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Cos 2 Theta Calculator

Created by Anna Szczepanek, PhD
Reviewed by Komal Rafay
Last updated: Jan 18, 2024


Omni's cos 2 theta calculator is here to help you whenever you have to deal with double angles and cosines.

In the article below we explain where the cos 2 theta identity comes from and what formula for cos 2 theta you should use depending on your data, i.e., whether you know the cosine of half the angle or rather its sine. We also discuss how to find cos 2 theta using this calculator and show you similar Omni tools, so that you can learn even more!

Formula for cos 2 theta

The formula for the cosine of a double angle is a trigonometric identity allowing us to quickly determine the value of the cosine of an angle if we know the cosine or sine value of half of the angle.

The cos 2 theta, or, to write it in a more beautiful way, cos(2θ)\cos(2\theta) identity reads

cos(2θ)=cos2(θ)sin2(θ)\small \cos(2\theta) = \cos^2(\theta)- \sin^2(\theta)

Using the Pythagorean identity sin2(θ)+cos2(θ)=1\sin^2(\theta) + \cos^2(\theta) =1, we can substitute cos2(θ)\cos^2(\theta) with 1sin2(θ)1-\sin^2(\theta), obtaining

cos(2θ)=12sin2(θ)\small \cos(2\theta) = 1- 2\sin^2(\theta)

Alternatively, we can substitute sin2(θ)\sin^2(\theta) with 1cos2(θ)1-\cos^2(\theta), which gives

cos(2θ)=2cos2(θ)1\small \cos(2\theta) = 2\cos^2(\theta)-1

You can choose any of these cos 2 theta identities to solve your cos of double angle problem. Make your choice based on what data is given or... just use Omni's cos 2 theta calculator!

How to use this cos 2 theta calculator?

To use this tool to find cos 2 theta, you only need to enter the angle θ and... admire the result. As for entering the angle, you have three options:

  • degrees;
  • radians; and
  • pi times radians.

Use the last option to input angles like π6\frac \pi6 or 34π\frac 3 4\pi.

🙋 Although the focus of this tool is on the cos 2 theta identity, you can easily switch it to compute the double angle trig formula for sine and tangent as well!

Omni double angle calculators

Happy with the cos 2 theta calculator? Omni features a whole collection of tools dedicated to the double angle trig identities:

FAQ

How do I find cos 4 theta given cos theta?

To calculate cos(4θ) from cos(θ), use the double angle formula twice:

  1. Calculate cos(2θ)= 2cos²(θ) - 1.
  2. Calculate cos(4θ)= 2cos²(2θ) - 1
  3. So the final formula reads cos(4θ)= 2(2cos²(θ) - 1)² - 1.
  4. If you're given sin(θ) instead of cos(θ), follow the same steps but use the formula cos(2θ)= 1- 2sin²(θ).

What is the value of cos(40°) given sin(20°)?

The value of cos(40°) is 0.766. To arrive at this result, we find cos 2 theta via the formula cos(2θ)= 1 - 2sin²(θ) for θ = 20°. Plugging in sin(20°) = 0.342, we obtain cos(2θ)= 1 - 2×0.342² = 0.766, as claimed.

Anna Szczepanek, PhD
Angle θ
deg
Compute...
Double cosine - cos(2θ)
Result
Input the angle...
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