Using the double angle formulae - Solving trigonometric equations - Higher Maths Revision

Trigonometric Equations using the double angle formulae

You can revise your knowledge of double angle formulae as part of Expressions and Functions.

Solve the equation \(5\sin 2x^\circ + 7\cos x^\circ = 0\), for \(0^\circ \le x^\circ \le 360^\circ\).

\(5\sin 2x^\circ + 7\cos x^\circ = 0\)

Replace \(\sin 2x^\circ\) with \(2\sin x^\circ \cos x^\circ\)

\(5(2\sin x^\circ \cos x^\circ ) + 7\cos x^\circ = 0\)

Multiply out the brackets:

\(10\sin x^\circ \cos x^\circ + 7\cos x^\circ = 0\)

Take out \(\cos x^\circ\) as the common factor.

\(\cos x^\circ (10\sin x^\circ + 7) = 0\)

Two possible solutions are:

\(\cos x^\circ = 0\)

\(10\sin x^\circ + 7 = 0\)

Solve each equation in turn:

\(\cos x^\circ = 0\)

\(x^\circ = 90^\circ\) or \(270^\circ\)

And:

\(10\sin x^\circ + 7 = 0\)

\(10\sin x^\circ = - 7\)

\(\sin x^\circ = - \frac{7}{{10}}\)

\(x^\circ = 224.4^\circ\) or \(315.6^\circ\)

Which gives solutions of \(90^\circ ,\,224.4^\circ ,\,270^\circ ,\,315.6^\circ\)