Angle transformation

$$ 1. \ \sin(A + B) = \sin A \cos B + \cos A \sin B $$

$$ 2. \ \sin(A − B) = \sin A \cos B − \cos A \sin B $$

$$ 3. \ \cos(A + B) = \cos A \cos B − \sin A \sin B $$

$$ 4. \ \cos(A − B) = \cos A \cos B + \sin A \sin B $$

$$ 5. \ \tan(A + B) = \frac {(\tan A + \tan B)}{(1 − \tan A \tan B)} $$

$$ 6. \ \tan(A − B) = \frac {(\tan A − \tan B)}{(1 + \tan A \tan B)} $$

$$ 6. \ \cot (A + B) = \frac {(\cot A.\cot B – 1)}{(\cot B + \cot A)} $$

$$ 6. \ \cot (A - B) = \frac {(\cot A.\cot B + 1)}{(\cot B - \cot A)} $$

Example:

$$ \text{Find value of} \sin 15^\circ $$

Solution:

$$ \sin 15^\circ= \sin (45^\circ - 30^\circ) $$

$$ = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ $$

$$ = \frac {1}{\sqrt {2}} \times \frac {\sqrt {3}}{2} - \frac {1}{\sqrt {2}} \times \frac {1}{2} $$

$$ = \frac {\sqrt {3}}{2 \sqrt {2}} - \frac {1}{2 \sqrt {2}} $$

$$ = \frac {\sqrt {3}-1}{2 \sqrt {2}} $$

Example:

$$ \text{Find value of} \sin 75^\circ $$

Solution:

$$ \sin 15^\circ= \sin (45^\circ + 30^\circ) $$

$$ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ $$

$$ = \frac {1}{\sqrt {2}} \times \frac {\sqrt {3}}{2} + \frac {1}{\sqrt {2}} \times \frac {1}{2} $$

$$ = \frac {\sqrt {3}}{2 \sqrt {2}} + \frac {1}{2 \sqrt {2}} $$

$$ = \frac {\sqrt {3}+1}{2 \sqrt {2}} $$