When we add or subtract angles, the result is called a compound angle. For example, $$30^\circ + 120^\circ$$ is a compound angle. Using a calculator, we find:

$\sin (30^\circ + 120^\circ ) = \sin (210^\circ ) = - 0.5$

$\sin (30^\circ ) + \sin (120^\circ ) = 1.366\,(to\,3\,d.p.)$

This shows that $$\sin (A + B)$$ is not equal to $$\sin A + \sin B$$. Instead, we can use the following identities:

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

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

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

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

• Addition formulae are given in a condensed form:
• $\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B$
• $\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B$

These formulae are used to expand trigonometric functions to help us simplify or evaluate trigonometric expressions of this form.

See how we approach this two-part question:

Question

1. By writing $$75^\circ = 45^\circ + 30^\circ$$ determine the exact value of $$\sin 75^\circ$$

2. Find the exact value of $$\cos \left( {\frac{{7\pi }}{{12}}} \right)$$

1. $$\sin 75^\circ = \sin (45 + 30)^\circ$$

Using the formula for $$\sin (A + B)$$

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

Using exact values that you should know:

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

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

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

2. Since $$\frac{{7\pi }}{{12}} = \frac{\pi }{3} + \frac{\pi }{4}$$ then:

$\cos \left( {\frac{{7\pi }}{{12}}} \right) = \cos \left( {\frac{\pi }{3} + \frac{\pi }{4}} \right)$

Using the formula for $$\cos (A + B)$$

$= \cos \frac{\pi }{3}\cos \frac{\pi }{4} - \sin \frac{\pi }{3}\sin \frac{\pi }{4}$

Using exact values that you should know:

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

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