E: Trigonometry: Understand and use double angle formulae; use of formulae for sin(A ± B), cos(A ± B) and tan(A ± B); understand geometrical proofs of these formulae. Understand and use expressions for a cos θ + b sin θ in the equivalent forms of R cos(θ ± α) or R sin(θ ± α).

The idea

This topic covers compound angle formulae, double angle formulae, their geometrical proofs, and the harmonic form R cos(θ ± α) or R sin(θ ± α). You must derive and apply sin(A±B), cos(A±B), tan(A±B), and the double angle results for sin 2θ, cos 2θ, tan 2θ. Geometrical proofs often use a unit circle or right-angled triangles to establish these identities. For a cos θ + b sin θ, you rewrite it as R cos(θ ± α) or R sin(θ ± α), where R = √(a² + b²) and α satisfies cos α = a/R, sin α = b/R (for R cos(θ - α)). This form helps solve equations and find maxima/minima. For example, 3 cos θ + 4 sin θ = 5 cos(θ - 53.13°). You must handle signs and choose the correct form.

What earns the marks

• Derive and state the compound angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B).
• Use double angle formulae to express sin 2θ, cos 2θ and tan 2θ in terms of sin θ, cos θ and tan θ.
• Apply geometrical proofs, such as using a unit circle or right-angled triangles, to establish the compound angle formulae.
• Convert a cos θ + b sin θ into R cos(θ ± α) or R sin(θ ± α), finding R and α correctly.
• Use the harmonic form to solve equations, find maximum/minimum values, or sketch graphs.

Common mistakes and how to put them right

• Incorrectly writing sin(A + B) = sin A + sin B.

  Put it right: Use the correct formula sin(A + B) = sin A cos B + cos A sin B.

• Forgetting the sign in cos(A + B) = cos A cos B - sin A sin B.

  Put it right: The sign change for cosine.

• Mixing up the expressions for R cos(θ ± α) and R sin(θ ± α).

  Put it right: For R cos(θ - α), use cos α = a/R and sin α = b/R; for R sin(θ + α), use sin α = a/R and cos α = b/R.

Exam tips

• Learn the compound angle formulae and double angle formulae thoroughly; they are not always given in the exam.
• When finding α, ensure it is in the correct quadrant by checking the signs of a and b.
• For geometrical proofs, draw a clear diagram and label angles carefully to avoid confusion.