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:

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...
Check out 22 similar trigonometry calculators 📐
ArccosArcsinArctan… 19 more
People also viewed…

Cubic inches of a cylinder

The cubic inches of a cylinder calculator is an easy tool that helps you determine volume of a cylinder in cubic inches.

Perimeter of a rectangle with given area

Our perimeter of a rectangle with a given area calculator lets you find the perimeter of a rectangle once you know the area and a side (length or width).

Pizza size

Make the best pizza choice with our Pizza Size Calculator – compare sizes and prices for the perfect order!

Secretary problem (Valentine's day)

Use the dating theory calculator to enhance your chances of picking the best lifetime partner.
Copyright by Omni Calculator sp. z o.o.
Privacy, Cookies & Terms of Service