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[parent] goniometric formulas (Topic)

The word 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. 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
    • $ \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. Formulas involving radicals
    • $ \displaystyle \sin{x} = \pm\frac{\tan{x}}{\sqrt{1+\tan^2{x}}}$
    • $ \displaystyle \cos{x} = \pm\frac{1}{\sqrt{1+\tan^2{x}}}$
  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. 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. Addition formulas and subtraction formulas
    • $ \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. 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. 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. 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. Opposite angle formulas
    • $ \sin(-x) = -\sin{x}$
    • $ \cos(-x) = \cos{x}$
    • $ \tan(-x) = -\tan{x}$
  12. Periodicity 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}$
    • $ \displaystyle \tan(2x) = \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}$
    • $ \displaystyle \tan(3x) = \frac{3\tan{x}-\tan^3{x}}{1-3\tan^2{x}}$
  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. Prosthaphaeresis formulas
    • $ \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. Product 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. 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. Linearization formulas
  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. Exponential formulas
    • $ 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. 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^2x-\sin^2y$



"goniometric formulas" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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See Also: trigonometry, definitions in trigonometry, example of solving a functional equation, integration of rational function of sine and cosine, Weierstrass substitution formulas, tangent of halved angle, example of telescoping sum, derivative for parametric form, complementary angles, supplementary angles, cycloid, hyperbolic identities

Other names:  trigonometric identities, goniometric formulae
Also defines:  supplement formula, complement formula, half angle formula, product formula, Pythagorean identities

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Cross-references: power, third power, second power, differences, sums, Prosthaphaeresis formulas, double angle formulas, angle, complex number, purely imaginary number, Weierstrass substitution formulas, identities, arguments, real, functions, trigonometric functions
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This is version 35 of goniometric formulas, born on 2007-04-28, modified 2008-02-23.
Object id is 9293, canonical name is GoniometricFormulae.
Accessed 3743 times total.

Classification:
AMS MSC: 26A09 (Real functions :: Functions of one variable :: Elementary functions)
 33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)
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