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 $\beta$.