goniometric formulas
The goniometric (from Greek γωνíα “angle” and μετϱικóς “measuring”) concerns the trigonometric functions and their mutual connections. There are a great amount of formulas involving these functions
(usually for real arguments
).
-
1.
-
–
sin2x+cos2x=1
-
–
tan2x+1=sec2x
-
–
1+cot2x=csc2x
-
–
-
2.
Fractional identities
-
–
tanx=sinxcosx
-
–
cotx=cosxsinx
-
–
cotx=1tanx
-
–
tanx=1cotx
-
–
cscx=1sinx
-
–
secx=1cosx
-
–
-
3.
Formulas involving radicals
(http://planetmath.org/Radical6)
-
–
sinx=±tanx√1+tan2x
-
–
cosx=±1√1+tan2x
-
–
-
4.
Weierstrass substitution formulas and related formula for tanx
-
–
sinx=2tan(x2)1+tan2(x2)
-
–
cosx=1-tan2(x2)1+tan2(x2)
-
–
tanx=2tan(x2)1-tan2(x2)
-
–
-
5.
Trigonometric functions of a purely imaginary number
-
–
sin(ix)=isinhx
-
–
cos(ix)=coshx
-
–
tan(ix)=itanhx
-
–
cot(ix)=icothx
-
–
csc(ix)=icschx
-
–
sec(ix)=sechx
-
–
-
6.
Addition formulas
and subtraction formulas (http://planetmath.org/AdditionFormulasForSineAndCosine)
-
–
sin(x±y)=sinxcosy±cosxsiny
-
–
cos(x±y)=cosxcosy∓sinxsiny
-
–
tan(x±y)=tanx±tany1∓tanxtany
-
–
-
7.
Formulas for trigonometric functions of a complex number
-
–
sin(x+iy)=sinxcoshy+icosxsinhy
-
–
cos(x+iy)=cosxcoshy-isinxsinhy
-
–
tan(x+iy)=tanx+itanhy1-itanxtanhy
-
–
-
8.
Complement formulas
-
–
sin(π2-x)=cosx
-
–
cos(π2-x)=sinx
-
–
tan(π2-x)=cotx
-
–
-
9.
Supplement formulas
-
–
sin(π-x)=sinx
-
–
cos(π-x)=-cosx
-
–
tan(π-x)=-tanx
-
–
-
10.
Explement formulas
-
–
sin(2π-x)=-sinx
-
–
cos(2π-x)=cosx
-
–
tan(2π-x)=-tanx
-
–
-
11.
angle formulas
-
–
sin(-x)=-sinx
-
–
cos(-x)=cosx
-
–
tan(-x)=-tanx
-
–
-
12.
Periodicity (http://planetmath.org/Periodic) formulas
-
–
sin(x+2π)=sinx
-
–
cos(x+2π)=cosx
-
–
tan(x+π)=tanx
-
–
-
13.
-
–
sin(2x)=2sinxcosx
-
–
cos(2x)=cos2x-sin2x=2cos2x-1=1-2sin2x
-
–
tan(2x)=2tanx1-tan2x
-
–
-
14.
Triple angle formulas
-
–
sin(3x)=3sinx-4sin3x=(4cos2x-1)sinx
-
–
cos(3x)=4cos3x-3cosx=(1-4sin2x)cosx
-
–
tan(3x)=3tanx-tan3x1-3tan2x
-
–
-
15.
Half angle formulas
-
–
sin(x2)=±√1-cosx2
-
–
cos(x2)=±√1+cosx2
-
–
tan(x2)=sinx1+cosx=1-cosxsinx=±√1-cosx1+cosx
-
–
-
16.
-
–
sinx+siny=2sin(x+y2)cos(x-y2)
-
–
sinx-siny=2sin(x-y2)cos(x+y2)
-
–
cosx+cosy=2cos(x+y2)cos(x-y2)
-
–
cosx-cosy=-2sin(x+y2)sin(x-y2)
-
–
-
17.
formulas
-
–
sinxsiny=cos(x-y)-cos(x+y)2
-
–
cosxsiny=sin(x+y)-sin(x-y)2
-
–
cosxcosy=cos(x-y)+cos(x+y)2
-
–
-
18.
Other sums and differences
-
–
tanx±tany=sin(x±y)cosxcosy
-
–
cotx±coty=sin(y±x)sinxsiny
-
–
cosx±sinx=√2sin(π4±x)=√2cos(π4∓x)
-
–
-
19.
formulas
-
–
-
*
sin2x=1-cos(2x)2
-
*
cos2x=1+cos(2x)2
-
*
tan2x=1-cos(2x)1+cos(2x)
-
*
-
–
Third power
-
*
sin3x=3sinx-sin(3x)4
-
*
cos3x=3cosx+cos(3x)4
-
*
tan3x=3sinx-sin(3x)3cosx+cos(3x)
-
*
-
–
-
*
sin4x=cos(4x)-4cos(2x)+38
-
*
cos4x=cos(4x)+4cos(2x)+38
-
*
tan4x=cos(4x)-4cos(2x)+3cos(4x)+4cos(2x)+3
-
*
-
–
-
20.
Recursion formulas
-
–
sin[(n+1)x]=2cosxsin(nx)-sin[(n-1)x]
-
–
cos[(n+1)x]=2cosxcos(nx)-cos[(n-1)x]
-
–
-
21.
Exponential formulas (http://planetmath.org/ExponentialFunction)
-
–
eix=cosx+isinx
-
–
e-ix=cosx-isinx
-
–
cosx=eix+e-ix2
-
–
sinx=eix-e-ix2i
-
–
tanx=eix-e-ixi(eix+e-ix)
-
–
-
22.
Some special formulas
-
–
tan(x+π4)=cosx+sinxcosx-sinx=±√1+sin2x1-sin2x
-
–
tanx+secx=tan(x2+π4)
-
–
tan(x±y2)=sinx±sinycosx+cosy=cosy-cosxsinx∓siny
-
–
tan(x+y2)tan(x-y2)=cosy-cosxcosy+cosx
-
–
sin(x+y)sin(x-y)=sin2x-sin2y
-
–
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 |