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:

\[= \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}\]

\[= \frac{{1 - \sqrt 3 }}{{2\sqrt 2 }}\]