Double Angle Calculator With Steps

Article: Understanding double-angle identities

1. What double-angle means

A double-angle expression uses 2θ instead of θ. These identities convert sin(2θ), cos(2θ), and tan(2θ) into values at θ, which is useful when you already know sin(θ), cos(θ), or tan(θ).

2. Core identity for sine

The key sine identity is sin(2θ)=2sin(θ)cos(θ). If sin(θ)=0.5 and cos(θ)=0.8660254, then sin(2θ)=2×0.5×0.8660254≈0.8660254. This matches the familiar case where θ=30° gives 2θ=60°.

3. Three equivalent cosine forms

Cosine has three common options: cos(2θ)=cos²(θ)−sin²(θ), cos(2θ)=1−2sin²(θ), and cos(2θ)=2cos²(θ)−1. This calculator lets you pick which one appears in the steps, so you can match your textbook method.

4. Tangent and undefined cases

Tangent uses tan(2θ)=2tan(θ)/(1−tan²(θ)). The denominator can be zero. For example, if tan(θ)=1 (θ=45° plus periods), then 1−tan²(θ)=0 and tan(2θ) is undefined because cos(2θ)=0.

5. Degrees, radians, and quick conversion

Degrees and radians describe the same angle with different scales. Use rad=deg×π/180. For instance, 30° equals π/6≈0.523599 rad, and doubling gives 60° or π/3≈1.047198 rad. Switching units changes input meaning, not the underlying trigonometry.

6. Using sin and cos inputs

When you enter sin(θ) and cos(θ), the tool checks sin²(θ)+cos²(θ). Perfect inputs give 1, but rounded values may produce results like 0.9999 or 1.0001. Small drift is normal and affects the last few decimal places.

7. Using tan-only input

Tangent alone does not uniquely determine sin(θ) and cos(θ) signs, but double-angle values can still be computed using tan-based identities: sin(2θ)=2t/(1+t²) and cos(2θ)=(1−t²)/(1+t²) where t=tan(θ).

8. Where these identities help

Double-angle formulas simplify integrals, solve trigonometric equations, and reduce expression complexity in geometry and physics. They also power quick checks: if θ=15°, then 2θ=30°, so sin(2θ)=0.5 and cos(2θ)=0.8660254. Use the step list to verify each substitution line by line.

FAQs

1) Which formula for cos(2θ) is best?

The best form depends on what you know. If sin(θ) is known, use 1−2sin²(θ). If cos(θ) is known, use 2cos²(θ)−1. Otherwise, use cos²(θ)−sin²(θ).

2) Why can tan(2θ) be undefined?

tan(2θ)=sin(2θ)/cos(2θ). If cos(2θ)=0, division by zero occurs and the tangent is undefined. This happens at 2θ=90°+180°k, such as θ=45°+90°k.

3) Can I compute using only tan(θ)?

Yes. The calculator uses sin(2θ)=2t/(1+t²) and cos(2θ)=(1−t²)/(1+t²), then tan(2θ)=2t/(1−t²), where t=tan(θ). This avoids needing θ itself.

4) What if my sin and cos do not satisfy sin²+cos²=1?

Rounded or measured inputs often deviate slightly. The tool shows the check value so you can judge quality. Large deviations suggest inconsistent inputs, which can noticeably change the computed double-angle values.

5) Does changing decimals change the true result?

No. Decimals only control rounding of displayed numbers and steps. Internally the calculator uses full floating-point precision, then formats the output to your chosen decimal places or scientific notation.

6) How do I convert degrees to radians quickly?

Multiply degrees by π/180. For example, 60°×π/180=π/3≈1.047198. To convert radians to degrees, multiply by 180/π.

7) Are results always between −1 and 1?

sin(2θ) and cos(2θ) stay within −1 to 1. tan(2θ) can be any real number and can blow up near angles where cos(2θ) is close to zero, making it undefined at exact zeros.