Double Angle Formula Calculator
Free Double angle formula Calculator for trigonometry. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
sin(2A) = 2sin(A)cos(A), cos(2A) = cos2(A) - sin2(A), tan(2A) = 2tan(A)/(1 - tan2(A))
The double angle formulas express sin, cos, and tan of twice an angle using the original angle values. cos(2A) has three equivalent forms using Pythagorean substitutions.
Worked Examples
Example 1: Double Angle of 30 Degrees
Problem:Calculate sin(60), cos(60), and tan(60) using the double angle formulas with A = 30 degrees.
Solution:sin(30) = 0.5, cos(30) = 0.86602540\nsin(60) = 2 sin(30) cos(30) = 2(0.5)(0.86602540) = 0.86602540\ncos(60) = cos2(30) - sin2(30) = 0.75 - 0.25 = 0.50000000\ntan(30) = 0.57735027\ntan(60) = 2(0.57735027)/(1 - 0.333333) = 1.15470054/0.666667 = 1.73205081\nAll match direct computation: sin(60) = 0.86602540, cos(60) = 0.5, tan(60) = 1.73205081
Result:sin(60) = 0.86602540 | cos(60) = 0.5 | tan(60) = 1.73205081
Example 2: Double Angle of 45 Degrees
Problem:Use double angle formulas with A = 45 degrees to find sin(90), cos(90), and tan(90).
Solution:sin(45) = cos(45) = sqrt(2)/2 = 0.70710678\nsin(90) = 2 sin(45) cos(45) = 2(0.70710678)(0.70710678) = 1.0\ncos(90) = cos2(45) - sin2(45) = 0.5 - 0.5 = 0.0\ntan(45) = 1.0\ntan(90) = 2(1)/(1 - 1) = 2/0 = undefined\nAll results confirmed: sin(90) = 1, cos(90) = 0, tan(90) is undefined.
Result:sin(90) = 1.0 | cos(90) = 0.0 | tan(90) = undefined
Frequently Asked Questions
What are the double angle formulas?
The double angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. The three main formulas are: sin(2A) = 2sin(A)cos(A), cos(2A) = cos2(A) - sin2(A), and tan(2A) = 2tan(A)/(1 - tan2(A)). The cosine double angle formula has two additional equivalent forms: cos(2A) = 2cos2(A) - 1 and cos(2A) = 1 - 2sin2(A). These formulas are derived from the angle addition formulas by setting both angles equal. They are among the most frequently used identities in trigonometry and appear throughout calculus, physics, and engineering applications.
How is the double angle formula for sine derived?
The double angle formula for sine is derived from the angle addition formula sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By setting B = A, we get sin(A + A) = sin(A)cos(A) + cos(A)sin(A) = 2sin(A)cos(A). Therefore sin(2A) = 2sin(A)cos(A). This elegant formula shows that the sine of a doubled angle depends on both the sine and cosine of the original angle multiplied together and doubled. Geometrically, this can be visualized using the unit circle: doubling the angle creates a specific relationship between the original and doubled positions. The formula is particularly useful in integration, where products of sine and cosine can be converted to a single sine term.
Why are there three forms of the cosine double angle formula?
The three forms of cos(2A) arise from the Pythagorean identity sin2(A) + cos2(A) = 1. The base form cos(2A) = cos2(A) - sin2(A) comes directly from the addition formula. Replacing sin2(A) with 1 - cos2(A) gives cos(2A) = 2cos2(A) - 1, which is useful when you only know the cosine. Replacing cos2(A) with 1 - sin2(A) gives cos(2A) = 1 - 2sin2(A), useful when you only know the sine. Having three equivalent forms provides flexibility in solving problems. In calculus, the form 2cos2(A) - 1 is rearranged to cos2(A) = (1 + cos(2A))/2 for the power reduction formula used extensively in integration of even powers of cosine.
When is the tangent double angle formula undefined?
The tangent double angle formula tan(2A) = 2tan(A)/(1 - tan2(A)) is undefined in two situations. First, when tan(A) itself is undefined, which happens at A = 90 degrees plus multiples of 180 degrees (odd multiples of pi/2). Second, when the denominator 1 - tan2(A) equals zero, which occurs when tan2(A) = 1, meaning tan(A) = 1 or tan(A) = -1. This happens at A = 45 degrees plus multiples of 90 degrees. At these angles, 2A equals 90 degrees plus multiples of 180 degrees, where tangent is indeed undefined. Understanding these restrictions is important for avoiding division-by-zero errors when applying the formula in calculations and proofs.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy