### 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.

### 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:

### 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:

**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 2*x*

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 2*x* must be less than or equal to 1.

This concept is represented by the following formula:

sin 2*x* = 1

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

sin 2*x* = 1

4 × sin 2*x* = 1 × 4

4 × sin 2*x* = 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**.

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