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Math

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Math

Double Angle Formula Calculator.

Calculates the sine, cosine, or tangent of double an angle using the double angle identities.

Results update live as you type.
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Your inputs

How it works

  1. 1

    Enter the angle in degrees.

  2. 2

    Select the trigonometric function (sin, cos, or tan).

  3. 3

    The calculator computes the double angle (2θ) and evaluates the chosen function.

  4. 4

    The result is displayed with four decimal places.

function == 'sin' ? sin(2 * angle_deg * pi / 180) : (function == 'cos' ? cos(2 * angle_deg * pi / 180) : tan(2 * angle_deg * pi / 180))

Frequently asked questions

What are the double angle formulas?

sin(2θ) = 2 sinθ cosθ, cos(2θ) = cos²θ - sin²θ, tan(2θ) = 2 tanθ / (1 - tan²θ).

Why is the result for cos(2*45°) exactly 0?

Because cos(90°) = 0, and 2*45° = 90°.

Can I use angles in radians?

This calculator uses degrees. Convert radians to degrees by multiplying by 180/π.

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Results

Formula checked

Result

0.866

Ask Anything

Estimate for general guidance only — verify important decisions with an appropriate professional.

How it works

Calculates the sine, cosine, or tangent of double an angle using the double angle identities.

  1. Enter the angle in degrees.
  2. Select the trigonometric function (sin, cos, or tan).
  3. The calculator computes the double angle (2θ) and evaluates the chosen function.
  4. The result is displayed with four decimal places.

Formulas

The math behind this calculator, written out so you can verify the result.

Sine double angle

sin(2θ) = 2 sinθ cosθ

The sine of twice an angle equals twice the product of sine and cosine of the original angle.

Example:

Input: θ = 30°

Calculation: 2 * sin(30°) * cos(30°) = 2 * 0.5 * 0.8660

Result: ≈ 0.8660

Cosine double angle

cos(2θ) = cos²θ - sin²θ

The cosine of twice an angle equals the difference between the squares of cosine and sine.

Example:

Input: θ = 45°

Calculation: cos²(45°) - sin²(45°) = 0.5 - 0.5

Result: = 0

Real-world use cases

Where this calculation shows up in everyday life.

Simplifying trigonometric expressions

Use double angle identities to rewrite expressions in simpler forms.

Example: Rewrite sin(2x) as 2 sin x cos x.

Solving trigonometric equations

Convert double angle terms to single angles to solve equations.

Example: Solve sin(2θ) = 0.5 for θ.

Physics and engineering

Model wave interference and harmonic motion where double angles appear.

Example: Calculating the resultant amplitude of two waves.

Tips and common mistakes

Tips

  • Remember that the double angle formula works for any angle, not just acute ones.
  • For tangent, the formula is undefined when tanθ = ±1 (i.e., θ = 45° + 90°k).
  • Use the calculator to check your manual calculations.
  • Angles can be negative; the formulas still hold.

Common Mistakes to Avoid

  • Forgetting to convert degrees to radians when using a calculator in radian mode.
  • Using the wrong sign for cosine double angle: cos(2θ) = cos²θ - sin²θ, not cos²θ + sin²θ.
  • Assuming tan(2θ) = 2 tanθ, which is incorrect.

Assumptions and limitations

  • Use the stated inputs and units.
  • Results are estimates for planning and education.
  • Check measurements and source data before making an important decision.