Trigonometry (Pure 3)

You will be able to

• Calculate both sides of sec²A ≡ 1 + tan²A, cosec²A ≡ 1 + cot²A, sin(A ± B), cos(A ± B), tan(A ± B), sin 2A, cos 2A and tan 2A for a chosen angle, and compare them.
• Deduce the double-angle formulae from the compound-angle formulae by putting B = A, and find exact values using 30°, 45° and 60°.
• Determine R and α (acute, to 1 d.p.) so that a sin θ + b cos θ ≡ R sin(θ ± α) or R cos(θ ± α), by comparing coefficients.
• Explain why an identity must hold wherever both sides are defined, and why a tempting slip such as sin(A + B) = sin A + sin B fails.

You will be able to

• Solve equations with sec, cosec, cot, sin 2θ, cos 2θ and tan 2θ by reducing them to one trigonometric function with the identities, then factorising.
• Determine every solution of a sin, cos or tan equation in a stated interval, using the principal value, the symmetry of the graph and the period, and reject the values outside it.
• Solve a sin θ + b cos θ = k by writing the left side as R sin(θ ± α) or R cos(θ ± α), and state when there is no solution (|k| > R).
• Explain why cos θ = 3 is rejected, and why dividing by sin θ or tan θ loses solutions.