Solving Equations

Video lesson · from 1:11:512 worked examples

One lesson video covers all of Trigonometry, so it opens at 1:11:51 for this topic — not from the beginning.

Theory

A composite trigonometric equation can be solved using the wave function format. Once compiled as $k\cos(x - \alpha) = c$ , it acts like solving standard basic trigonometric equations.

⚠️ Common Examiner Traps

Convert first, then solve: you cannot solve $a\sin x + b\cos x = c$ directly. Rewrite it as a single wave, then treat it as a basic equation.
Adjust the domain for the bracket: if you are solving for $x - \alpha$ , shift the domain by $\alpha$ before finding solutions, then shift back.
Give every solution: there are normally two in a revolution. Use CAST on the bracket, not on $x$ .
Check for no solutions: if $c$ is larger than $k$ , there are none — say so rather than forcing an answer.

Worked examples

Example 1

Solve $5\cos x^\circ + \sin x^\circ = 2$ for $0 \leq x^\circ \leq 360$ .

Write $5\cos x^\circ + \sin x^\circ$ using $k\cos(x - \alpha)^\circ$ .

\begin{aligned} k\cos\alpha &= 5 \\ k\sin\alpha &= 1 \end{aligned}

k = \sqrt{5^2 + 1^2} = \sqrt{26}

\tan\alpha = \frac{1}{5} \implies \alpha \approx 11.3^\circ

So the equation becomes:

\begin{aligned} \sqrt{26}\cos(x - 11.3)^\circ &= 2 \\ \cos(x - 11.3)^\circ &= \frac{2}{\sqrt{26}} \end{aligned}

Base angle: $\cos^{-1}\left(\frac{2}{\sqrt{26}}\right) \approx 66.9^\circ$ . Q1, Q4 (Cosine is positive).

\begin{aligned} x - 11.3 &= 66.9 \\ x &= 78.2^\circ \\ \\ x - 11.3 &= 360 - 66.9 = 293.1 \\ x &= 304.4^\circ \end{aligned}

Example 2

Solve $2\cos 2x + \sin 2x = 1$ for $0 \leq x \leq 2\pi$ .

Write $2\cos 2x + \sin 2x$ as $k\cos(2x - \alpha)$ .

\begin{aligned} k\cos\alpha &= 2 \\ k\sin\alpha &= 1 \end{aligned}

k = \sqrt{2^2 + 1^2} = \sqrt{5}

\tan\alpha = \frac{1}{2} \implies \alpha \approx 0.464 \text{ rad}

So the equation becomes:

\begin{aligned} \sqrt{5}\cos(2x - 0.464) &= 1 \\ \cos(2x - 0.464) &= \frac{1}{\sqrt{5}} \end{aligned}

Since Domain $0 \leq x \leq 2\pi$ , then $2x$ goes up to $4\pi$ (two cycles).

Base angle: $\cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \approx 1.107$ rad. Q1, Q4.

\begin{aligned} 2x - 0.464 &= 1.107, 2\pi - 1.107 \\ &= 1.107, 5.176 \\ 2x - 0.464 &= 1.107 + 2\pi, \quad 5.176 + 2\pi \\ &= 7.390, 11.459 \\ \\ 2x &= 1.571 , 5.64 , 7.854 , 11.923 \\ x &= 0.785 (\text{or } \frac{\pi}{4}), 2.82, 3.927 (\text{or } \frac{5\pi}{4}), 5.96 \end{aligned}