How do I find cos 4 theta given cos theta?

To calculate cos(4θ) from cos(θ), use the double angle formula twice: Calculate cos(2θ)= 2cos²(θ) - 1. Calculate cos(4θ)= 2cos²(2θ) - 1 So the final formula reads cos(4θ)= 2(2cos²(θ) - 1)² - 1. 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.

Cos 2 Theta Calculator

Last updated: May 24, 2025

Loading calculator…

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 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, $cos (2 θ)$ cos(2θ) identity reads

cos (2 θ) = cos^{2} (θ) - sin^{2} (θ)

cos(2θ)=cos2(θ)−sin2(θ)

Using the Pythagorean identity $sin^{2} (θ) +$ cos2(θ)=1sin2(θ)+cos2(θ)=1, we can substitute $cos^{2} (θ)$ cos2(θ) with $1 -$ sin2(θ)1−sin2(θ), obtaining

cos (2 θ) = 1 - 2 sin^{2} (θ)

cos(2θ)=1−2sin2(θ)

Alternatively, we can substitute $sin^{2} (θ)$ sin2(θ) with $1 -$ cos2(θ)1−cos2(θ), which gives

cos (2 θ) = 2 cos^{2} (θ) - 1

cos(2θ)=2cos2(θ)−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 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 $\frac{π}{6}$ 6π or $\frac{3}{4} π$ 43π.

🙋 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!

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

The double angle formula calculator;
The double angle calculator;
The double angle identities calculator; and
The sin 2 theta calculator.

FAQs

How do I find cos 4 theta given cos theta?: To calculate cos(4θ) from cos(θ), use the double angle formula twice: Calculate cos(2θ)= 2cos²(θ) - 1. Calculate cos(4θ)= 2cos²(2θ) - 1 So the final formula reads cos(4θ)= 2(2cos²(θ) - 1)² - 1. 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.

Cos 2 Theta Calculator

Formula for cos 2 theta

How to use this cos 2 theta calculator?

FAQs

Related calculators

Remainder

Slope

Average

Circumference