# Tangent

### Formula for Tangent – ACT Math

This page explains the trigonometric formula for the calculation of tangent for ACT math problems.

Tangent is calculated as “opposite over adjacent.”

So in our fraction, the length of the opposite side is placed in the numerator and the length of the adjacent side is placed in the denominator.

$\text{tangent = } \dfrac{opposite}{adjacent}$

### Tangent of Angle a – Example 1

In the illustration below , side Z is the opposite side since it is on the other side of angle a.

The adjacent side is side X because it is next to angle a and it is not the hypotenuse.

So, the formula for tan of angle a is:

$\text{tan a = } \dfrac{Z}{X}$

### Tangent of Angle b – Example 2

Now let’s look at how to calculate the tangent of the other angle (angle b) in our example triangle.

In the illustration below, side X is the opposite side since it is on the other side of angle b.

The adjacent side is side Z because it is next to angle b and it is not the hypotenuse.

So, the formula for tan of angle b is:

$\text{tan b = } \dfrac{X}{Z}$

Sample question:

If cos A = y/z and sin A = x/z then tan A = ?

A. z/x

B. z/y

C. xy/z

D. x/y

E. y/x

In this problem, sin A = x/z and cos A = Y/z

Using the rules for sine and cosine, we know that:

$\text{sin A = } \dfrac{opposite}{hypotenuse} \text{= } \dfrac{X}{Z}$

So, side X is the opposite side and side Z is the hypotenuse.

We can then conclude that side Y is the adjacent side.

Finally, we can solve for tan:

$\text{tan A = } \dfrac{opposite}{adjacent} \text{= } \dfrac{X}{Y}$

You should also have a look at the exercises for sine and cosine.

More trigonometry exercises and examples can be found in this section of the website.