Trigonometric identities
Specialist Mathematics · Unit 2 — Complex numbers, further proof, trigonometry, functions and transformations · Trigonometry and functions
Learning objectives (6)
LO-1Convert sums 𝑎 cos(𝑥) + 𝑏 sin(𝑥) to 𝑅 cos(𝑥 ± 𝛼) or 𝑅 sin(𝑥 ± 𝛼) and apply these to sketch graphs.LO-2Model and solve problems that involve equations of the form 𝑎 cos(𝑥) + 𝑏 sin(𝑥) = 𝑐.LO-3Prove and apply multi-angle trigonometric identities up to angles of 4𝑥 using the identities listed above, e.g. cos(4𝑥) = 8 cos4(𝑥) − 8 cos2(𝑥) + 1 and cosec(2𝑥) − cot(2𝑥) = tan(𝑥). Specialist Mathematics 2025 v1.4LO-4Prove and apply the angle sum, difference and double-angle identities for sines and cosines. sin(𝐴 + 𝐵) = sin(𝐴) cos(𝐵) + cos(𝐴) sin(𝐵) sin(𝐴 − 𝐵) = sin(𝐴) cos(𝐵) − cos(𝐴) sin(𝐵) cos(𝐴 + 𝐵) = cos(𝐴) cos(𝐵) − sin(𝐴) sin(𝐵) cos(𝐴 − 𝐵) = cos(𝐴) cos(𝐵) + sin(𝐴) sin(𝐵) sin(2𝐴) = 2 sin(𝐴) cos(𝐴) cos(2𝐴) = cos2(𝐴) − sin2(𝐴) = 1 − 2 sin2(𝐴) = 2 cos2(𝐴) − 1LO-5Prove and apply the identities for products of sines and cosines expressed as sums and differences. sin(𝐴) sin(𝐵) = 1 2 (cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)) cos(𝐴) cos(𝐵) = 1 2 (cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)) sin(𝐴) cos(𝐵) = 1 2 (sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)) cos(𝐴) sin(𝐵) = 1 2 (sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵))LO-6Prove and apply the Pythagorean identities. sin2(𝐴) + cos2(𝐴) = 1 tan2(𝐴) + 1 = sec2(𝐴) cot2(𝐴) + 1 = cosec2(𝐴)
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