PRIME

Platonic

Realms

Interactive

Mathematics

Encyclopedia

 

trigonometric identity

The so-called trigonometric identities are a useful set of equations that often allow one to make substitutions in an expression containing trigonometric functions, in order to simplify the expression or to put it in a more useful form. Most trig identities are actually quite easy to derive algebraically from their definitions, and every student of mathematics should derive them all at least once. Thereafter, it will no longer be necessary to memorize them; you will be able to derive them whenever they are needed.

We demonstrate one identity here—a form of the Pythagorean identity—to help you get started.

\[ \begin{eqnarray*} \sin^2 \varphi + \cos^2 \varphi& = & 1 \\ & & \\ & & \\ \left(\frac{\mbox{opp}}{\mbox{rad}}\right)^2 + \left(\frac{\mbox{adj}}{\mbox{rad}}\right)^2 & = & 1 \\ & & \\ & & \\ \frac{\mbox{opp}^2}{\mbox{rad}^2} + \frac{\mbox{adj}^2}{\mbox{rad}^2} & = & 1 \\ & & \\ & & \\ \mbox{opp}^2 + \mbox{adj}^2 & = & \mbox{rad}^2 \\ \end{eqnarray*} \]

…and the last equation follows from the Pythagorean Theorem. The proofs of the other identities listed below are similar. These identities are also available, together with other valuable trig stuff, on a downloadable Trig Reference Sheet.

Pythagorean Identities

\[\begin{eqnarray*} \sin^2 \varphi + \cos^2 \varphi& = & 1 \\ & & \\ & & \\ 1 + \tan^2 \varphi & = & \sec^2 \varphi \\ & & \\ & & \\ 1 + \cot^2 \varphi & = & \csc^2 \varphi \\ \end{eqnarray*} \]

Cofunction Identities

\[ \begin{eqnarray*} \sin\left(\frac{\pi}{2} - \varphi\right) & = & \cos \varphi \\ & & \\ & & \\ \csc\left(\frac{\pi}{2} - \varphi\right) & = & \sec \varphi \\ & & \\ & & \\ \sec\left(\frac{\pi}{2} - \varphi\right) & = & \csc \varphi \\ & & \\ & & \\ \cos\left(\frac{\pi}{2} - \varphi\right) & = & \sin \varphi \\ & & \\ & & \\ \tan\left(\frac{\pi}{2} - \varphi\right) & = & \cot \varphi \\ & & \\ & & \\ \cot\left(\frac{\pi}{2} - \varphi\right) & = & \tan \varphi \\ \end{eqnarray*} \]

Reduction Formulas

\[ \begin{eqnarray*} \sin( - \varphi ) & = & -\sin \varphi \\ & & \\ & & \\ \csc( - \varphi ) & = & -\csc \varphi \\ & & \\ & & \\ \sec( - \varphi ) & = & \sec \varphi \\ & & \\ & & \\ \cos( - \varphi ) & = & \cos \varphi \\ & & \\ & & \\ \tan( - \varphi ) & = & -\tan \varphi \\ & & \\ & & \\ \cot( - \varphi ) & = & -\cot \varphi \\ \end{eqnarray*} \]

Sum and Difference Formulas

\[ \begin{eqnarray*} \sin( \varphi \pm \theta ) & = & \sin \varphi \cos \theta \pm \cos \varphi \sin \theta \\ & & \\ & & \\ \cos( \varphi \pm \theta ) & = & \cos \varphi \cos \theta \mp \sin \varphi \sin \theta \\ & & \\ & & \\ \tan( \varphi \pm \theta ) & = & \frac{\tan \varphi \pm \tan \theta}{1 \mp \tan \varphi \tan \theta} \\ \end{eqnarray*} \]

Double-Angle Formulas

\[ \begin{eqnarray*} \sin 2\varphi & = & 2 \sin \varphi \cos \varphi \\ & & \\ & & \\ \cos 2\varphi & = & \cos^2 \varphi - \sin^2 \varphi \\ & & \\ & & \\ & = & 2 \cos^2 \varphi - 1 \\ & & \\ & & \\ & = & 1 - 2 \sin^2 \varphi \\ & & \\ & & \\ \tan 2\varphi & = & \frac{2 \tan \varphi}{1 - \tan^2 \varphi} \\ \end{eqnarray*} \]

Power-Reducing Formulas

\[ \begin{eqnarray*} \sin^2 \varphi & = & \frac{1 - \cos 2 \varphi}{2} \\ & & \\ & & \\ \cos^2 \varphi & = & \frac{1 + \cos 2 \varphi}{2} \\ & & \\ & & \\ \tan^2 \varphi & = & \frac{1 - \cos 2 \varphi}{1 + \cos 2 \varphi} \\ \end{eqnarray*} \]

Sum-to-Product Formulas

\[ \begin{eqnarray*} \sin \varphi + \sin \theta & = & 2 \sin \left(\frac{\varphi + \theta}{2}\right) \cos \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \sin \varphi - \sin \theta & = & 2 \cos \left(\frac{\varphi + \theta}{2}\right) \sin \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \cos \varphi + \cos \theta & = & 2 \cos \left(\frac{\varphi + \theta}{2}\right) \cos \left(\frac{\varphi - \theta}{2}\right) \\ & & \\ & & \\ \cos \varphi - \cos \theta & = & -2 \sin \left(\frac{\varphi + \theta}{2}\right) \sin \left(\frac{\varphi - \theta}{2}\right) \\ \end{eqnarray*} \]

Product-to-Sum Formulas

\[ \begin{eqnarray*} \sin \varphi \sin \theta & = & \frac{1}{2}\left( \cos (\varphi - \theta) - \cos (\theta + \varphi) \right) \\ & & \\ & & \\ \cos \varphi \cos \theta & = & \frac{1}{2}\left( \cos (\varphi - \theta) + \cos (\theta + \varphi) \right) \\ & & \\ & & \\ \sin \varphi \cos \theta & = & \frac{1}{2}\left( \sin (\varphi + \theta) + \sin (\theta - \varphi) \right) \\ & & \\ & & \\ \cos \varphi \sin \theta & = & \frac{1}{2}\left( \sin (\varphi + \theta) - \sin (\theta - \varphi) \right) \\ \end{eqnarray*} \]

Contributors

  • B. Sidney Smith, author

Citation Info

  • [MLA] Smith, B. Sidney. "trigonometric identity." Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 2 Mar 2014. Web. 2 Mar 2014. <http://platonicrealms.com/>
  • [APA] Smith, B. Sidney (2 Mar 2014). trigonometric identity. Retrieved 2 Mar 2014 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/trigonometric-identity/

Advertisement

Get the ultimate math study-guide Math & Me: Embracing Successproduct thumbnail image Available in the Math Store
detail from Escher pic Belvedere

Are you a mathematical artist?

Platonic Realms is preparing an online gallery space to showcase and market the works of painters, sculptors, and other artists working in a tangible medium.

If your work celebrates mathematical themes we want to hear from you!

Please let us know about yourself using the contact page.

Ask A Nerd | Homework Help

Advertisement