Need to compute sine, cosine, or tangent of twice an angle? This double angle formula calculator computes all six trigonometric functions for 2θ given any input angle θ.
Double Angle Calculator
Using the double angle calculator
Enter your angle in the Angle (θ) field and choose Degrees or Radians. The calculator instantly computes all six trigonometric functions for twice that angle: sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
The result title shows the computed angle – enter 30° and you'll see results for 60°.
What are the double angle formulas?
The double angle formulas express trigonometric functions of 2θ in terms of functions of θ. They're among the most frequently used trig identities.
Double angle formula for sine
Double angle formulas for cosine
Cosine has three equivalent forms (pick your favorite):
Which form you use depends on what information you have. If you only know sine, use the third form. If you only know cosine, use the second.
Double angle formula for tangent
Note that tan(2θ) is undefined when tan²θ = 1, which occurs at θ = 45° + 90°n.
Example: double angle of 30°
Let's compute sin(60°) using the double angle formula with θ = 30°:
This matches the known value of sin(60°) = √3/2.
Deriving the double angle formulas
The double angle formulas follow directly from the angle addition formulas. For sine:
Set A = B = θ:
Similarly, from cos(A + B) = cos A cos B - sin A sin B with A = B = θ:
The other cosine forms come from substituting sin²θ = 1 - cos²θ or cos²θ = 1 - sin²θ.
When to use double angle formulas
Double angle formulas appear frequently in:
- Calculus: Integrating
sin²xorcos²x(rewrite usingcos(2x)) - Physics: Projectile motion (range formula involves
sin(2θ)) - Signal processing: Frequency doubling and modulation
- Solving trig equations: Reducing equations to simpler forms
The power reduction connection
Rearranging the double angle cosine formulas gives the power reduction identities:
These are essential for integrating even powers of trig functions.
