cos(2θ) Calculator — Double Angle Formula for Cosine
Enter angle θ in degrees or radians to compute cos(2θ) — shown in all three equivalent double-angle forms — along with sin(2θ) and tan(2θ).
°
Angle unit
Double-angle cosine — always in [−1, 1]
- 1
Convert to radians
30 × π ÷ 180 = 0.523599 rad - 2
cos(θ)
cos(30°) = 0.866025 - 3
cos²(θ)
0.866025² = 0.75 - 4
cos(2θ) = 2cos²(θ) − 1
2 × 0.75 − 1 = 0.500000Equivalent to cos²θ − sin²θ and to 1 − 2sin²θ.
How does this calculator work?
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. All three forms are equal — choose by which trig values are available. sin(2θ) = 2sinθ cosθ. cos(2θ) oscillates at twice the frequency of cos(θ) and always lies in [−1, 1].
Formula
How this is calculated
The double-angle formulas express trigonometric functions of 2θ in terms of functions of θ. For cosine, three equivalent forms exist. Starting from the angle-addition identity cos(A+B) = cosA cosB − sinA sinB, set A = B = θ: cos(2θ) = cos²θ − sin²θ. Substituting sin²θ = 1 − cos²θ gives 2cos²θ − 1 (useful when only cos(θ) is known). Substituting cos²θ = 1 − sin²θ gives 1 − 2sin²θ (useful when only sin(θ) is known).
All three forms yield identical numerical results — the calculator shows all three as a cross-check and for algebraic convenience. sin(2θ) = 2sin(θ)cos(θ) follows from the same angle-addition identity. These identities are fundamental in calculus (integrating cos²θ and sin²θ), signal processing (double-frequency analysis), and solving trigonometric equations.
Note that cos(2θ) has twice the frequency of cos(θ): while cos(θ) completes a cycle every 360°, cos(2θ) completes two full cycles over the same span. The cosine wave visualisation is phase-shifted to show where 2θ falls on the cycle.
Frequently asked questions
All three are algebraically identical: cos²θ − sin²θ, 2cos²θ − 1, or 1 − 2sin²θ. In practice: use 2cos²θ − 1 when only cos(θ) is known, 1 − 2sin²θ when only sin(θ) is known, and cos²θ − sin²θ when simplifying an expression that involves both.
Like all cosine values, cos(2θ) lies in [−1, 1]. It reaches +1 when 2θ = 0°, 360°, … (i.e., θ = 0°, 180°) and −1 when 2θ = 180°, 540°, … (i.e., θ = 90°, 270°).
tan(2θ) is undefined when cos(2θ) = 0, i.e., when 2θ = 90° + n·180°, meaning θ = 45°, 135°, 225°, 315°, etc. At these angles the calculator displays "undefined".
TG we-Calculate Editorial Team. (2026). cos(2θ) Calculator — Double Angle Formula for Cosine [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cos-2-theta-calculator
TG we-Calculate Editorial Team. "cos(2θ) Calculator — Double Angle Formula for Cosine." TG we-Calculate. 2026. https://we-calculate.com/calculator/cos-2-theta-calculator.
TG we-Calculate Editorial Team, "cos(2θ) Calculator — Double Angle Formula for Cosine," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cos-2-theta-calculator
@misc{wecalculate_cos_2_theta_calculator, title = {cos(2θ) Calculator — Double Angle Formula for Cosine}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cos-2-theta-calculator}}, year = {2026}, note = {TG we-Calculate} }
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