Trigonometric Identities in Physics Problems
Trigonometry in a maths classroom is about proving identities; trigonometry in a physics problem is about using a handful of the same identities to simplify a formula or resolve a vector — and it is exactly the same identities, just applied rather than proved. This guide works through four genuinely physical problems that each lean on a different identity: projectile range, resolving a velocity into components, average AC power, and the small-angle approximation that makes the simple pendulum formula possible.
The identities that matter for physics
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
sin2θ = 2 sinθ cosθ
cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Small angle (θ in radians): sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1
The sign in cos(A±B) is deliberately opposite to the sign in sin(A±B) — this ∓ is the single most common place students copy the formula wrong.
Worked example 1 — the range formula and sin2θ
Projectile range: R = u²sin2θ / g. A ball is launched at u = 20 m/s, θ = 30°, with g = 9.8 m/s². Find R, and verify sin2θ using the double-angle identity.
Verify: sin2θ = 2 sinθ cosθ = 2 × sin30° × cos30° = 2 × 0.5 × 0.8660 = 0.8660, which matches sin60° = 0.8660 directly — the identity checks out.
R = (20² × 0.8660) ÷ 9.8 = (400 × 0.8660) ÷ 9.8 = 346.4 ÷ 9.8 = 35.35 m
Worked example 2 — resolving a vector, and sin²+cos² as a check
A velocity v = 50 m/s points at θ = 37° above the horizontal. Using the standard approximation sin37° ≈ 0.6, cos37° ≈ 0.8 (the familiar 3-4-5 triangle), find the components.
vx = v cosθ = 50 × 0.8 = 40 m/s
vy = v sinθ = 50 × 0.6 = 30 m/s
Check using sin²θ + cos²θ = 1: 0.6² + 0.8² = 0.36 + 0.64 = 1.00 ✓, and using Pythagoras on the components themselves: 40² + 30² = 1600 + 900 = 2500 = 50² ✓ — both checks confirm the split is consistent.
Worked example 3 — average AC power from the double-angle identity
An AC voltage and current in phase, V = V₀ sin(ωt) and I = I₀ sin(ωt), give instantaneous power p = V₀I₀ sin²(ωt). Using cos2θ = 1 − 2sin²θ, rearrange to sin²θ = (1 − cos2θ)/2. Averaged over a full cycle, the average of cos2(ωt) is zero, so the average of sin²(ωt) is exactly 1/2 — this single identity is the reason average AC power is V₀I₀/2, not V₀I₀.
With peak voltage V₀ = 310 V and peak current I₀ = 2 A:
Pavg = V₀I₀ × (1/2) = 310 × 2 × 0.5 = 310 W
Worked example 4 — small-angle approximation and why the pendulum formula needs it
The simple pendulum period T = 2π√(L/g) is only valid because sinθ ≈ θ (in radians) for small oscillations. Check the approximation at two angles.
At θ = 5°: in radians, θ = 5 × π/180 = 0.0873 rad. Actual sin5° = 0.0872. Difference ≈ 0.0001 — an error of about 0.1%, safely small.
At θ = 30°: in radians, θ = 30 × π/180 = 0.5236 rad. Actual sin30° = 0.5000. Difference = 0.0236, a relative error of 0.0236 ÷ 0.5 = 4.7% — no longer negligible, which is exactly why the simple pendulum formula is stated as valid only for "small" angular amplitude (conventionally under about 10°).
Common mistakes that cost marks
- Using degrees in the small-angle approximation. sinθ ≈ θ is only true when θ is in radians — 30° is 0.5236 rad, not 30, and substituting the number 30 directly gives a meaningless result.
- Writing sin2θ = 2sinθ instead of expanding it fully as 2 sinθ cosθ — a very common shortcut error.
- Mixing up the sign in cos(A±B) — it is the opposite sign to sin(A±B), not the same one.
- Applying the pendulum's small-angle formula at large amplitude and expecting the same accuracy — as example 4 shows, the error grows quickly past a few degrees.
- Forgetting sin²θ + cos²θ = 1 is a free check on any resolved vector — always worth a quick verification before moving on.
Where these identities appear in exams
| Exam | Typical use |
|---|---|
| CBSE Class 11 Physics | Projectile motion, vector resolution, oscillations (SHM), AC circuits |
| CBSE Class 11 Maths | Trigonometric Functions chapter — the identities themselves, proved directly |
| JEE Main & Advanced | Heavy use throughout mechanics, SHM and alternating-current numericals |
| NEET | Projectile range and component-resolution questions |
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