:

$\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$
$\tan 45^\circ = 1$
$\tan 60^\circ = \sqrt{3} \approx 1.732$

These exact values come from side ratios in right triangles with angles of $30^\circ$ , $45^\circ$ , or $60^\circ$ .

Calculating Tangent Values

A tangent value follows from the two legs of the right triangle, so the example below reads the opposite side and the adjacent side from the drawing and then divides them.

Example of Calculating Tangent

Consider a right triangle with a $30^\circ$ angle, opposite side $15 \text{ cm}$ , adjacent side $15\sqrt{3} \text{ cm}$ , and hypotenuse $30 \text{ cm}$ .

Example Triangle with

30^\circ

Angle

Ratio of opposite side (

15 \text{ cm}

) to adjacent side (

15\sqrt{3} \text{ cm}

The tangent value of this angle is:

\begin{aligned}\tan 30^\circ &= \frac{\text{opposite side}}{\text{adjacent side}} \\ &= \frac{15}{15\sqrt{3}} \\ &= \frac{1}{\sqrt{3}} \\ &= \frac{\sqrt{3}}{3} \approx 0.577\end{aligned}

Measuring Slopes and Heights

Tangent relates a slope angle to the vertical and horizontal changes in a right triangle. This relationship can be used to:

Calculate an object's height from its angle of elevation and horizontal distance
Calculate the slope of a road or staircase from its vertical rise and horizontal run
Calculate a roof angle from the height and width of its cross-section

Practice Exercise

Consider the following right-angled triangle. The acute angle is $\theta$ , the opposite side is $15 \text{ cm}$ , and the adjacent side is $7 \text{ cm}$ .

Triangle with Angle

\theta

Triangle with opposite side

15 \text{ cm}

and adjacent side

7 \text{ cm}

Find $\tan \theta$ and explain which two sides determine the ratio.

Worked Solution

Relative to $\theta$ , the side of length $15 \text{ cm}$ is the opposite side and the side of length $7 \text{ cm}$ is the adjacent side. Tangent compares exactly these two sides:

\begin{aligned} &\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \\ &= \frac{15 \text{ cm}}{7 \text{ cm}} \\ &= \frac{15}{7} \approx 2.14 \end{aligned}

The centimetre units cancel because this is a ratio. Therefore, $\tan \theta \approx 2.14$ . The value is greater than $1$ , which is reasonable because the opposite side is longer than the adjacent side.

As a consistency check, $\theta=\arctan(15/7)\approx65.0^\circ$ , matching the acute angle shown in the diagram.