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Trigonometry Cheat Sheet

 

Basic Identities

tan(x)=sin(x)cos(x)  tan(x)=1cot(x) 
cot(x)=1tan(x)  cot(x)=cos(x)sin(x) 
sec(x)=1cos(x)  csc(x)=1sin(x) 


Pythagorean Identities

cos2(x)+sin2(x)=1 sec2(x)tan2(x)=1
csc2(x)cot2(x)=1


Double-Angle Identities

sin(2x)=2sin(x)cos(x) cos(2x)=12sin2(x)
cos(2x)=2cos2(x)1 cos(2x)=cos2(x)sin2(x)
tan(2x)=2tan(x)1tan2(x) 


Sum/Difference Identities

sin(s+t)=sin(s)cos(t)+cos(s)sin(t)
sin(st)=sin(s)cos(t)cos(s)sin(t)
cos(s+t)=cos(s)cos(t)sin(s)sin(t)
cos(st)=cos(s)cos(t)+sin(s)sin(t)
tan(s+t)=tan(s)+tan(t)1tan(s)tan(t) 
tan(st)=tan(s)tan(t)1+tan(s)tan(t) 


Product-To-Sum Identities

cos(s)cos(t)=cos(st)+cos(s+t)2 
sin(s)sin(t)=cos(st)cos(s+t)2 
sin(s)cos(t)=sin(s+t)+sin(st)2 
cos(s)sin(t)=sin(s+t)sin(st)2 


Triple-Angle Identities

sin(3x)=sin3(x)+3cos2(x)sin(x)
sin(3x)=4sin3(x)+3sin(x)
cos(3x)=cos3(x)3sin2(x)cos(x)
cos(3x)=4cos3(x)3cos(x)
tan(3x)=3tan(x)tan3(x)13tan2(x) 
cot(3x)=3cot(x)cot3(x)13cot2(x) 


Function Ranges

y=sin(x) 1y1
y=cos(x) 1y1
y=tan(x) <y<
y=cot(x) <y<
y=csc(x) <y1 1y<
y=sec(y) <y1 1y<
y=arcsin(x) π 2  y π 2 
y=arccos(x) 0 y π
y=arctan(x) π 2 < y<π 2 :
y=arccot(x) 0<x<π
y=arccsc(x) 0y<π2  πy<3π2 
y=arcsec(x) π<yπ2  0<y<π2 <


Function Values

sin(x) cos(x) tan(x) cot(x)
0 0 1 0 Undefined
π6  12  32  33  3
π4  22  22  1 1
π3  32  12  3 33 
π2  1 0 Undefined 0
2π3  32  12  3 33 
3π4  22  22  1 1
5π6  12  32  33  3
π 0 1 0 Undefined
7π6  12  32  33  3
5π4  22  22  1 1
4π3  32  12  3 33 
3π2  1 0 Undefined 0
5π3  32  12  3 33 
7π4  22  22  1 1
11π6  12  32  33  3
2π 0 1 0 Undefined