Trig Identities and Equations: How to Score in 0606

Trigonometry questions in 0606 cluster into three types, identity proofs, equation solving, and the R-formula, and each has a fixed routine that converts study time directly into marks. The underlying content is in our trigonometry topic notes; this page is the exam technique.

Type 1. Proving identities

The command is usually “show that” or “prove”, meaning every step must be visible and justified (command words decoded). The routine:

1. Choose the messier side. Simplifying complexity is mechanical; inventing complexity needs inspiration.
2. Convert to $\sin$ and $\cos$ when no path is obvious, $\sec$, $\csc$, $\cot$ and $\tan$ all reduce. This single move resolves most 0606 proofs.
3. Use the Pythagorean identities, $\sin^2\theta + \cos^2\theta = 1$ and its two divided forms ($1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$). The skill is spotting rearrangements: $\cos^2\theta = 1 - \sin^2\theta$ appearing inside a fraction is the syllabus’s favourite trick.
4. Never move terms across the equals sign. An identity proof transforms one side into the other; treating it like an equation scores zero method marks even with a correct-looking final line.

Write each transformation on its own line. The mark scheme awards per-step, cramming three manipulations into one line risks all of them.

Type 2. Solving trig equations

The mark-killer here is missing solutions. The routine that prevents it:

1. Check the range and its units first. $0 \le x \le 360°$ and $0 \le x \le 2\pi$ demand different mode discipline; radian ranges are common in 0606 (see circular measure).
2. Rearrange to a single trig function $=$ constant, factorising if quadratic in $\sin$/$\cos$/$\tan$. Never divide both sides by $\cos\theta$, dividing deletes the $\cos\theta = 0$ family of solutions; factorise instead: $\sin\theta\cos\theta = \cos\theta \Rightarrow \cos\theta(\sin\theta - 1) = 0$.
3. Find the reference angle, then sweep all four quadrants using the CAST/unit-circle picture to collect every solution in range.
4. Watch compound arguments. For $\sin(2x - 30°)$ with $0 \le x \le 180°$, the argument runs over a doubled range, solve for the argument across its full range first, then convert back to $x$.

On Paper 1, solutions come from the exact-value table, instant recall of $\sin$/$\cos$/$\tan$ at $0°$, $30°$, $45°$, $60°$, $90°$ is assumed.

Type 3. The R-formula

“Express $a\sin\theta + b\cos\theta$ in the form $R\sin(\theta + \alpha)$” is among the most predictable questions in the syllabus, and it’s a gift once drilled: $R = \sqrt{a^2 + b^2}$, $\tan\alpha = \frac{b}{a}$ (with $\alpha$ placed in the correct quadrant). The follow-ups are always one of: solve the equation (now a single-function equation. Type 2), or state the maximum/minimum ($\pm R$) and where it occurs. Full derivation and practice in the trig notes, formulas on the memorise list.

Making it stick

Trig technique decays fast without use. Two identity proofs, two equations and one R-formula question per week from past papers, marked against real schemes, keeps the routines warm through to exam day. The 8-week revision plan schedules exactly this.

If trig is the topic that’s been costing your grade, it responds quickly to guided practice: in 1-to-1 online classes, Teacher Rig diagnoses which of the three types leaks marks and drills that one. The first hour is a free trial, message us on WhatsApp.