Law of Sines

\[\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\]

Variables

aside opposite angle A (m)

bside opposite angle B (m)

cside opposite angle C (m)

Atriangle angle A (rad or °)

Btriangle angle B (rad or °)

Ctriangle angle C (rad or °)

The Law of Sines is a fundamental trigonometric relationship that connects the sides of a triangle with the sines of their opposite angles.

It states that the ratio between a side length and the sine of its opposite angle is constant for all three sides of a triangle.

This formula is widely used to solve oblique triangles where standard right triangle trigonometry cannot be directly applied.

Use the Law of Sines when solving triangles in cases such as:

- Two angles and one side are known (AAS or ASA)

- Two sides and a non-included angle are known (SSA)

- Determining unknown sides or angles

- Surveying and navigation calculations

It is especially useful for non-right triangles.

If:

A = 30°

B = 45°

a = 10 m

Then:

b = a*sin(B)/sin(A)

b = 10*sin(45°)/sin(30°)

b = 14.14 m

- Trigonometry and geometry

- Civil engineering

- Navigation and surveying

- Astronomy

- Physics and mechanics

- Computer graphics