# Trigonometric Functions of Compound Angles

A compound angle is the one, which is formed by algebraic addition of 2 or more angles.

In algebraic addition, we consider the sign of the angle. Thus, $\angle A$ and $\angle B$ give $\angle (A+B)$ and $\angle A$ and $- \angle B$ give $\angle A + \ -B$ i.e. $\angle (A-B)$.

Let $A$ and $B$ be any 2 angles.

• ### Sum and Difference

${sin (A+B) = sin(A)cos(B) \ + \ cos(A)sin(B)}$

${sin (A-B) = sin(A)cos(B) \ - \ cos(A)sin(B)}$

${cos (A+B) = cos(A)cos(B) \ - \ sin(A)sin(B)}$

***${cos (A-B) = cos(A)cos(B) \ + \ sin(A)sin(B)}$

The formula *** is in some sense the base formula for all others. So, if we prove that this formula is true, rest of them can be proved very easily.

${tan (A+B) = \frac {tan(A)+tan(B)}{1- tan(A)tan(B)}}$

${tan (A-B) = \frac {tan(A)-tan(B)}{1+ tan(A)tan(B)}}$

• ### Allied Angles

Whenever the sum/ difference of any 2 angles is either zero or an integral multiple of $\frac {\pi}{2}$, they are known as allied angles. Using the sum and difference formulas and the ratios of standard angles, we can obtain the ratios of allied angles.

Let $\theta$ be any angle. Then $- \theta$, $\frac {\pi}{2} \pm \theta$, $\pi \pm \theta$, $\frac {3 \pi}{2} \pm \theta$, $2 \pi - \theta$ are its allied angles.

• ### Double Angle Formulas

${sin(2A) = 2sin(A)cos(A)}$

${cos(2A) = cos^2(A)- sin^2 (A)}$

${tan(2A) = \frac {2 tan(A)}{1- tan^2(A)}}$

Recall, ${sin^2(A)+cos^2(A) =1}$.

Also, ${sin (2A) = \frac {2 tan (A)}{1+ tan^2 (A)}}$

${cos (2A) = \frac {1 - tan^2 (A)}{1+ tan^2 (A)}}$

• ### Triple Angle Formulas

${sin(3A)= 3sin(A)-4sin^3(A)}$

${cos(3A)= 4cos^3(A)- 3cos (A)}$

${tan(3A)= \frac {3tan(A)-tan^3(A)}{1-3tan^2(A)}}$

• ### Half Angle Formulas

Let $\theta$ be any angle. Then,

${sin(\theta) = 2 sin (\frac \theta 2) cos (\frac \theta 2)}$

${cos(\theta) = cos^2 (\frac \theta 2) - sin^2 (\frac \theta 2)}$

${tan (\theta)= \frac {2 tan (\frac \theta 2)}{1 - tan^2 \frac \theta 2}}$