Sine: in a Right Triangle

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

Formula

Formula

Sine is the ratio of
[Opposite side]/[Hypotenuse]
in a right triangle.

The opposite side means
the side opposite to ∠A.

To remember the ratio,
remember SOH:
Sine, Opposite side, and Hypotenuse.

Example 1

Example

Solution

Sine is SOH:
Sine, Opposite side, and Hypotenuse.

The Opposite side is 3.
The Hypotenuse is 5.

So,
S, sin A
is equal to,
O: opposite side, 3
over,
H: hypotenuse, 5.

So sin A = 3/5.

Example 2

Example

Solution

Sine is SOH:
Sine, Opposite side, and Hypotenuse.

The Opposite side is 12.
The Hypotenuse is 13.

So,
S, sin A
is equal to,
O: opposite side, 12
over,
H: hypotenuse, 13.

So sin A = 12/13.

Example 3

Example

Solution

First, find sin A
from the given right triangle.

Sine is SOH:
Sine, Opposite side, and Hypotenuse.

The Opposite side is x.
The Hypotenuse is 10.

So,
S, sin A
is equal to,
O: opposite side, x
over,
H: hypotenuse, 10.

Next, it says
sin A = 4/5.

So write
[ = 4/5].

So sin A = x/10 = 4/5.

Solve x/10 = 4/5.

Multiply 10 to both sides.
Then x = [4/5]⋅10.

Cancel the denominator 5
and reduce the numerator 10 to, 10/5, 2.

4⋅2 = 8

So x = 8.