# Law of Sines

How to use the law of sines to find the side of the given triangle: formula, 2 examples, and their solutions.

## Formula

### Formula

a/(sin A) = b/(sin B) = c/(sin C)

a, b, c: Sides of a triangle
∠A, ∠B, ∠C: Angles of a triangle.

Use the law of sines for the below cases:
1 side, 2 angles → 1 side
2 sides, 1 angle → 1 angle.

## Example 1

### Solution

Sides: x, 12
Opposite angles: 45º, 60º

So x/(sin 45º) = 12/(sin 60º).

Multiply sin 45º to both sides.

To find sin 45º and sin 60º,

draw a 45-45-90 triangle
whose sides are 1, 1, √2

and draw a 30-60-90 triangle
whose sides are 1, √3, 2.

Write 12 and the main fraction bar.

Find sin 45º.

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).

So sin 45º = 1/√2.

Find sin 60º.

Sine is SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).

So sin 60º = √3/2.

So 12⋅[(sin 45º) / (sin 60º)]
= 12⋅[(1/√2) / (√3/2)].

Solve the complex fraction.

The numerator is,
multiply the outer numbers,
1⋅2 = 2.

The denominator is,
multiply the inner numbers,
2⋅√3 = √6.

To rationalize the denominator6,
multiply [√6/√6].

2√6 = 2√6

6⋅√6 = 6

Cancel the denominator 6
and reduce 12 to, 12/6, 2.

2⋅2√6 = 4√6

So x = 4√6.

## Example 2

### Solution

Set the unknown angle θ.

Then θ + 30 + 105 = 180.

Triangle: Interior Angles

30 + 105 = 135

Move +135 to the right side.

Then θ = 45.

So θ = 45º.

See the given triangle.

Sides: x, 4
Opposite angles: 30º, 45º

So x/(sin 30º) = 4/(sin 45º).

Multiply sin 30º to both sides.

To find sin 30º and sin 45º,

draw a 30-60-90 triangle
whose sides are 1, √3, 2,

and draw a 45-45-90 triangle
whose sides are 1, 1, √2.

Write 4 and the main fraction bar.

Find sin 30º.

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (2).

So sin 30º = 1/2.

Find sin 45º.

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).

So sin 45º = 1/√2.

So 4⋅[(sin 30º) / (sin 45º)]
= 4⋅[(1/2) / (1/√2)].

Solve the complex fraction.

The numerator is,
multiply the outer numbers,
1⋅√2 = √2.

The denominator is,
multiply the inner numbers,
2⋅1 = 2.

Cancel the denominator 2
and reduce 4 to, 4/2, 2.

So x = 2√2.