The Double Angle Formula for Cosine
The double angle identity for cosine expresses cos(2θ) in terms of either cosine or sine of the original angle θ. This identity has three equivalent forms, each useful depending on what information you already have.
cos(2θ) = cos²(θ) − sin²(θ)
cos(2θ) = 2cos²(θ) − 1
cos(2θ) = 1 − 2sin²(θ)
θ— The original angle in degrees or radianscos(θ)— The cosine of angle θsin(θ)— The sine of angle θ
Understanding the Three Forms
All three formulas are mathematically equivalent, derived from the Pythagorean identity sin²(θ) + cos²(θ) = 1. Choose whichever form matches your input data:
- First form: cos(2θ) = cos²(θ) − sin²(θ) is most direct when you know both cosine and sine of θ.
- Second form: cos(2θ) = 2cos²(θ) − 1 works best if you only have cos(θ).
- Third form: cos(2θ) = 1 − 2sin²(θ) is ideal when sin(θ) is your only known value.
Each rearrangement eliminates one trigonometric function, making calculations more practical depending on what the problem provides.
How to Use the Calculator
Enter your angle θ using one of three input modes: degrees (e.g., 30), radians (e.g., π/6), or a multiple of π (enter 1/6 for π/6). You can also input known values of sin(θ) or cos(θ) directly, and the calculator will compute cos(2θ) along with sin(2θ) and tan(2θ) if needed.
The calculator automatically recognises special angles like 30°, 45°, and 60°, displaying exact values (surds) rather than decimal approximations. This precision is valuable when solving equations or simplifying algebraic expressions.
Extending to Higher Multiples
Need cos(4θ) or cos(6θ)? Apply the double angle formula repeatedly. For cos(4θ), first calculate cos(2θ) = 2cos²(θ) − 1, then treat this result as your new input and apply the formula again: cos(4θ) = 2cos²(2θ) − 1. This nested approach generalises to any power of 2.
For example, if cos(θ) = 0.8, then cos(2θ) = 2(0.8)² − 1 = 0.28, and cos(4θ) = 2(0.28)² − 1 ≈ −0.843.
Common Pitfalls and Tips
Avoid these mistakes when working with the double angle formula:
- Confusing angle units — Always check whether your angle is in degrees or radians before entering it. cos(2θ) in radians differs from cos(2θ) when θ is measured in degrees. The calculator provides both units to prevent this error.
- Mixing forms of the formula — Use only one form per calculation. Blending forms—like computing cos²(θ) from one formula and sin²(θ) from another—introduces errors if rounding occurs between steps. Stick with the form that matches your inputs.
- Rounding intermediate values — When working by hand, avoid rounding sin(θ) or cos(θ) until the final step. For instance, using sin(20°) = 0.342 gives cos(40°) ≈ 0.766, but premature rounding to 0.34 shifts the answer noticeably. Maintain extra decimal places throughout.
- Sign changes near 90° — Cosine of a doubled angle can oscillate rapidly between positive and negative values. At θ = 45°, cos(90°) = 0. Just beyond, cos(2θ) becomes negative. Verify the sign of your result makes sense geometrically.