# goniometric formulas

The goniometric (from Greek $\gamma\omega\nu$í$\alpha$ “angle” and $\mu\varepsilon\tau\varrho\iota\kappa$ó$\varsigma$ “measuring”) concerns the trigonometric functions and their mutual connections. There are a great amount of formulas involving these functions (usually for real arguments).

1. 1.
• $\sin^{2}{x}+\cos^{2}{x}=1$

• $\tan^{2}{x}+1=\sec^{2}{x}$

• $1+\cot^{2}{x}=\csc^{2}{x}$

2. 2.

Fractional identities

• $\displaystyle\tan{x}=\frac{\sin{x}}{\cos{x}}$

• $\displaystyle\cot{x}=\frac{\cos{x}}{\sin{x}}$

• $\displaystyle\cot{x}=\frac{1}{\tan{x}}$

• $\displaystyle\tan{x}=\frac{1}{\cot{x}}$

• $\displaystyle\csc{x}=\frac{1}{\sin{x}}$

• $\displaystyle\sec{x}=\frac{1}{\cos{x}}$

3. 3.

• $\displaystyle\sin{x}=\pm\frac{\tan{x}}{\sqrt{1+\tan^{2}{x}}}$

• $\displaystyle\cos{x}=\pm\frac{1}{\sqrt{1+\tan^{2}{x}}}$

4. 4.

Weierstrass substitution formulas and related formula for $\tan x$

• $\displaystyle\sin{x}=\frac{\displaystyle 2\tan\left(\frac{x}{2}\right)}{% \displaystyle 1+\tan^{2}\left(\frac{x}{2}\right)}$

• $\displaystyle\cos{x}=\frac{\displaystyle 1-\tan^{2}\left(\frac{x}{2}\right)}{% \displaystyle 1+\tan^{2}\left(\frac{x}{2}\right)}$

• $\displaystyle\tan{x}=\frac{\displaystyle 2\tan\left(\frac{x}{2}\right)}{% \displaystyle 1-\tan^{2}\left(\frac{x}{2}\right)}$

5. 5.

Trigonometric functions of a purely imaginary number

• $\sin(ix)=i\sinh x$

• $\cos(ix)=\cosh x$

• $\tan(ix)=i\tanh x$

• $\cot(ix)=i\coth x$

• $\csc(ix)=i\operatorname{csch}x$

• $\sec(ix)=\operatorname{sech}x$

6. 6.

• $\sin(x\pm y)=\sin{x}\cos{y}\pm\cos{x}\sin{y}$

• $\cos(x\pm y)=\cos{x}\cos{y}\mp\sin{x}\sin{y}$

• $\displaystyle\tan(x\pm y)=\frac{\tan{x}\pm\tan{y}}{1\mp\tan{x}\tan{y}}$

7. 7.

Formulas for trigonometric functions of a complex number

• $\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y$

• $\cos(x+iy)=\cos x\cosh y-i\sin x\sinh y$

• $\displaystyle\tan(x+iy)=\frac{\tan x+i\tanh y}{1-i\tan x\tanh y}$

8. 8.

Complement formulas

• $\displaystyle\sin\left(\frac{\pi}{2}-x\right)=\cos{x}$

• $\displaystyle\cos\left(\frac{\pi}{2}-x\right)=\sin{x}$

• $\displaystyle\tan\left(\frac{\pi}{2}-x\right)=\cot{x}$

9. 9.

Supplement formulas

• $\sin(\pi-x)=\sin{x}$

• $\cos(\pi-x)=-\cos{x}$

• $\tan(\pi-x)=-\tan{x}$

10. 10.

Explement formulas

• $\sin(2\pi-x)=-\sin{x}$

• $\cos(2\pi-x)=\cos{x}$

• $\tan(2\pi-x)=-\tan{x}$

11. 11.

angle formulas

• $\sin(-x)=-\sin{x}$

• $\cos(-x)=\cos{x}$

• $\tan(-x)=-\tan{x}$

12. 12.

Periodicity (http://planetmath.org/Periodic) formulas

• $\sin(x+2\pi)=\sin{x}$

• $\cos(x+2\pi)=\cos{x}$

• $\tan(x+\pi)=\tan{x}$

13. 13.
• $\sin(2x)=2\sin{x}\cos{x}$

• $\cos(2x)=\cos^{2}{x}-\sin^{2}{x}=2\cos^{2}{x}-1=1-2\sin^{2}{x}$

• $\displaystyle\tan(2x)=\frac{2\tan{x}}{1-\tan^{2}{x}}$

14. 14.

Triple angle formulas

• $\sin(3x)=3\sin{x}-4\sin^{3}{x}=(4\cos^{2}{x}-1)\sin{x}$

• $\cos(3x)=4\cos^{3}{x}-3\cos{x}=(1-4\sin^{2}{x})\cos{x}$

• $\displaystyle\tan(3x)=\frac{3\tan{x}-\tan^{3}{x}}{1-3\tan^{2}{x}}$

15. 15.

