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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 $\beta$.

The comparison clearly indicates that the tangent function is defined as being the ratio obtained when we divide the length of the side opposite the mentioned reference angle by the length of the adjacent side.

Here:

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

Thus,

${\displaystyle \tan \alpha ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

Similarly,

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

Thus,

${\displaystyle \tan \beta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

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

How to remember the formula of the tangent function?

We created the SOH – CAH – TOA table in previous lessons to remember the formulae of the trigonometric functions. You must memorize the third portion — TOA — of the code-word SOH – CAH – TOA to remember the formula of the tangent function.

Here is the table:

SOH

CAH

TOA

Sine

Cosine

Tangent

Opposite by Hypotenuse

Adjacent by Hypotenuse

Opposite by Adjacent

${\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}$

${\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}$

${\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

                                                                                     

                                                                       YOU ARE HERE!

Example $1$

Given a right-angled triangle with the reference angle $\alpha$. What is the tangent of angle $\alpha$?

Solution:

Looking at the diagram, it is clear that the length of the side $12$ is the adjacent side that lies right next to the reference angle $\alpha$. The side of length $5$ is the opposite side that lies exactly opposite the reference angle $\alpha$, and the side of length $13$ is the hypotenuse. Thus,

Adjacent = $12$

Opposite = $5$

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

${\displaystyle \tan \alpha ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

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

${\displaystyle \tan \alpha ={\frac {5}{12}}}$

Example 2

Given a right-angled triangle with the reference angle $\alpha$. What is the tangent of angle $\alpha$?

Solution:

Looking at the diagram, it is clear that the side of length $20$ is the opposite side that lies exactly opposite the reference angle $\alpha$. Also, the length of the side $29$ is the hypotenuse.

We have to determine the tangent of angle $\alpha$. All right, here comes the tricky part.

We know that the tangent function is the ratio of the opposite side to the adjacent side, but the length of the adjacent side is missing. What should we do?

Step 1: Determine the unknown but relevant side — the adjacent.

To determine the adjacent side, we need to use the Pythagoras theorem,

$c^{2}=a^{2}+b^{2}$

looking at the diagram, we have:

Opposite $a = 20$

Hypotenuse $c = 29$

Adjacent $b =$?

Substitute $a = 20$ and $c = 29$ in the formula

$29^{2}=20^{2}+b^{2}$

$841=400+b^{2}$

$b^{2}=441$

$b = 21$ units

Thus, the length of the adjacent is $21$ units.

Step 2: Determine the tangent of angle $\alpha$.

Now, we have:

Adjacent $= 21$

opposite $= 20$

Using the formula of the tangent function

${\displaystyle \tan \alpha ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

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

${\displaystyle \tan \alpha ={\frac {20}{21}}}$

Example 3

Given a right-angled triangle with the reference angle $\alpha$. Which option represents the trigonometric ratio of ${\frac {7}{24}}$?

       a) $\sin \alpha$

       b) $\cos \alpha$

       c) $\tan \alpha$

       d) $\cot \alpha$

Solution:

Looking at the diagram, it is clear that the side of length $7$ is the opposite side that lies exactly opposite the reference angle $\alpha$, and the side of length $24$ is the adjacent side that lies right next to the reference angle $\alpha$.

Thus,

Opposite = $7$

Adjacent = $24$

We know that formula of the tangent function is

${\displaystyle \tan \alpha ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

Thus,

${\displaystyle \tan \alpha ={\frac {7}{24}}}$

Therefore, option c) is the true choice.

Example 4

A right-angled triangle is shown in the figure.

       a) What is the missing side here from the perspective of angle $\beta$

       b) Determine the tangent of angle $\beta$

Solution:

Part a) Determining the missing side from the perspective of angle $\beta$

The diagram clearly indicates that $15$ the opposite side that lies exactly opposite the reference angle $\beta$, and $17$ is the hypotenuse.

Thus, the adjacent side is missing from the perspective of the reference angle \beta$.

Let $x$ be the missing adjacent side.

Using the Pythagorean theorem to solve the missing side.

$c^{2}=a^{2}+b^{2}$

looking at the diagram, we have:

Opposite $b = 15$

Hypotenuse $c = 17$

Adjacent $a =$?

Substitute $b = 15$ and $c = 17$ in the formula

$17^{2}=a^{2}+15^{2}$

$289=a^{2}+225$

$a^{2}=64$

$a = 8$ units

Thus, the length of the adjacent is $8$ units.

Part b) Determining the tangent of angle $\beta$

Now, we have:

Adjacent = $8$

Opposite = $15$

We know that formula of the tangent function is

${\displaystyle \tan \alpha ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}$

substitute adjacent = $8$, opposite = $15$ in the formula

Therefore, the tangent of angle $\beta$ is

${\displaystyle \tan \beta ={\frac {15}{8}}}$

 

Practice Questions

1. Given the right triangle $LMN$ with respect to the reference angle $L$, what is the tangent of angle $L$?

2. Given the right triangle $PQR$ with respect to the reference angle $P$, which option represents the trigonometric ratio of $\dfrac {QR}{PR}$?

3. Given a right-angled triangle with the reference angle $\alpha$, what is the tangent of angle $\alpha$?

4. Given a right-angled triangle with the reference angle $\beta$. What is the value of $\tan \alpha$?

5. Given a right-angled triangle with the reference angle $\beta$. What is the value of $\tan \beta$?


 

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