Question

When do we use the addition and subtraction formulas for sine or cosine?

Answer 1

When we need the cosine of the sum of two angles, we use the addition formula for cosine.

When we need the sine of the difference of two angles, we use the difference formula for sine.

Let's try one.  Say we needed to know

 $\cos 15^\circ$

We're only expected to know the trig functions of 30/60/90 and 45/45/90 triangles.  Fortunately, $15^\circ = 45^\circ - 30^\circ$ so

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

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

$= \dfrac{\sqrt{2}}{2} \cdot \dfrac{ \sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2}$

$\cos 15^\circ = \frac{1}{4} (\sqrt{6} + \sqrt{2})$