Key takeaways
- cos 2θ = cos²θ - sin²θ.
- It uses the ORIGINAL angle's sine and cosine, not the doubled angle's.
- Equivalent forms: cos 2θ = 2cos²θ - 1 = 1 - 2sin²θ.
- cos(2θ) when θ = 45° is 0, since cos²(45°) - sin²(45°) = 0.5 - 0.5.
In this guide
What is the double angle formula for cosine?
It's a trigonometric identity that rewrites cos(2θ) using sin θ and cos θ of the original, un-doubled angle. It's one of the most-used identities in calculus, since it helps simplify expressions involving squared sines and cosines.
Why does the double angle formula matter?
Just like with sine, recalculating cos(2θ) from scratch wastes effort you've already put in. If sin θ and cos θ are already known, this identity converts them directly into cos(2θ) with two squarings and a subtraction.
Three equivalent versions of this formula exist —cos²θ - sin²θ,2cos²θ - 1, and1 - 2sin²θ— all give the identical answer, since sin²θ + cos²θ = 1 links them together.
The formula
| Symbol | What it means | Example |
|---|---|---|
| θ (theta) | The original angle, in degrees | 20° |
| cos θ, sin θ | Cosine and sine of the original angle | 0.9397, 0.3420 |
| cos 2θ | Cosine of the doubled angle | 0.7660 |
How to find cos 2θ step by step
- 1Find sin θ and cos θ for your original angle, for example with the sin, cos, tan calculator.
- 2Square the cosine value.
cos²θ. - 3Square the sine value.
sin²θ. - 4Subtract.
cos 2θ = cos²θ - sin²θ.
Enter your own angle below to see cos 2θ calculated step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Squaring the doubled angle instead of the original one. Both
cos θandsin θcome from the original, un-doubled angle θ. - Forgetting other equivalent forms.
cos 2θalso equals2cos²θ - 1or1 - 2sin²θ— all three give the same answer, just written differently. - Confusing cos²θ with cos(θ²). They are not the same calculation at all.
Tips and tricks
- It's the cosine counterpart to the double angle formula for sine, which uses the same original angle to find sin(2θ) instead.
- Sanity-check with θ = 0°:
cos(0°) = 1, andcos²0° - sin²0° = 1 - 0 = 1matches exactly. - The result should always stay between -1 and 1, since it's still a cosine value.
Where you'll use it in real life
- Calculus: simplifying integrals involving squared sine or cosine terms.
- Physics: wave equations and oscillation analysis.
- Engineering: signal processing wherever a frequency or phase angle needs doubling.
- Electronics: power and energy calculations in AC circuit analysis.
Quick summary
cos 2θ = cos²θ - sin²θ reuses the sine and cosine of the original angle to find the cosine of double that angle. Three equivalent forms exist, so pick whichever is easiest with the values you already have. Try the calculator above with your own angle.
Worked example
A physicist has already measured sin(20°) and cos(20°) for a wave calculation and now needs cos(40°) without measuring the doubled angle directly.
- 01cos 2 θ = cos²θ - sin²θ
- 02= cos²20° - sin²20°
- 03= 0.9397² - 0.342²
- 04= 0.883 - 0.117
- 05cos 40° = 0.766
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the double angle formula for cosine?
cos 2θ = cos²θ - sin²θ. It gives the cosine of a doubled angle using only the sine and cosine of the original angle.
Are there other versions of the cosine double angle formula?
Yes — cos 2θ also equals 2cos²θ - 1 and 1 - 2sin²θ. All three are mathematically identical, just rearranged using sin²θ + cos²θ = 1.
What is cos(2θ) when θ = 45°?
cos(90°) = 0, which matches cos²(45°) - sin²(45°) = 0.5 - 0.5 = 0.
How is this different from the double angle formula for sine?
The sine version multiplies sin θ by cos θ; the cosine version subtracts sin²θ from cos²θ. Both start from the same original angle θ.
