04 // Interactive Formula Compendium // GEOMETRY-TRIGsin(2θ) = 2 sin(θ) cos(θ), cos(2θ) = cos²(θ) − sin²(θ)
Trigonometric Double-Angle, Half-Angle & Angle-Sum Identities Sheet
The double-angle trigonometric identities express functions of 2θ in terms of θ: sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ).
Formula Reference
Main Formula
sin(2θ) = 2 sin(θ) cos(θ), cos(2θ) = cos²(θ) − sin²(θ)
Rearranged Forms
cos(2θ) = 2 cos²(θ) − 1 = 1 − 2 sin²(θ)
sin²(θ) = (1 − cos(2θ)) / 2, cos²(θ) = (1 + cos(2θ)) / 2 (Power-Reduction)
tan(2θ) = 2 tan(θ) / (1 − tan²(θ))
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| sin(2θ) | Double-Angle Sine Value sin(2θ) | dimensionless |
| theta_deg | Base Angle θ (degrees) | deg |
Inputs & Parameters
[deg]
Calculated Result (Double-Angle Sine Value sin(2θ))
sin(2θ) =0.866025dimensionless
Formula Notes & Background
- From Euler's formula, e^(i(α+β)) = e^(iα)e^(iβ) = (cos α + i sin α)(cos β + i sin β).
- Matching real and imaginary parts gives cos(α+β) = cos α cos β − sin α sin β and sin(α+β) = sin α cos β + cos α sin β.
- Setting α = β = θ immediately produces sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) − sin²(θ).
Step-by-Step Examples
#1: Find the exact values of sin(60°) and cos(60°) from θ = 30°.
θ = 30°sin(30°) = 0.5cos(30°) = √3/2
sin(60°) = 2(0.5)(√3/2) = √3/2; cos(60°) = (3/4) − (1/4) = 1/2
→ sin(60°) ≈ 0.86603, cos(60°) = 0.50000
#2: Find the exact value of sin(15°) using the half-angle formula with 2θ = 30°.
θ = 15°cos(30°) = √3/2
sin(15°) = √[(1 − cos 30°) / 2] = √[(2 − √3) / 4] = (√6 − √2) / 4
→ sin(15°) ≈ 0.25882
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Frequently Asked Questions
Why are the power-reduction formulas sin²θ = ½(1 − cos 2θ) essential in calculus?+
They convert squared trig functions into first-power cosines of 2θ that can be integrated immediately (e.g., computing RMS AC voltage).
Why does projectile range R = (v₀² sin 2θ)/g peak at θ = 45°?+
Because horizontal range is proportional to 2 sin(θ)cos(θ) = sin(2θ), which attains its maximum value of 1 when 2θ = 90°, or θ = 45°.
What is the Weierstrass tangent half-angle substitution t = tan(θ/2)?+
Setting t = tan(θ/2) expresses sin θ = 2t/(1+t²) and cos θ = (1−t²)/(1+t²), turning any rational trigonometric equation into a polynomial in t.