60
°
0.5
0.8660254
0.57735027
0.8660254
0.5
1.73205081
0.52359878
rad
60
°
0.5
0.8660254
0.57735027
0.8660254
0.5
1.73205081
0.52359878
rad
The Double Angle Calculator computes the trigonometric functions of twice a given angle using the double-angle identities. These formulas express $$\sin(2\theta)$$, $$\cos(2\theta)$$, and $$\tan(2\theta)$$ in terms of functions of $$\theta$$ alone, making them indispensable tools in algebra, calculus, and applied sciences.
The three core double-angle identities are:
$$\sin(2\theta) = 2\sin\theta\cos\theta$$
$$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$
$$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$
The cosine double-angle formula has two equivalent alternative forms obtained by applying the Pythagorean identity $$\sin^2\theta + \cos^2\theta = 1$$:
$$\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$
The double-angle formulas follow directly from the angle addition identities by setting both angles equal to $$\theta$$:
$$\sin(\theta + \theta) = \sin\theta\cos\theta + \cos\theta\sin\theta = 2\sin\theta\cos\theta$$
$$\cos(\theta + \theta) = \cos\theta\cos\theta - \sin\theta\sin\theta = \cos^2\theta - \sin^2\theta$$
This derivation highlights that double-angle formulas are special cases of the more general sum formulas.
Double-angle identities are essential for integrating even powers of sine and cosine. The identity $$\cos(2\theta) = 1 - 2\sin^2\theta$$ can be rearranged to:
$$\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$$
This power-reduction formula converts a squared trigonometric function into a linear expression, making integration straightforward. Without it, integrals like $$\int \sin^2\theta\,d\theta$$ would be much harder to evaluate.
In physics, the double-angle identity for sine determines the optimal launch angle for projectile range. The range formula $$R = \frac{v_0^2 \sin(2\theta)}{g}$$ shows that maximum range occurs when $$\sin(2\theta) = 1$$, giving $$\theta = 45°$$.
In signal processing, double-angle formulas help analyze frequency doubling, harmonics, and modulation. In electrical engineering, they appear in power calculations involving AC circuits where voltage and current oscillate sinusoidally.
Enter an angle $$\theta$$ in degrees. The calculator computes $$\sin\theta$$, $$\cos\theta$$, and $$\tan\theta$$, then applies the double-angle formulas to display $$\sin(2\theta)$$, $$\cos(2\theta)$$, and $$\tan(2\theta)$$. The doubled angle value is also shown for reference.
The calculator converts $$\theta$$ to radians and evaluates $$\sin\theta$$ and $$\cos\theta$$ directly. Then $$\sin(2\theta) = 2\sin\theta\cos\theta$$, $$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$, and $$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$. When $$\tan^2\theta = 1$$ (at 45° or 135°), $$\tan(2\theta)$$ is undefined.
The outputs show both the original trig values and their doubled-angle counterparts. Compare $$\sin(2\theta)$$ with the product $$2\sin\theta\cos\theta$$ to verify the identity. A result of NaN for $$\tan(2\theta)$$ means the tangent of the doubled angle is undefined (vertical asymptote).
Inputs
Results
sin(60°) = 2·sin(30°)·cos(30°) = 2(0.5)(0.866) ≈ 0.866. The double of 30° is 60°, and all identities check out.
Inputs
Results
Doubling 22.5° gives 45°. Since sin(45°) = cos(45°) = √2/2 ≈ 0.7071, and tan(45°) = 1, the results confirm the double-angle formulas perfectly.
The double angle formula for sine is $$\sin(2\theta) = 2\sin\theta\cos\theta$$. It expresses the sine of twice an angle as twice the product of the sine and cosine of the original angle.
The three forms—$$\cos^2\theta - \sin^2\theta$$, $$2\cos^2\theta - 1$$, and $$1 - 2\sin^2\theta$$—are all equivalent because $$\sin^2\theta + \cos^2\theta = 1$$. Different forms are convenient depending on whether you know sine, cosine, or both.
$$\tan(2\theta)$$ is undefined when $$1 - \tan^2\theta = 0$$, i.e., when $$\tan\theta = \pm 1$$. This occurs at $$\theta = 45°, 135°, 225°, 315°$$ (and their coterminal equivalents), where $$2\theta$$ equals 90° or 270°.
The horizontal range of a projectile is $$R = \frac{v_0^2 \sin(2\theta)}{g}$$, where $$\theta$$ is the launch angle. The factor $$\sin(2\theta)$$ is maximized when $$2\theta = 90°$$, giving the optimal launch angle of 45° for maximum range.
Power-reduction formulas are rearrangements of the double-angle identities: $$\sin^2\theta = \frac{1 - \cos(2\theta)}{2}$$ and $$\cos^2\theta = \frac{1 + \cos(2\theta)}{2}$$. They reduce the power of trigonometric functions, which is essential for integration.
Yes. By applying the addition formula to $$\sin(2\theta + \theta)$$, you get $$\sin(3\theta) = 3\sin\theta - 4\sin^3\theta$$ and $$\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$$. These are the triple-angle formulas.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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