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:

\(\sin (30^\circ + 120^\circ ) = \sin 150^\circ = 0.5\)

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

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

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

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

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

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

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 \(\sin 75^\circ\)

Question

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

��ý

Trigonometric expressionsAddition formulae