Half angle formulas

• $\displaystyle\sin\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos{x}}{2}}$

• $\displaystyle\cos\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1+\cos{x}}{2}}$

• $\displaystyle\tan\left(\frac{x}{2}\right)=\frac{\sin{x}}{1+\cos{x}}=\frac{1-% \cos{x}}{\sin{x}}=\pm\sqrt{\frac{1-\cos{x}}{1+\cos{x}}}$

16. 16.
• $\displaystyle\sin{x}+\sin{y}=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-% y}{2}\right)$

• $\displaystyle\sin{x}-\sin{y}=2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+% y}{2}\right)$

• $\displaystyle\cos{x}+\cos{y}=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-% y}{2}\right)$

• $\displaystyle\cos{x}-\cos{y}=-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x% -y}{2}\right)$

17. 17.

formulas

• $\displaystyle\sin{x}\,\sin{y}=\frac{\cos(x-y)-\cos(x+y)}{2}$

• $\displaystyle\cos{x}\,\sin{y}=\frac{\sin(x+y)-\sin(x-y)}{2}$

• $\displaystyle\cos{x}\,\cos{y}=\frac{\cos(x-y)+\cos(x+y)}{2}$

18. 18.

Other sums and differences

• $\displaystyle\tan{x}\pm\tan{y}=\frac{\sin(x\pm y)}{\cos{x}\,\cos{y}}$

• $\displaystyle\cot{x}\pm\cot{y}=\frac{\sin(y\pm x)}{\sin{x}\,\sin{y}}$

• $\displaystyle\cos{x}\pm\sin{x}=\sqrt{2}\sin\!\left(\frac{\pi}{4}\pm x\right)=% \sqrt{2}\cos\!\left(\frac{\pi}{4}\mp x\right)$

19. 19.

formulas

• *

$\displaystyle\sin^{2}{x}=\frac{1-\cos(2x)}{2}$

• *

$\displaystyle\cos^{2}{x}=\frac{1+\cos(2x)}{2}$

• *

$\displaystyle\tan^{2}{x}=\frac{1-\cos(2x)}{1+\cos(2x)}$

• Third power

• *

$\displaystyle\sin^{3}{x}=\frac{3\sin{x}-\sin(3x)}{4}$

• *

$\displaystyle\cos^{3}{x}=\frac{3\cos{x}+\cos(3x)}{4}$

• *

$\displaystyle\tan^{3}{x}=\frac{3\sin{x}-\sin(3x)}{3\cos{x}+\cos(3x)}$

• *

$\displaystyle\sin^{4}{x}=\frac{\cos(4x)-4\cos(2x)+3}{8}$

• *

$\displaystyle\cos^{4}{x}=\frac{\cos(4x)+4\cos(2x)+3}{8}$

• *

$\displaystyle\tan^{4}{x}=\frac{\cos(4x)-4\cos(2x)+3}{\cos(4x)+4\cos(2x)+3}$

20. 20.

Recursion formulas

• $\displaystyle\sin[(n\!+\!1)x]=2\cos{x}\,\sin(nx)-\sin[(n\!-\!1)x]$

• $\displaystyle\cos[(n\!+\!1)x]=2\cos{x}\,\cos(nx)-\cos[(n\!-\!1)x]$

21. 21.

Exponential formulas (http://planetmath.org/ExponentialFunction)

• $e^{ix}=\cos x+i\sin x$

• $e^{-ix}=\cos x-i\sin x$

• $\displaystyle\cos x=\frac{e^{ix}+e^{-ix}}{2}$

• $\displaystyle\sin x=\frac{e^{ix}-e^{-ix}}{2i}$

• $\displaystyle\tan x=\frac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}$

22. 22.

Some special formulas

• $\displaystyle\tan{\left(x\!+\!\frac{\pi}{4}\right)}=\frac{\cos{x}+\sin{x}}{% \cos{x}-\sin{x}}=\pm\sqrt{\frac{1+\sin{2x}}{1-\sin{2x}}}$

• $\displaystyle\tan x+\sec x=\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)$

• $\displaystyle\tan\left(\frac{x\pm y}{2}\right)=\frac{\sin{x}\pm\sin{y}}{\cos{x% }+\cos{y}}=\frac{\cos{y}-\cos{x}}{\sin{x}\mp\sin{y}}$

• $\displaystyle\tan\left(\frac{x+y}{2}\right)\,\tan\left(\frac{x-y}{2}\right)=% \frac{\cos{y}-\cos{x}}{\cos{y}+\cos{x}}$

• $\displaystyle\sin(x+y)\,\sin(x-y)=\sin^{2}x-\sin^{2}y$

 Title goniometric formulas Canonical name GoniometricFormulas Date of creation 2013-03-22 17:00:40 Last modified on 2013-03-22 17:00:40 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 39 Author Wkbj79 (1863) Entry type Topic Classification msc 33B10 Classification msc 26A09 Synonym trigonometric identities Synonym goniometric formulae Related topic Trigonometry Related topic DefinitionsInTrigonometry Related topic ExampleOnSolvingAFunctionalEquation Related topic IntegrationOfRationalFunctionOfSineAndCosine Related topic WeierstrassSubstitutionFormulas Related topic TangentOfHalvedAngle Related topic ExampleOfTelescopingSum Related topic DerivativeForParametricForm Related topic ComplementaryAngles Related topic S Defines supplement formula Defines complement formula Defines half angle formula Defines product formula Defines Pythagorean identities