Further Trigonometry

This lesson extends trigonometry with addition formulae, double angle formulae, and the R cos/sin form — powerful tools for simplifying expressions and solving equations.

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Addition Formulae

Formula
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
tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
tan(A − B) = (tan A − tan B)/(1 + tan A tan B)

Worked Example 1

Find the exact value of sin 75°.

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4

= (√6 + √2)/4

Worked Example 2

Given sin A = 3/5 (A acute) and cos B = 5/13 (B acute), find cos(A − B).

cos A = 4/5 (from Pythagoras), sin B = 12/13.