Properties of triangles

The sine, cosine, and tangent rules are used to solve problems involving triangles.
The sine rule relates the length of a side of a triangle to the sine of the opposite angle.
The cosine rule relates the length of a side of a triangle to the lengths of the other two sides and the cosine of the included angle.
The tangent rule relates the length of a side of a triangle to the tangent of the opposite angle and the lengths of the other two sides.
The projection rule, also known as the law of sines, can be used to find the height of a triangle given the length of the opposite side and the angle opposite to it.

Sine rule: \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \)
Cosine rule: \( a^2=b^2+c^2-2bc\cos A \)
Tangent rule: \( \frac{a-b}{a+b}=\frac{\tan\frac{1}{2}(A-B)}{\tan\frac{1}{2}(A+B)} \)
Projection rule: \( a=b\cos C+c\cos B \)