# Sine

### Calculating Sine

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

The correct answer is C.

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.

Click here for more trigonometry exercises and solutions.

In the trigonometry part of our website, you will find explanations of the trigonometric functions and formulas for sine, cos, and tan.

Click here for the formulas for cosine and tangent.