# sin^3(theta) what is the formula?

**URL:** <https://boards.straightdope.com/t/sin-3-theta-what-is-the-formula/63869>\
**Category:** Factual Questions\
**Created:** [April 30, 2001, 5:03pm UTC](https://boards.straightdope.com/t/sin-3-theta-what-is-the-formula/63869 "2001-04-30T17:03:01Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![WarmNPrickly](https://avatars.discourse-cdn.com/v4/letter/w/d78d45/32.png) [@WarmNPrickly](https://boards.straightdope.com/u/WarmNPrickly)\
**Post date:** [April 30, 2001, 5:03pm UTC](https://boards.straightdope.com/t/sin-3-theta-what-is-the-formula/63869/1 "2001-04-30T17:03:01Z")

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I know that sin^2(theta) = 1 - cos 2Xtheta. Is there a similar formula relating sin^3(theta) to sin(3Xtheta) or cos(3Xtheta)?

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**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\
**Post date:** [April 30, 2001, 5:54pm UTC](https://boards.straightdope.com/t/sin-3-theta-what-is-the-formula/63869/2 "2001-04-30T17:54:22Z")

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Christopher:

You left off a factor of 1/2 in your formula for sin^2. Your right hand side gives _twice_ the value of sin^2.

The formulas you’re looking for are:

sin^3(A)=(3/4)sin(A)-(1/4)sin(3A)

cos^3(A)=(3/4) cos(A)+(1/4)cos(3A)  
You can find these formulas in **Fundamental Formulas of Physics** Donald Menzel, ed. Dover Books section 2.5. It’s also in **Table of Integral, Series, and Products** ed. I.S. Gradshteyn and I.M. Ryzhik, Academic Press, section 1.31.

Or you can write the mutiple angle out in complex form and extract the real and imaginary parts.