# Sine

### Calculating Sine – ACT Math

You will need to know how to use sine in trigonometry problems on the ACT math test.

When expressed in the form of a fraction, sine is calculated as “opposite over hypotenuse.”

The length of the opposite side goes in the numerator and the length of the hypotenuse goes in the denominator.

$\text{sine = } \dfrac{opposite}{hypotenuse}$

### Example 1: Sine of Angle a

As in our illustration of cosine, side Y is the hypotenuse in our triangle since side Y is on the other side of the right angle.

The side that is opposite to angle a is side Z because side Z is directly across from angle a.

So, the sine of angle a in our illustration above is calculated as follows:

$\text{sine a = } \dfrac{Z}{Y}$

### Example 2: Cosine of Angle b

Remember that if you measure a different angle, the opposite side will also be different.

Let’s look at the sine of angle b in our example triangle.

In the illustration below, the opposite side is now side X because it is opposite to angle b.

So, the formula for sin of angle b is:

$\text{sine b = } \dfrac{X}{Y}$

Example Problem:

Now try this problem in order to practice calculating sine.

If x represents a real number, what is the greatest possible value of:

4 × sin 2x

A. 2

B. 3

C. 4

D. 6

E. 12

Remember that the greatest possible value of sine is 1.

Therefore, sin 2x must be less than or equal to 1.

This concept is represented by the following formula:

sin 2x = 1

Now, multiply each side of the equation by 4 in order to get 4 × sin 2x.

sin 2x = 1

4 × sin 2x = 1 × 4

4 × sin 2x = 4

So, the greatest possible value is 4.