Grade 12 Trigonometry — How to Identify the Question Type
Grade 12 trigonometry combines the Grade 11 topics (reduction formulae, identities, equations, sine/cosine rules) with compound and double angle identities. The instruction word and the angle expression in the question tell you which technique is needed.
Type 1: Compound and double angle identities
Trigger words: "Prove", "Simplify", an expression containing sin(A ± B), cos(A ± B), sin 2A, cos 2A
Trigger structure: The expression contains a compound angle like cos 75° = cos(45° + 30°), or a double angle like sin 2x = 2 sin x cos x.
Do not confuse with: Reduction formulae — those handle angles like sin(180° − x); compound identities handle sums and differences of two angles.
Method (no numbers — just the steps)
- Recognise the compound or double angle form
- Expand using the correct identity: sin(A + B) = sin A cos B + cos A sin B etc.
- Simplify using Pythagorean, co-function, or reduction identities as needed
See the progression — same type, increasing difficulty
Type 2: Solving trigonometric equations (general solutions)
Trigger words: "Solve for x", "General solution", "for x ∈ [0° ; 360°]"
Trigger structure: A trig equation is given. You must find all angles satisfying it, either in a given interval or as a general solution.
Do not confuse with: Proving an identity — if the question says 'prove' or 'show that', you may not solve; instead, simplify one side to match the other.
Method (no numbers — just the steps)
- Isolate the trig ratio
- Find the reference angle (always positive, acute) using the inverse function
- Determine the quadrant(s) where the sign is correct
- Write specific solutions for the given interval OR general solution: x = ref + 360°k or x = (180° − ref) + 360°k etc.
See the progression — same type, increasing difficulty
Type 3: Proving trigonometric identities
Trigger words: "Prove that", "Show that LHS = RHS"
Trigger structure: An equation is given with a complex trig expression on one or both sides. You must verify it is always true.
Do not confuse with: Solving — you cannot cross-multiply or move terms across the equals sign when proving an identity.
Method (no numbers — just the steps)
- Work on ONE side only (usually the more complex side)
- Use known identities to rewrite: sin²x + cos²x = 1, tan x = sin x/cos x, double angle formulas
- Simplify until you reach the other side exactly
- Never assume the result — do not write what you are trying to prove at any intermediate step
See the progression — same type, increasing difficulty
Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.