(A Level) Compound-angle formulae and the R-form

Learning objectives

Use the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) to find exact values such as cos 75° or tan(π/12) and to simplify expressions

Use sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A and tan 2A = 2 tan A/(1 − tan²A) to prove identities and to write sin²x or cos²x in terms of cos 2x

Express a sin θ + b cos θ in the form R sin(θ ± α) or R cos(θ ± α), with R > 0 and α acute, giving α exactly or to a stated accuracy

Use the R-form to solve equations and to find the greatest and least values of expressions such as 5 + 3 sin x − 4 cos x or 1/(7 − 2 cos x + 3 sin x), and where they occur

Select the identities that suit an equation such as cos 2θ = 5 sin θ − 2 or sin(θ + 30°) = 2 cos θ, and find all solutions in a stated interval