Double Angle Formula Calculator
Calculate sin(2x), cos(2x), and tan(2x) using double angle identities — enter an angle in degrees or radians for instant results.
Enter an angle, select the unit, and choose which double angle formulas to display.
Double Angle Formula Calculator
Calculate sin(2x), cos(2x), and tan(2x) using double angle identities — enter an angle in degrees or radians for instant results.
About the double angle formula calculator
Double angle formulas are trigonometric identities that express sin(2x), cos(2x), and tan(2x) in terms of sin(x) and cos(x). They are among the most frequently used identities in trigonometry, calculus, physics, and engineering, because they let you reduce the argument of a trigonometric function by a factor of two.
The three core double angle identities are: sin(2x) = 2 sin(x) cos(x); cos(2x) = cos²(x) − sin²(x), which can also be written as 2cos²(x) − 1 or 1 − 2sin²(x); and tan(2x) = 2tan(x) / (1 − tan²(x)), which is undefined whenever cos(2x) = 0, i.e., when 2x = 90°, 270°, etc.
The sine double angle formula sin(2x) = 2 sin(x) cos(x) follows directly from the angle addition formula sin(a + b) = sin(a)cos(b) + cos(a)sin(b) when you set a = b = x. The same approach applied to cos(a + b) = cos(a)cos(b) − sin(a)sin(b) yields cos(2x) = cos²(x) − sin²(x). Using the Pythagorean identity sin²(x) + cos²(x) = 1, you can substitute sin²(x) = 1 − cos²(x) to get cos(2x) = 2cos²(x) − 1, or substitute cos²(x) = 1 − sin²(x) to get cos(2x) = 1 − 2sin²(x). All three forms of the cosine double angle formula are equivalent and useful in different contexts.
In calculus, double angle formulas are essential for integrating products of sine and cosine. For example, the integral of sin(x)cos(x) is simplified by recognizing that the integrand equals (1/2)sin(2x), making it straightforward to anti-differentiate. Similarly, integrals of sin²(x) and cos²(x) are handled by rewriting them using the half-angle forms derived from the double angle cosine formula.
In physics, double angle identities appear in wave mechanics, optics, and mechanics. The range formula for projectile motion, R = (v²/g)sin(2θ), uses the sine double angle to express the maximum range as a function of launch angle. Optical interference patterns, harmonic oscillators, and rotating machinery all involve combinations of trigonometric functions where double angle identities simplify the analysis.
This double angle formula calculator accepts any angle in degrees or radians. It converts the input to radians internally, computes sin(x) and cos(x), then applies the identities to produce sin(2x), cos(2x), and tan(2x). When tan(2x) is undefined (when the double angle is an odd multiple of 90°), the calculator displays an explicit "Undefined" message rather than a large or misleading number. Results are displayed to ten significant digits for precision.
Double angle formula examples
Common reference angles with their exact or highly precise double angle values.
| Angle (x) | sin(2x) / cos(2x) / tan(2x) | Notes |
|---|---|---|
| x = 30° | sin(60°) = 0.866, cos(60°) = 0.5, tan(60°) = 1.732 | sin(2×30°) = 2 sin30° cos30° = 2 × 0.5 × 0.866 = 0.866. A common reference angle with exact values. |
| x = 45° | sin(90°) = 1, cos(90°) = 0, tan(90°) = Undefined | Doubling 45° gives 90°. sin(90°) = 1, cos(90°) = 0. Tangent is undefined since cos(90°) = 0. |
| x = 60° | sin(120°) = 0.866, cos(120°) = −0.5, tan(120°) = −1.732 | The double angle 120° is in the second quadrant: sine is positive, cosine is negative, tangent is negative. |
| x = π/6 rad (≈ 0.5236) | sin(π/3) ≈ 0.866, cos(π/3) = 0.5, tan(π/3) ≈ 1.732 | π/6 radians equals 30°. Result is identical to the first example, confirming the unit conversion. |
How to use the double angle formula calculator
- Enter the angle value x in the Angle field. Any real number is accepted — positive, negative, or zero.
- Select the unit: Degrees for common angles like 30°, 45°, 60°, or Radians for values like π/6.
- Choose the Formula Type: All Formulas displays sin(2x), cos(2x), and tan(2x); or select a single formula if you need only one result.
- Click Calculate. The results panel shows the computed values for the selected formulas and marks tan(2x) as Undefined when applicable.
- Click Reset to clear the inputs, or change the angle and unit to explore different values.
Double angle formula calculator FAQ
What is the double angle formula for sine?
The double angle formula for sine is sin(2x) = 2 sin(x) cos(x). It is derived from the sine addition formula sin(a + b) = sin(a)cos(b) + cos(a)sin(b) by setting both a and b equal to x. This identity is used in integration, physics, and signal processing.
Why are there three versions of the cosine double angle formula?
All three versions — cos(2x) = cos²x − sin²x, cos(2x) = 2cos²x − 1, and cos(2x) = 1 − 2sin²x — are equivalent. The first comes directly from the cosine addition formula. The other two are obtained by substituting the Pythagorean identity sin²x + cos²x = 1. Different forms are useful in different integration and simplification contexts.
When is tan(2x) undefined?
tan(2x) is undefined whenever cos(2x) = 0, which occurs when 2x = 90° + 180°k for any integer k, i.e., when x = 45° + 90°k. At these angles, the formula tan(2x) = 2tan(x)/(1 − tan²x) involves division by zero, and the tangent function itself approaches ±∞.
How are double angle formulas used in calculus?
Double angle formulas are essential for evaluating integrals of powers of trigonometric functions. For example, ∫sin²(x)dx = ∫(1 − cos(2x))/2 dx, which is straightforward to integrate. Without these identities, such integrals would require much more complex techniques.
Can double angle formulas be applied to negative angles?
Yes. Since sin and cos are defined for all real numbers, the double angle formulas work for negative angles too. For example, sin(2 × (−30°)) = sin(−60°) = −sin(60°) ≈ −0.866. The calculator accepts any real number for the angle input.
What is the relationship between double angle formulas and half-angle formulas?
Half-angle formulas are derived by replacing x with x/2 in the double angle formulas. For instance, from cos(2x) = 1 − 2sin²x, substituting x → x/2 gives cos(x) = 1 − 2sin²(x/2), which rearranges to sin²(x/2) = (1 − cos x)/2. These half-angle formulas are useful for computing trigonometric values of angles that are half of familiar reference angles.