Addition formulae

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

curriculum-key-fact
  • 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 }}