Trigonometry — common mistakes

A very common exam question involves an equation with both $\cos(2\theta)$ and a term in $\cos\theta$ or $\sin\theta$. This is a strong signal to immediately substitute the appropriate double angle formula for $\cos(2\theta)$ to create a quadratic equation.

When using harmonic form, be meticulous with your signs. If you are expressing $a\sin\theta - b\cos\theta$ as $R\sin(\theta - \alpha)$, the expansion is $R(\sin\theta\cos\alpha - \cos\theta\sin\alpha)$. Comparing coefficients correctly is vital. Always state the values of $R$ and $\alpha$ clearly.

The question will always specify the interval for the solution. If it's given in degrees (e.g., $0^\circ \le x \le 360^\circ$), your answer must be in degrees. If it's given in radians (e.g., $0 \le x \le 2\pi$), your answer must be in radians. Make sure your calculator is in the correct mode!

Look for clues. If an equation has different trig functions (like $\sin\theta$ and $\cot\theta$), use an identity to write everything in terms of one function. If it has different angles (like $2\theta$ and $\theta$), use a double angle formula. If you see a sum like $a\sin\theta + b\cos\theta$, think of the harmonic form.

It simplifies expressions. An expression like $3\sin x + 4\cos x$ is difficult to analyse. By converting it to $5\sin(x+53.1^\circ)$, you can immediately see its amplitude (maximum value) is 5, and you can easily solve equations like $3\sin x + 4\cos x = 2$.

Use the symmetry of the trigonometric graphs or a CAST diagram. For an angle $\alpha$:

• For $\sin x = k$: the other solution is $180^\circ - \alpha$ (or $\pi - \alpha$).
• For $\cos x = k$: the other solution is $360^\circ - \alpha$ (or $2\pi - \alpha$, or simply $-\alpha$).
• For $\tan x = k$: the other solution is $180^\circ + \alpha$ (or $\pi + \alpha$). Then, add or subtract multiples of $360^\circ$ (or $2\pi$) to find all solutions within the required range.