Addition formulae
These addition formulae let you find the ratio of a sum or difference of two angles from the ratios of each angle. Note the sign pattern: cosine "flips" the sign.
Key formula /
sin(A ± B) = sin A cos B ± cos 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)
Double angle formulae
Putting B = A gives the double angle formulae: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1, and tan 2A = 2 tan A/(1 − tan²A).
Rearranging cos 2A gives two further useful forms, sin²A = (1 − cos 2A)/2 and cos²A = (1 + cos 2A)/2, which turn a squared ratio into a first-power expression and are helpful before integrating. The addition formulae also let you write exact values for non-standard angles such as 15° and 75° by splitting each one into a sum or difference of the special angles 30°, 45° and 60°.
Worked example
Given sin A = 3/5 and cos A = 4/5, find sin 2A and cos 2A. sin 2A = 2 sin A cos A = 2 × (3/5)(4/5) = 24/25. cos 2A = 1 − 2 sin²A = 1 − 2(9/25) = 1 − 18/25 = 7/25.
Also cos 75° = cos(45° + 30°) = cos45 cos30 − sin45 sin30 = (√6 − √2)/4.
Remember
- Cosine addition uses the opposite sign of the bracket.
- cos 2A has three equivalent forms — pick the one that fits the given data.
- Keep exact surd form for special angles.