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:
1. By writing \(75^\circ = 45^\circ + 30^\circ\) determine the exact value of \(\sin 75^\circ\)
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. Find the exact value of \(\cos {\frac{{7\pi }}{{12}}}\)
2. Since \(\frac{{7\pi }}{{12}} = \frac{\pi }{3} + \frac{\pi }{4}\) then:
\(\cos {\frac{{7\pi }}{{12}}} = \cos \left( {\frac{\pi }{3} + \frac{\pi }{4}} \right)\)
Using the formula for \(\cos (A + B)\)
\(= \cos \frac{\pi }{3}\cos \frac{\pi }{4} - \sin \frac{\pi }{3}\sin \frac{\pi }{4}\)
\(= \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}\)
\(= \frac{{1 - \sqrt 3 }}{{2\sqrt 2 }}\)