What are the double-angle formulae?
sin 2A = 2 sinA cosA; cos 2A = cos²A − sin²A (also 2cos²A − 1 and 1 − 2sin²A); and tan 2A = 2tanA/(1 − tan²A). They come from putting B = A in the compound-angle formulae.
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Double-Angle Formulae
Evaluate sin 2A, cos 2A and tan 2A with full working, switching between the three equivalent forms of cos 2A, in degrees or radians.
sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A)
Angle unit
Frequently asked questions
Why are there three forms of cos 2A?
Starting from cos²A − sin²A and using sin²A + cos²A = 1 you can replace one term to get 2cos²A − 1 or 1 − 2sin²A. Pick the form that leaves only the function you need.
Can you show a worked example?
For A = 30°, sin 2A = 2 sin30 cos30 = 2 × 0.5 × 0.8660 ≈ 0.8660 = sin 60°. The same angle in radians is 30° = π/6 ≈ 0.5236 rad; the toggle re-derives in radians.
When is tan 2A undefined?
When 1 − tan²A = 0, i.e. A is an odd multiple of 45° (π/4 rad), or when 2A is an odd multiple of 90°. The calculator flags these instead of returning a wrong number.
Where are they used?
Lowering powers for integration, deriving the half-angle formulae, and modelling frequency doubling in optics, AC power and signal processing.