About

Errors in double-angle trigonometric evaluation propagate through signal processing pipelines, Fourier decompositions, and mechanical linkage analyses. A mis-signed cos(2θ) term in a vibration model can shift predicted resonant frequencies by octaves, risking structural fatigue or control-loop instability. This calculator evaluates cos(2θ) using all three standard double-angle identities simultaneously: cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ. Cross-verification across all three forms catches input mistakes immediately. The tool accepts both degrees and radians and displays intermediate values (sin θ, cos θ) so you can trace each step. Note: results are IEEE 754 double-precision approximations. Values near 10⁻¹⁶ should be treated as zero.

Formulas

The cosine double-angle identity expresses cos(2θ) in three algebraically equivalent forms derived from the angle addition formula cos(α + β) = cos α cos β − sin α sin β by setting α = β = θ.

cos(2θ) = cos²θ − sin²θ

cos(2θ) = 2cos²θ − 1

cos(2θ) = 1 − 2sin²θ

The second and third forms are obtained by substituting the Pythagorean identity sin²θ + cos²θ = 1 into the first form. Each is preferred depending on which single function value is known.

Degree to radian conversion used internally:

θ_rad = θ_deg × π180

Where θ = the input angle, sin θ = sine of the angle, cos θ = cosine of the angle, and π ≈ 3.14159265.

Reference Data

Angle θ (°)	Angle θ (rad)	sin θ	cos θ	cos(2θ)
0	0	0	1	1
15	0.2618	0.2588	0.9659	0.8660
30	π/6	0.5	0.8660	0.5
45	π/4	0.7071	0.7071	0
60	π/3	0.8660	0.5	−0.5
75	1.3090	0.9659	0.2588	−0.8660
90	π/2	1	0	−1
120	2π/3	0.8660	−0.5	−0.5
135	3π/4	0.7071	−0.7071	0
150	5π/6	0.5	−0.8660	0.5
180	π	0	−1	1
210	7π/6	−0.5	−0.8660	0.5
225	5π/4	−0.7071	−0.7071	0
240	4π/3	−0.8660	−0.5	−0.5
270	3π/2	−1	0	−1
300	5π/3	−0.8660	0.5	−0.5
315	7π/4	−0.7071	0.7071	0
330	11π/6	−0.5	0.8660	0.5
360	2π	0	1	1

Frequently Asked Questions

All three forms are algebraically identical. They are derived from the single identity cos²θ − sin²θ by substituting sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ via the Pythagorean identity. The choice depends on which value you already know: use 2cos²θ − 1 when only cos θ is available, or 1 − 2sin²θ when only sin θ is available.

IEEE 754 double-precision floating-point arithmetic cannot represent π exactly. When you enter 45° the internal radian value is an approximation, so cos(90°) yields approximately 6.12 × 10⁻¹⁷ rather than 0. Any result with absolute value below 10⁻¹⁴ should be treated as zero for practical purposes.

Cosine is an even function: cos(−x) = cos(x). Therefore cos(2(−θ)) = cos(−2θ) = cos(2θ). Entering a negative angle yields the same result as the corresponding positive angle. This property holds for both degree and radian inputs.

The cos²θ − sin²θ form factors as (cos θ + sin θ)(cos θ − sin θ), which simplifies integration in certain Fourier analysis contexts. The 2cos²θ − 1 form is standard in power-reduction identities used to lower the degree of trigonometric polynomials. The 1 − 2sin²θ form appears frequently in deriving the half-angle formulas.

Yes. The identity cos(2θ) = cos²θ − sin²θ holds for all θ ∈ ℂ. For purely imaginary θ = iy, cos(2iy) = cosh(2y), which grows exponentially. This calculator handles real-valued inputs only; for complex arguments, the Euler formula e^(iθ) = cos θ + i sin θ must be used.

The function cos(θ) has period 2π (360°). Multiplying the argument by 2 halves the period to π (180°). This means cos(2θ) completes two full cycles in the interval [0, 2π]. The frequency doubling is why the double-angle identity appears in harmonic analysis and signal processing.