### 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.

### 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:

### 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:

### Cosine Rules

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

sin^{2} + cos^{2} = 1

cos^{2} = 1 − sin^{2}

sin^{2} = 1 − cos^{2}

**Sample Question:**

Consider the laws of sines and cosines. Sin^{2} *A* = ?

A. 1 − cos^{2} *A*

B. cos^{2} *A* − 1

C. tan^{2} *A*

D. 1 − tan^{2} *A*

E. tan^{2} *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, sin^{2 }of any angle is always equal to 1 – cos^{2} of that angle.

Now try the exercises for **sine** and **tangent**.

Go back to the **Trigonometry Equations Page**.

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