Trigonometry
A compact reference of high-frequency trigonometry identities for physics: angle formulas, product-sum, R-formula, Euler form, and small-angle approximations.
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The core idea
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Trigonometry is everywhere in physics: oscillations, waves, circular motion, and any time you resolve vectors into components.
Many standard results (especially series expansions like sin x ≈ x) assume x is in radians.
Basic identities
Angle addition (compound angles)
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)
Product-to-sum
sin u sin v = (1/2)[cos(u-v) - cos(u + v)]; cos u cos v = (1/2)[cos(u-v) + cos(u + v)]; sin u cos v = (1/2)[sin(u + v) + sin(u-v)]; cos u sin v = (1/2)[sin(u + v) - sin(u-v)]
Double-angle
sin 2A = 2 sin A cos A; ; cos 2A = cos² A- sin² A; = 2 cos² A-1; = 1-2 sin² A; ; tan 2A = (2 tan A)/(1- tan² A)
Sum-to-product (“factor” formulas)
sin S + sin T = 2 sin(S + T)/2 cos(S-T)/2; sin S- sin T = 2 cos(S + T)/2 sin(S-T)/2; cos S + cos T = 2 cos(S + T)/2 cos(S-T)/2; cos S- cos T = -2 sin(S + T)/2 sin(S-T)/2
Euler form
cos x = (e^ix + e^(-ix))/2; sin x = (e^ix-e^(-ix))/2i
R-formula (combine sine + cosine)
a cos θ ± b sin θ = R cos(θ ∓ α); a sin θ ± b cos θ = R sin(θ ± α)
where:
One consequence: the maximum possible value of a cos θ + b sin θ is R and the minimum is -R.
Worked example
Write 3 cos θ-4 sin θ in the form R cos(θ + α).
We want:
Match coefficients:
So R = square root of (3² + 4²) = 5 and tan α = 4/3, giving:
Small-angle approximations (radians)
From Taylor series:
Even/Odd Functions
sin(-θ) = - sin θ; cos(-θ) = cos θ; tan(-θ) = - tan θ
csc(-θ) = - csc θ; sec(-θ) = sec θ; cot(-θ) = - cot θ
Cofunction Identities
sin(π/2-θ) = cos θ; csc(π/2-θ) = sec θ; tan(π/2-θ) = cot θ
cos(π/2-θ) = sin θ; sec(π/2-θ) = csc θ; cot(π/2-θ) = tan θ