Law of Sines
How to solve the law of sines problems: formula, proof, examples, and their solutions.
a/(sin A) = b/(sin B) = c/(sin C)
a, b, c: Sides of a triangle
A, B, C: Angles of a triangle
The area of a triangle
can be written in three ways:
(by choosing the different angles)
(area) = (1/2)⋅bc sin A
= (1/2)⋅ac sin B
= (1/2)⋅ab sin C
Area of the triangle (using sine)
Divide each side by (1/2)⋅abc.
Then the dark gray factors will be cancelled.
So (sin A)/a = (sin B)/b = (sin C)/c.
Switch the numerators and the denominators.
Then a/(sin A) = b/(sin B) = c/(sin C).
The law of sines is used when
[1 side, 2 opposite angles] → [other opposite side]
[2 sides, 1 opposite angle] → [other opposite angle].
([Given] → [Find])
Sides: x, 12
Opposite angles: 45º, 60º
x/(sin 45º) = 12/(sin 60º)
To use the law of sines,
find the measure of the brown angle.
(= the angle opposite to the known side: 4)
m∠(brown) = 45º.
Interior angles of a triangle
Sides: x, 4
Opposite angles: 30º, 45º
x/(sin 30º) = 4/(sin 45º)