Cosine: in a Right Triangle

How to find cosine in a right triangle (trigonometry): formula, 3 examples, and their solutions.

Formula

Formula

Cosine is the ratio of
[Adjacent side]/[Hypotenuse]
in a right triangle.

The adjacent side means
the side adjancent to ∠A
(which is not the hypotenuse).

To remember the ratio,
remember CAH:
Cosine, Adjacent side, and Hypotenuse.

Example 1

Example

Solution

Cosine is CAH:
Cosine, Adjacent side, and Hypotenuse.

The Adjacent side is 4.
The Hypotenuse is 5.

So,
C, cos A
is equal to,
A: adjacent side, 4
over,
H: hypotenuse, 5.

So cos A = 4/5.

Example 2

Example

Solution

Cosine is CAH:
Cosine, Adjacent side, and Hypotenuse.

The Adjacent side is 5.
The Hypotenuse is 13.

So,
C, cos A
is equal to,
A: adjacent side, 5
over,
H: hypotenuse, 13.

So cos A = 5/13.

Example 3

Example

Solution

First, find cos A
from the given right triangle.

Cosine is CAH:
Cosine, Adjacent side, and Hypotenuse.

The Adjacent side is 6.
The Hypotenuse is x.

So,
C, cos A
is equal to,
A: adjacent side, 6
over,
H: hypotenuse, x.

Next, it says
cos A = 3/5.

So write
[ = 3/5].

So cos A = 6/x = 3/5.

Solve 6/x = 3/5.

Then 3x = 30.

Divide both sides by 10.
Then x = 10.

So x = 10.