Cos 2 Theta Calculator

Enter an angle θ and instantly get cos(2θ), sin(2θ), and tan(2θ) using the double-angle identities.

Quick Facts

Double-angle identity
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
All three forms are equivalent because sin²θ + cos²θ = 1.
Range
−1 ≤ cos(2θ) ≤ 1
Like any cosine value, cos(2θ) never leaves this range.

Your Results

Calculated
cos(2θ)
-
cos²θ − sin²θ
sin(2θ)
-
2 sinθ cosθ
tan(2θ)
-
sin(2θ) ÷ cos(2θ)
Double angle (2θ)
-
Shown in degrees and radians

Ready

Enter an angle θ, then press Calculate.

Understanding Cos 2 Theta

Cos 2 theta, written cos(2θ), is the cosine of a doubled angle. It comes from the cosine addition formula cos(A + B) = cosA·cosB − sinA·sinB. Setting A = B = θ gives cos(2θ) = cosθ·cosθ − sinθ·sinθ = cos²θ − sin²θ. Combined with the Pythagorean identity sin²θ + cos²θ = 1, this identity can be rewritten in two more useful forms.

The double-angle formula (three equivalent forms)

  • cos(2θ) = cos²θ − sin²θ — the direct definition, useful when both sinθ and cosθ are known.
  • cos(2θ) = 2cos²θ − 1 — obtained by substituting sin²θ = 1 − cos²θ; useful when only cosθ is known.
  • cos(2θ) = 1 − 2sin²θ — obtained by substituting cos²θ = 1 − sin²θ; useful when only sinθ is known.

A fourth form uses the tangent: cos(2θ) = (1 − tan²θ) / (1 + tan²θ). This calculator evaluates cos(2θ) directly from θ by doubling the angle and applying the cosine function, then reports sin(2θ) = 2 sinθ cosθ and tan(2θ) = sin(2θ) / cos(2θ) alongside it.

Degrees vs. radians

Choose the unit that matches your problem before reading the result. Conversion: radians = degrees × π/180, and a full circle is 2π radians = 360°. The calculator converts your angle to radians internally regardless of which unit you pick, so the trigonometric functions are always evaluated correctly.

Common applications

  • Power-reduction and half-angle formulas in calculus, which are derived directly from cos(2θ) = 1 − 2sin²θ and cos(2θ) = 2cos²θ − 1
  • Simplifying trigonometric integrals and Fourier series terms that involve cos²θ or sin²θ
  • Physics and engineering problems with doubled phase angles, such as wave interference, AC power calculations, and rotational motion

Frequently Asked Questions

What is the formula for cos 2 theta?
cos(2θ) can be found three equivalent ways: cos²θ − sin²θ, 2cos²θ − 1, or 1 − 2sin²θ. All three use only θ (or its sine and cosine) and always agree because sin²θ + cos²θ = 1.
How do I find cos 2 theta if I know tan theta?
Use cos(2θ) = (1 − tan²θ) / (1 + tan²θ). This form is handy when tan θ is known but sin θ and cos θ are not, and it is undefined only where tan θ itself is undefined, such as θ = 90°.
Does this calculator use degrees or radians?
Either — choose the unit next to the angle field. Internally the angle is converted to radians before applying the cosine and sine functions, then the doubled angle is converted back for display in both units.
Why is cos(2θ) sometimes negative even when cos(θ) is positive?
Doubling the angle can push 2θ into a different quadrant than θ. For example, θ = 60° has a positive cosine, but 2θ = 120° lands in Quadrant II, where cosine is negative, so cos(2θ) = −0.5.

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