Trigonometry in A Level Math Paper 3: common mistakes and examiner advice

Paper 3 trigonometry goes beyond Paper 1: addition and double-angle formulae, the R-formula, proofs of identities, equations in 2θ, and calculus with trigonometric functions. The mistakes are mostly slips in formulae and in dealing with a doubled angle. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Quoting a formula incorrectly. Examples include cos(x − 30°) = cos x − cos 30°, an incorrect half-angle formula, or the wrong expansion for tan(α + β). The reports say that if a formula is misquoted and then applied, it is treated as a method error.
• Sign errors and factors. Sign errors in the compound-angle formulae, including or excluding the factors of ½ and 2 in the double-angle identity, and 45° being used instead of 60°.
• Forgetting R. In R sin(θ + α) or R cos(θ − α) questions, some students wrote cos α = 3 and sin α = 1 instead of R cos α and R sin α, then got the right tan α but the wrong R. Others found R and α but then continued to use θ where 2x was needed.
• The doubled angle. In equations such as tan 2θ = k, students solved for 2θ and never halved the solutions, replaced 2θ with θ, or wrote 2(θ − 23.41) in place of 2θ − 23.41.
• Missing solutions. Reports record students who found only some of the four solutions of sin x = ±(√3/2) in the given interval, or who gave only 0° and 180° from sin x = 0 and omitted −180°. Another common lapse was overlooking the negative square root.
• Proving identities. Students who worked from the right-hand side often introduced a mixture of sign errors. Others made the proof longer than needed, confused sec x and cosec x, or did not recognise cos²x/sin²x as cot²x. Solutions that tried to work on both sides did not meet in the middle, which is not a complete proof.
• Notation. Not carefully distinguishing cos²θ from cos 2θ, leaving out brackets, or forgetting to write the angle.
• Calculus with trigonometry. Quotient rule errors (a missing denominator, or terms subtracted in the wrong order), sign errors, and not applying the chain rule, for example d/dx (sin 3x) = cos 3x. In integration, a missing minus sign and difficulty with integrals such as sec θ tan θ.
• Radians. Using the wrong angle in sector area, and not working in radians when the question uses radians.
• Rounding. Premature approximation of decimal values, or giving an over-specified answer.

What examiners advise

• Learn the compound-angle and double-angle formulae exactly, and double-check arithmetic.
• Work from one side of an identity and show every step. Quote the formula you are using.
• Tell the stages apart in a "hence" question. If a later part has 2θ where an earlier part had θ, replace θ by 2θ in the result.
• Include all the solutions in the range, and be sure to halve if the angle was doubled.
• Use brackets with each trigonometric expression to avoid losing a denominator or a sign.

Habits that help

• When solving for 2θ, change the interval to the interval for 2θ first, find all the values, and only then halve them.
• Write the identity you are using as a short note beside the line.
• Use the video on proving identities for practice with standard patterns.

Questions students ask

Why do I lose marks on R sin(θ + α) questions?

Students forget the R, writing cos α = 3 and sin α = 1 instead of R cos α and R sin α, so they get the right tan α but the wrong R. Others find R and α and then use θ where 2x was needed.

How do I solve an equation such as tan 2θ = k?

Change the interval to the interval for 2θ first, find all the values, and only then halve them. Reports record students who solved for 2θ and never halved, replaced 2θ with θ, or wrote 2(θ − 23.41) in place of 2θ − 23.41.

How do I prove a trigonometric identity properly?

Work from one side only, quote the formula you use, and show every step. Solutions that work on both sides and meet in the middle are not a complete proof. Know that cos²x/sin²x is cot²x, and do not confuse sec x with cosec x.

Is cos(x − 30°) equal to cos x − cos 30°?

No. Misquoting a compound-angle formula is a recorded error, and examiners treat a misquoted formula that is then applied as a method error. Learn the compound-angle and double-angle formulae exactly.

How do I avoid missing solutions in Paper 3 trigonometry?

Check how many solutions to expect in the interval. Students found only some of the four solutions of sin x = ±(√3/2), or gave only 0° and 180° from sin x = 0 and omitted −180°. Also remember the negative square root.

What do I need to remember when differentiating trigonometric functions?

The chain rule: d/dx (sin 3x) is 3 cos 3x, not cos 3x. In the quotient rule include the denominator and subtract the terms in the right order. Work in radians when the question uses radians.