Double Angle
Free online double angle calculator computes sin(2theta), cos(2theta), and tan(2theta) instantly. Interactive graph shows double-angle functions from 0 deg to 360 deg for trigonometry students and engineers.
About This Calculator
This double angle calculator computes all three primary double-angle trigonometric identities -- sin(2theta), cos(2theta), and tan(2theta) -- for any angle entered in degrees. Double-angle formulas are fundamental in trigonometry, used extensively in calculus, physics, engineering, and signal processing to simplify expressions and solve equations involving double angles.
The calculator uses the standard double-angle identities: sin(2theta) = 2 sin theta cos theta, cos(2theta) = cos^2theta - sin^2theta, and tan(2theta) = 2 tan theta / (1 - tan^2theta). It converts your input from degrees to radians internally, applies the formulas, and displays all results simultaneously. An interactive chart plots the selected function across 0 deg to 360 deg to help visualize the periodic behavior of these identities.
How to use
Enter any angle in degrees (e.g. 30, 45, 60, 90) and click Calculate. The three primary double-angle values appear immediately. Use the chart tabs to toggle between viewing sin(2theta), cos(2theta), or both functions on one graph. Share your calculation via the share button to save or send the link with pre-filled values.
Frequently Asked Questions
What is the double angle formula?
The double angle formulas are trigonometric identities that express sin(2theta), cos(2theta), and tan(2theta) in terms of sin(theta), cos(theta), and tan(theta): sin(2theta) = 2 sin(theta) cos(theta), cos(2theta) = cos^2(theta) - sin^2(theta) = 2 cos^2(theta) - 1 = 1 - 2 sin^2(theta), and tan(2theta) = 2 tan(theta) / (1 - tan^2(theta)).
How do you find sin(2theta) given sin(theta) and cos(theta)?
Use the identity sin(2theta) = 2 sin(theta) cos(theta). Multiply your known sin(theta) and cos(theta) values together, then double the result. For example, if sin(30 deg) = 0.5 and cos(30 deg) = 0.8660, then sin(60 deg) = 2 x 0.5 x 0.8660 = 0.8660.
What are the three forms of cos(2theta)?
cos(2theta) has three equivalent forms: cos^2(theta) - sin^2(theta), 2 cos^2(theta) - 1, and 1 - 2 sin^2(theta). Each form is useful in different contexts -- the first is the standard form, the second helps when you know cos(theta), and the third helps when you know sin(theta).
When is tan(2theta) undefined?
tan(2theta) is undefined when 1 - tan^2(theta) = 0, which occurs at theta = 45 deg + nx90 deg for integer n. At these angles, the denominator of the double-angle tangent formula equals zero, causing the function to approach infinity.
How do double-angle identities relate to the unit circle?
On the unit circle, sin(2theta) represents the y-coordinate and cos(2theta) the x-coordinate of the point at angle 2theta. The double angle formulas show how these coordinates relate to the coordinates at angle theta, effectively describing the geometric effect of doubling an angle on the circle.
Why are double-angle identities important in calculus?
Double-angle identities are essential in calculus for simplifying trigonometric integrals, derivatives, and series expansions. They reduce powers of trig functions, make u-substitutions possible, and help integrate products like sin^2(x) or cos^2(x) by rewriting them as (1 - cos(2x))/2 or (1 + cos(2x))/2.