sin +
cos +
tan +
cot +
sec +
csc +
sin -
cos +
tan -
cot -
sec +
csc +
Degrees Radians !"#$ %&!$ '(#$ %&'$ !)%$ %!%$
0 *+,-.+ 0 1 0 -- 1 --
30/0
12
32
33
3 2 33
2
45/4
22
22
1 1 2 2
60/5
32
12
3 33
2 2 33
90/. 1 0 -- 0 -- 1
120./5
32
−12
− 3 −33
-2 2 33
1355/4
22
−22
-1 -1 − 2 2
1507/0
12 −
32
−33
− 3 −2 33
2
180 + 0 -1 0 -- -1 --
2108/0
−12 −
32
33
3 −2 33
-2
2257/4 −
22
−22
1 1 − 2 − 2
2404/5 −
32
−12
3 33
-2 −2 33
2705/.
-1 0 -- 0 -- -1
3007/5 −
32
12
− 3 −33
2 −2 33
3158/4 −
22
22
-1 -1 2 − 2
330 99/0
−12
32
−33
− 3 2 33
-2
Tangent and Cotangent Identities
tan= =sin=cos=
cot = =cos=sin=
Reciprocal Identities
csc= =1
sin=sin= =
1csc=
sec= =1
cos=cos= =
1sec=
cot = =1
tan=tan= =
1cot =
Pythagorean Identities
sinD = + cosD = = 1
tanD = + 1 = secD =
1 + cotD = = cscD =
Sum Formulas
sin F + G = sinF cosG + cosF sinG
cos F + G = cosF cosG − sinF sinG
tan F + G =tanF + tanG1− tanF tanG
Difference Formulas
sin F − G = sinF cosG − cosF sinG
cos F − G = cosF cosG + sinF sinG
tan F − G =tanF − tanG1+ tanF tanG
Half Angle Identities
sin=2= ±
1− cos=2
cos=2= ±
1+ cos=2
tan=2=1 − cos=sin=
=sin=
1 + cos=
Double Angle Identities
sin2= = 2sin= cos=
cos2= = cosD = − sinD =
cos2= = 2cosD = − 1
cos2= = 1 − 2sinD =
tan2= =2 tan=
1 − tanD=
TRIGONOMETRYDegrees to Radians
Formula
If x is an angle in degrees and t is an angle in radians,
then
IJKL
= MN
O = INJKL P = JKLM
I
Quad IQuad II
Quad IV
hyp
adj
opp
=Sakai-Kawada, F.
2018
sin +
cos -
tan -
cot -
sec -
csc +
sin -
cos -
tan +
cot +
sec -
csc - Quad III
Even and Odd Identities
sin −= = −sin =
cos −= = cos =
Cofunction Identities
sinQ2− = = cos=
cosQ2− = = sin=
Product to Sum Identities
cos R cos S =12cos R + S + cos R − S
sin R sin S =12cos R − S − cos R + S
sin R cos S =12sin R + S + sin R − S
RsinF
=S
sinG=
TsinU
RD = SD + TD − 2ST cosFSD = RD + TD − 2RT cosGTD = RD + SD − 2RS cosU
b a
cA
C
B
Law of Sines
Law of Cosines
NOTE: Trigonometric functions are periodic, in that they repeat
exactly in regular cycles.The length of the cycle is a called a
period
sin= =VWWℎYW = = sinZJ
VWWℎYW
cos= =R[\ℎYW
= = cosZJR[\ℎYW
tan= =VWWR[\ = = tanZJ
VWWR[\
Periodic Formulas
sin = + 2Q] = sin=
cos = + 2Q] = cos =
tan = + Q] = tan=
