# Cosine

### How to Calculate Cosine

Cosine can be calculated as a fraction, expressed as “adjacent over hypotenuse.”

The length of the adjacent side is in the numerator and the length of the hypotenuse is in the denominator.

$\text{cosine = } \dfrac{adjacent}{hypotenuse}$

### Cosine of Angle a – Example 1

In the illustration below , side Y is the hypotenuse since it is on the other side of the right angle.

The adjacent side is side X because it is next to angle a.

So, the formula for cos of angle a is:

$\text{cosine a = } \dfrac{X}{Y}$

### Cosine of Angle b – Example 2

If you change the angle that you are measuring, the adjacent side will be different.

Suppose we want to measure the cosine of the other angle (angle b) in our example triangle.

In the illustration below, the adjacent side is now side Z because it is next to angle b.

So, the formula for cos of angle b is:

$\text{cosine b = } \dfrac{Z}{Y}$

### Cosine Rules

Remember the following useful trigonometric formulas. They are valid with respect to any angle:

sin2 + cos2 = 1

cos2 = 1 − sin2

sin2 = 1 − cos2

Sample Question:

Consider the laws of sines and cosines.  Sin2 A = ?

A. 1 − cos2 A

B. cos2 A − 1

C. tan2 A

D. 1 − tan2 A

E. tan2 A − 1

The correct answer is A.

Hopefully, you will have committed the above rules to memory before attempting the problem above.

According to the rules above,  sin2  of any angle is always equal to 1 – cos2 of that angle.

Now try the exercises for sine and tangent.

Go back to the Trigonometry Equations Page.

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