# Tangent – Explanation & Examples

In the context of a right triangle, we can simply define the tangent function or any other trigonometric function using the terms hypotenuse, opposite, and adjacent in a right-angled triangle. Sounds interesting? Yes, it is. But, how can we define the tangent function using a right-angled triangle?

The tangent function is defined by determining the ratio of the length of the side opposite a reference angle (acute angle) of a right triangle to the length of the adjacent side of a right triangle.

After studying this lesson, we are expected to learn the concepts driven by these questions and be qualified to address accurate, specific, and consistent answers to these questions.

• What is a tangent function?
• How can we determine the formula for tangent function from a right-angled triangle?
• How can we solve actual problems using trigonometric functions?

This lesson aims to clear up any confusion you might have about the concepts involving the tangent function.

## What is tangent?

In the context of a triangle, the tangent function is the ratio of the opposite side to the adjacent side. For an angle $\alpha$, the tangent function is denoted by $\tan \alpha$. In other words, the tangent is a trigonometric function of any given angle.

The following figure 5-1 represents a typical right triangle. The lengths of the three legs (sides) of the right triangle are named $a$, $b$, and $c$. The angles opposite the legs of lengths $a$, $b$, and $c$ are named $\alpha$, $\beta$, and $\gamma$. The tiny square with the angle $\gamma$ shows that it is a right angle.

Use the diagram in Figure 5-1 to determine the tangent function from the perspective of the angle $\alpha$.

Looking at Figure 5-1, we can determine the tangent function from the right-angled triangle if we divide the length of the side opposite the reference angle $\alpha$ (acute angle) by the length of the adjacent side.

The following figure 5-2 represents a tangent function.

Looking at Figure 5-2, we can identify that the side of length $a$ is the opposite side that lies exactly opposite the reference angle $\alpha$, and the side of length $b$ is the adjacent side that lies right next to the reference angle $\alpha$. Thus,

Opposite = $a$

Adjacent = $b$

Therefore, the tangent of an angle $\alpha$ is

${\displaystyle \tan \alpha ={\frac {a}{b}}}$

Therefore, we conclude that the tangent function is the ratio of the opposite side to the adjacent side.

Tangent function from the perspective of the angle $\beta$

We should be cautious when we apply the terms opposite and adjacent because the meaning of these terms is dependent on the reference angle we are using.

The following figure 5-3 represents a typical right triangle from the perspective of the angle $\beta$.

You can observe that now the roles of the sides have been shifted.

Looking at Figure 5-3, it is clear now the length of the side $a$ is right next to the reference angle $\beta$, and the length of the side $b$  lies exactly opposite the reference angle $\beta$. Thus, in relation to the angle measuring $\beta$, now we have

Adjacent = $a$

Opposite = $b$

While the hypotenuse $c$ remains the same. This is why the hypotenuse is very special in a right-angled triangle.

The following figure 5-4 represents a tangent function from the perspective of the angle $\beta$.

Looking at Figure 5-4, we can identify that the side of length $b$ is the opposite side that lies exactly opposite the reference angle $\beta$, and the length of the side $a$  lies right next to the reference angle $\beta$. Thus,

Opposite = $b$

Adjacent = $a$

We know that the tangent function is the ratio of the opposite side to the adjacent side.

Therefore, the tangent of an angle $\beta$ is

${\displaystyle \tan \beta ={\frac {b}{a}}}$

## What is the formula for the tangent?

The following figure 5-5 illustrates a clear comparison of how we determined the ratios of tangent function from the perspective of both the angles$\alpha$ and $\beta$.