Omni Calculator logo

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:


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
Related calculators
Angle θ
Double cosine - cos(2θ)
Input the angle...
Check out 21 similar trigonometry calculators 📐
ArccosArcsinArctan… 18 more
People also viewed…

Bilinear interpolation

The bilinear interpolation calculator helps you estimate the value of an unknown function based on the method of bilinear interpolation.

Center of ellipse

Using the center of ellipse calculator, you can find the center of an ellipse, if any of the following are known: equation, vertices, co-vertices, or foci.

Christmas tree

Welcome to the Christmas tree calculator, where you will find out how to decorate your Christmas tree in the best way. Take a look at the perfect Christmas tree formula prepared by math professors and improved by physicists. Plan in advance how many lights and decorations you'll need!

Korean age

If you're wondering what would your age be from a Korean perspective, use this Korean age calculator to find out.
Copyright by Omni Calculator sp. z o.o.
Privacy, Cookies & Terms of Service