Double Angle Formula
Free online double angle formula calculator computes sin(2theta), cos(2theta), and tan(2theta) using standard double-angle identities with formula derivation and interactive function graph.
About This Calculator
This double angle formula calculator evaluates all three primary double-angle trigonometric identities simultaneously. Enter any angle in degrees, and the calculator applies the standard formulas -- sin(2theta) = 2 sin theta cos theta, cos(2theta) = cos^2theta - sin^2theta, and tan(2theta) = 2 tan theta / (1 - tan^2theta) -- to compute precise results displayed to six decimal places.
Double-angle formulas are derived from the angle addition identities by setting both angles equal. They are foundational in trigonometry and are widely applied in calculus for integration techniques, in physics for projectile motion and wave analysis, and in engineering for signal processing and control systems.
Using the calculator
Enter an angle in degrees, click Calculate, and all three double-angle values appear in the results panel. Toggle the chart tabs to visualize sin(2theta), cos(2theta), or both functions plotted from 0 deg to 360 deg. The URL updates automatically so you can share or bookmark specific calculations.
Frequently Asked Questions
What is the double angle formula for sine?
The double angle formula for sine is sin(2theta) = 2 sin(theta) cos(theta). This identity is derived from the sine addition formula sin(a + b) = sin(a) cos(b) + cos(a) sin(b) by setting a = b = theta.
How is cos(2theta) derived from the addition formulas?
cos(2theta) is derived from cos(a+b) = cos(a) cos(b) - sin(a) sin(b). Setting a = b = theta gives cos(2theta) = cos^2(theta) - sin^2(theta). Using the identity cos^2(theta) + sin^2(theta) = 1, this can be rewritten as 2 cos^2(theta) - 1 or 1 - 2 sin^2(theta).
What is tan(2theta) formula?
The double angle formula for tangent is tan(2theta) = 2 tan(theta) / (1 - tan^2(theta)). This is derived from the tangent addition formula and is valid for all theta where tan(theta) is defined and tan^2(theta) ≠ 1.
Can double-angle formulas be used with radians?
Yes, double-angle formulas work identically for radians. Simply substitute the radian measure for theta. For example, sin(2 x pi/3) = 2 sin(pi/3) cos(pi/3). This calculator accepts degrees and converts internally, but you can also think of the output in radian terms.
How are double-angle formulas used in physics?
In physics, double-angle formulas appear in projectile motion (range equation R = (v^2 sin(2theta))/g), wave interference, simple harmonic motion, and alternating current analysis. They simplify expressions involving 2omegat in AC circuits and wave superposition problems.
What is the relationship between double-angle and half-angle formulas?
Double-angle and half-angle formulas are inverses of each other. Setting φ = 2theta in the double-angle formulas gives the half-angle formulas. For example, since cos(2theta) = 1 - 2 sin^2(theta), solving for sin(theta) gives sin(theta/2) = +/-sqrt((1 - cos(theta))/2).