Multiple Angles

Video lesson · from 1:00:581 worked example

One lesson video covers all of Trigonometry, so it opens at 1:00:58 for this topic — not from the beginning.

Theory

The wave function method works exactly the same if there are multiple angles inside the trigonometric functions, as long as both terms have the same multiple angle.

⚠️ Common Examiner Traps

The method is unchanged: treat $2x$ exactly as you would treat $x$ — compare coefficients and find $k$ and $\alpha$ in the usual way.
$\alpha$ belongs to the whole bracket: the answer is $k\cos(2x - \alpha)$ , not $k\cos 2(x - \alpha)$ , unless you deliberately factorise.
Widen the domain when solving: if you go on to solve an equation, the multiple angle needs the extended range just as it does elsewhere.
The period changes, the amplitude does not: $k$ is still $\sqrt{a^2+b^2}$ .

Worked examples

Example 1

Write $5\cos 2x^\circ + 12\sin 2x^\circ$ in the form $k\sin(2x + \alpha)^\circ$ where $k > 0$ and $0 \leq \alpha^\circ \leq 360$ .

Expand $k\sin(2x + \alpha)$ and equate coefficients:

k\sin(2x + \alpha)^\circ = k\sin 2x^\circ\cos\alpha^\circ + k\cos 2x^\circ\sin\alpha^\circ

Rearrange given: $12\sin 2x^\circ + 5\cos 2x^\circ$

\begin{aligned} k\cos\alpha &= 12 \\ k\sin\alpha &= 5 \end{aligned}

Find $k$ :

k = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = 13

Find $\alpha$ :

\tan\alpha = \frac{5}{12}

Base angle: $\tan^{-1}\left(\frac{5}{12}\right) \approx 22.6^\circ$ . Q1.

Final expression:

13\sin(2x + 22.6)^\circ