# 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, $\cos(2\theta)$ identity reads

$\small \cos(2\theta) = \cos^2(\theta)- \sin^2(\theta)$

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

$\small \cos(2\theta) = 1- 2\sin^2(\theta)$

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

$\small \cos(2\theta) = 2\cos^2(\theta)-1$

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;

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

🙋 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
Related calculators
Angle θ
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Double cosine - cos(2θ)
Result
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