Double Angle Formula Calculator
Enter an angle theta and choose your unit system to compute the sine, cosine, and tangent of its double angle 2theta. All three cosine double-angle variants are shown, along with the reciprocal functions, a step-by-step working panel, and a comparison chart so you can see how all the values relate at a glance.
Formula
Worked example
For theta = 30 degrees: sin(30) = 0.5, cos(30) = approx 0.866. So sin(60) = 2 x 0.5 x 0.866 = 0.866, which matches the known value of sin(60) = sqrt(3)/2. cos(60) = 0.866^2 - 0.5^2 = 0.75 - 0.25 = 0.5, matching the known value. tan(60) = 2 x tan(30) / (1 - tan^2(30)) = 2 x (1/sqrt(3)) / (1 - 1/3) = (2/sqrt(3)) / (2/3) = sqrt(3) approx 1.732.
What are double angle formulas?
Double angle formulas are trigonometric identities that express a trig function of twice an angle (2theta) entirely in terms of trig functions of the original angle (theta). They are derived directly from the angle-addition identities - for example, sin(A + B) = sin(A)cos(B) + cos(A)sin(B). When you set A = B = theta, you get sin(2theta) = 2 sin(theta) cos(theta). The same substitution applied to the cosine and tangent addition formulas produces the remaining identities. Because they reduce a doubled argument to a single argument, double angle formulas are essential tools in calculus integration, Fourier analysis, geometry proofs, and physics problems involving oscillations and wave equations.
The three cosine double-angle forms
The cosine identity is special because it has three equivalent forms, each useful in different contexts. The first, cos(2theta) = cos^2(theta) - sin^2(theta), comes directly from the addition formula. The second, cos(2theta) = 2cos^2(theta) - 1, is obtained by replacing sin^2(theta) with 1 - cos^2(theta) using the Pythagorean identity. The third, cos(2theta) = 1 - 2sin^2(theta), follows by replacing cos^2(theta) with 1 - sin^2(theta). In practice, you choose whichever form avoids an unknown: if you only know sin(theta), use the third; if you only know cos(theta), use the second; if you know both, any form works. The rearranged power-reduction forms (cos^2(theta) = (1 + cos(2theta)) / 2 and sin^2(theta) = (1 - cos(2theta)) / 2) are widely used in integral calculus to convert squared trig functions into simple first-power expressions.
When tan(2theta) is undefined
The tangent double-angle formula, tan(2theta) = 2tan(theta) / (1 - tan^2(theta)), has two potential sources of undefined output. First, tan(theta) itself is undefined when theta = 90° + n*180° (i.e., where the cosine of theta is zero). Second, even when tan(theta) is finite, the denominator 1 - tan^2(theta) equals zero whenever tan(theta) = plus or minus 1, which occurs at theta = 45° + n*90°. At those angles, 2theta lands exactly on 90° + n*180°, where the tangent function has a vertical asymptote. The calculator detects both conditions and returns an undefined result rather than a misleading large number.
Common exact values from double angle formulas
Double angle formulas let you derive many standard exact values from a few memorised base angles. Starting from sin(30°) = 1/2 and cos(30°) = sqrt(3)/2, the formula gives sin(60°) = 2 * (1/2) * (sqrt(3)/2) = sqrt(3)/2 and cos(60°) = 1 - 2*(1/2)^2 = 1/2. From sin(45°) = cos(45°) = 1/sqrt(2), you get sin(90°) = 2*(1/sqrt(2))*(1/sqrt(2)) = 1 and cos(90°) = 1 - 2*(1/sqrt(2))^2 = 0. Applying the formula a second time to theta = 22.5° produces the exact values for 45°, and so on. This cascade is the basis for computing trig values at any angle that is a binary fraction of 360°.
Double angle identities - all six functions
| Function | Double-angle identity | Alternative form |
|---|---|---|
| sin(2theta) | 2 sin(theta) cos(theta) | - |
| cos(2theta) | cos^2(theta) - sin^2(theta) | 2cos^2(theta) - 1 or 1 - 2sin^2(theta) |
| tan(2theta) | 2 tan(theta) / (1 - tan^2(theta)) | sin(2theta) / cos(2theta) |
| csc(2theta) | 1 / (2 sin(theta) cos(theta)) | 1 / sin(2theta) |
| sec(2theta) | 1 / (cos^2(theta) - sin^2(theta)) | 1 / cos(2theta) |
| cot(2theta) | (1 - tan^2(theta)) / (2 tan(theta)) | 1 / tan(2theta) |
All six double-angle trigonometric identities derived from the angle-addition formulas.
Frequently asked questions
How do I use the double angle calculator?
Select your angle unit (degrees, radians, or pi-radians) from the dropdown, then enter the base angle theta. The calculator instantly computes sin(2theta), cos(2theta), tan(2theta), and all three cosine variants, as well as the reciprocal functions csc, sec, and cot. The steps panel shows the full working so you can follow every substitution.
Why are there three formulas for cos(2theta)?
All three are equivalent - they give the same numerical answer - but each is derived by applying the Pythagorean identity (sin^2 + cos^2 = 1) differently to the base form cos^2(theta) - sin^2(theta). The version 2cos^2(theta) - 1 is convenient when only cos(theta) is known; the version 1 - 2sin^2(theta) is convenient when only sin(theta) is known. Rearranging either form also gives the power-reduction identities used heavily in integration.
Can I use radians or pi-radians instead of degrees?
Yes. Use the angle unit selector to choose degrees, radians (e.g. enter 0.5236 for 30 degrees), or pi-radians (enter 1/6 = 0.1667 for 30 degrees). All three options give identical results for the same geometric angle. Pi-radians mode is convenient when your angle is a simple fraction of pi, such as pi/3, pi/4, or pi/6, because you can enter the coefficient directly.
What happens when tan(2theta) is undefined?
tan(2theta) is undefined at angles where 2theta equals 90° + n*180° for any integer n. This happens when theta itself is at a vertical asymptote (90°, 270°, etc.) or when tan(theta) = plus or minus 1 (theta = 45°, 135°, etc.). The calculator returns an undefined marker in those cases. sin(2theta) and cos(2theta) are still well defined at those points.
How are double angle formulas derived?
They come from the angle-addition identities. For sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Setting A = B = theta gives sin(theta + theta) = sin(theta)cos(theta) + cos(theta)sin(theta) = 2sin(theta)cos(theta). For cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B), so cos(2theta) = cos^2(theta) - sin^2(theta). For tangent: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), which simplifies to 2tan(theta) / (1 - tan^2(theta)) when A = B = theta.
What is a practical use of the sin(2theta) formula?
One of the most common applications is in physics: the horizontal range of a projectile fired at speed v and angle theta on level ground is R = (v^2 sin(2theta)) / g. Because sin(2theta) reaches its maximum of 1 at theta = 45°, the formula immediately tells you that 45° maximises range. In calculus, sin(2theta) = 2sin(theta)cos(theta) is used to integrate products of sin and cos, and the double angle substitution simplifies many definite integrals over trig expressions.