goniometric formulas

The goniometric (from Greek $\gamma \omega \nu$í$\alpha$ “angle” and $\mu \epsilon \tau \varrho \iota \kappa$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. Pythagorean identities

   • ${\sin }^{2}x+{\cos }^{2}x=1$

   • ${\tan }^{2}x+1={\sec }^{2}x$

   • $1+{\cot }^{2}x={\csc }^{2}x$

2. Fractional identities

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

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

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

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

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

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

3. Formulas involving radicals (http://planetmath.org/Radical6)

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

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

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

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

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

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

5. Trigonometric functions of a purely imaginary number

   • $\sin (ix)=i\sinh x$

   • $\cos (ix)=\cosh x$

   • $\tan (ix)=i\tanh x$

   • $\cot (ix)=i\coth x$

   • $\csc (ix)=i\mathrm{csch}x$

   • $\sec (ix)=\mathrm{sech}x$

6. Addition formulas and subtraction formulas (http://planetmath.org/AdditionFormulasForSineAndCosine)

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

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

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

7. Formulas for trigonometric functions of a complex number

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

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

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

8. Complement formulas

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

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

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

9. Supplement formulas

   • $\sin (\pi -x)=\sin x$

   • $\cos (\pi -x)=-\cos x$

   • $\tan (\pi -x)=-\tan x$

10. Explement formulas

    • $\sin (2\pi -x)=-\sin x$

    • $\cos (2\pi -x)=\cos x$

    • $\tan (2\pi -x)=-\tan x$

11. angle formulas

    • $\sin (-x)=-\sin x$

    • $\cos (-x)=\cos x$

    • $\tan (-x)=-\tan x$

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. Double angle formulas

    • $\sin (2x)=2\sin x\cos x$

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

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

14. Triple angle formulas

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

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

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

15. Half angle formulas

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

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

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

16. Prosthaphaeresis formulas

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

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

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

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

17. formulas

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

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

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

18. Other sums and differences

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

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

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

19. formulas

    • Second power

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

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

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

    • Third power

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

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

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

    • Fourth power

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

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

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

20. Recursion formulas

    • $\sin [(n+1)x]=2\cos x\sin (nx)-\sin [(n-1)x]$

    • $\cos [(n+1)x]=2\cos x\cos (nx)-\cos [(n-1)x]$

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

    • $e^{ix}=\cos x+i\sin x$

    • $e^{-ix}=\cos x-i\sin x$

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

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

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

22. Some special formulas

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

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

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

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

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

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