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Trigonometric identities and addition formulae
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Identities are true for every angle. They let you swap one trig expression for another to simplify, prove or solve.
Two key identities
cos2 θ + sin2 θ = 1 (learn this)
tan θ = sin θcos θ
If sin θ = 35 and θ is acute, cos θ = √(1 − 925) = 45 and tan θ = 34.
tan θ = sin θcos θ
If sin θ = 35 and θ is acute, cos θ = √(1 − 925) = 45 and tan θ = 34.
Addition formulae
These are on the formula sheet:
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)
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)
Exact values
cos 75° = cos(45° + 30°)
= cos 45° cos 30° − sin 45° sin 30°
= 1√2 × √32 − 1√2 × 12 = (√6 − √2)4
= cos 45° cos 30° − sin 45° sin 30°
= 1√2 × √32 − 1√2 × 12 = (√6 − √2)4
sin A = 35 and cos B = 513, with A and B acute. Find sin(A + B).
- cos A = 45 and sin B = 1213
- sin A cos B + cos A sin B = 35 × 513 + 45 × 1213
- 1565 + 4865
Answer: 6365
cos2 θ + sin2 θ = 1 and tan θ = sin θcos θ. Use the addition formulae for sums and differences of angles, and for exact values like 75° or 15°.
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