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Precalculus Trigonometric Identities Cheat Sheet, Cheat Sheet of Trigonometry

trig identities and sum/product formulas

Typology: Cheat Sheet

2020/2021

Uploaded on 04/26/2021

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Pre-Calc
Trigonometric Identities
Trigonometric IdentitiesTrigonometric Identities
Trigonometric Identities
The six trigonometric functions
The six trigonometric functionsThe six trigonometric functions
The six trigonometric functions/
//
/Reciprocals
ReciprocalsReciprocals
Reciprocals
Identities
IdentitiesIdentities
Identities
xr
y
hyp
opp
xcsc
1
sin ===
xy
r
opp
hyp
xsin
1
csc ===
xr
x
hyp
adj
xsec
1
cos ===
xx
r
adj
hyp
xcos
1
sec ===
x
x
x
y
adj
opp
xcos
sin
tan ===
x
x
xy
x
opp
adj
xsin
cos
tan
1
cot ====
Pythagorean Identities
Pythagorean IdentitiesPythagorean Identities
Pythagorean Identities
sin
2
x + cos
2
x = 1 cos
2
x = 1 – sin
2
x sin
2
x = 1 - cos
2
x
tan
2
x – sec
2
x = -1 1 + tan
2
x = sec
2
x 1 + cot
2
x = csc
2
x
Sum or difference of two angles
Sum or difference of two angles Sum or difference of two angles
Sum or difference of two angles
sin(a + b) = sina cosb + cosa sinb cos(a + b) = cosa cosb - sina sinb
sin(a - b) = sina cosb - cosa sinb cos(a - b) = cosa cosb + sina sinb
b
a
ba
ba
tan
tan
1
tantan
)tan(
+
=+
b
a
ba
ba
tan
tan
1
tantan
)tan( +
=
Double Angle Formulas
Double Angle FormulasDouble Angle Formulas
Double Angle Formulas
Half
Half Half
Half Angle Formulas
Angle FormulasAngle Formulas
Angle Formulas
cos2x = 1- 2sin
2
x cos2x = 2cos
2
x - 1
2
2cos1
sin
2
x
x
=
2
2cos1
cos
2
x
x
+
=
x
x
x
2
cos
1
2cos1
tan
2
+
=
cos2x = cos
2
x – sin
2
x sin2x = 2sinx cosx
2
cos1
2
sin xx
±=
2
cos1
2
cos xx +
±=
x
x
x
2
tan
1
tan2
2tan
=
x
x
x
x
x
xx
sin
cos1
cos1
sin
cos1
cos1
2
tan
=
+
=
+
±=
Reduction Formulas
Reduction FormulasReduction Formulas
Reduction Formulas
Law of cosines
Law of cosinesLaw of cosines
Law of cosines
a
aa
a
2
22
2
= b
= b= b
= b
2
22
2
+ c
+ c+ c
+ c
2
22
2
2bc cos A
2bc cos A2bc cos A
2bc cos A
(where A is the angle of a scalene opposite side a)
sin(-x) = -sinx cos(-x) = cosx
sin(x) = -sin(x – π) cos(x) = -cos(x – π) Co
CoCo
Co-
--
-Functions
FunctionsFunctions
Functions
tan(-x) = -tanx tan(x) = tan(x – π) sinx = cos(90
o
– x) cosx = sin(90
o
– x)
±=± 2
cossin
π
xx
±=± 2
sincos
π
xx
tanx = cot(90
o
– x) secx = csc(90
o
– x)
Sum and Product Formulas
Sum and Product FormulasSum and Product Formulas
Sum and Product Formulas
sina cosb = ½[sin(a + b) + sin(a - b)] cosa cosb = ½[cos(a + b) + cos(a - b)]
cosa sinb = ½[sin(a + b) – sin(a - b)] sina sinb = ½[cos(a – b) – cos(a + b)]
+
=+ 2
cos
2
sin2sinsin yxyx
yx
+
= 2
sin
2
cos2sinsin yxyx
yx
+
=+ 2
cos
2
cos2coscos yxyx
yx
+
= 2
sin
2
sin2coscos yxyx
yx

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Pre-Calc Trigonometric IdentitiesTrigonometric IdentitiesTrigonometric IdentitiesTrigonometric Identities

The six trigonometric functionsThe six trigonometric functionsThe six trigonometric functionsThe six trigonometric functions////ReciprocalsReciprocalsReciprocals IdentitiesReciprocalsIdentitiesIdentitiesIdentities

r x

y

hyp

opp x csc

sin = = = y x

r

opp

hyp x sin

csc = = =

r x

x

hyp

adj x sec

cos = = = x x

r

adj

hyp x cos

sec = = =

x

x

x

y

adj

opp x cos

sin tan = = = x

x

y x

x

opp

adj x sin

cos

tan

cot = = = =

Pythagorean IdentitiesPythagorean Identities Pythagorean IdentitiesPythagorean Identities

sin^2 x + cos^2 x = 1 cos^2 x = 1 – sin^2 x sin^2 x = 1 - cos^2 x

tan 2 x – sec 2 x = -1 1 + tan 2 x = sec 2 x 1 + cot 2 x = csc 2 x

Sum or difference of two anglesSum or difference of two anglesSum or difference of two anglesSum or difference of two angles

sin(a + b) = sina cosb + cosa sinb cos(a + b) = cosa cosb - sina sinb

sin(a - b) = sina cosb - cosa sinb cos(a - b) = cosa cosb + sina sinb

a b

a b a b 1 tan tan

tan tan tan( ) −

a b

a b a b 1 tan tan

tan tan tan( )

Double Angle FormulasDouble Angle FormulasDouble Angle FormulasDouble Angle Formulas HalfHalf Angle FormulasHalfHalfAngle FormulasAngle FormulasAngle Formulas

cos2x = 1- 2sin 2 x cos2x = 2cos 2 x - 1 2

1 cos 2 sin

2 x x

1 cos 2 cos

2 x x

x

x x 1 cos 2

1 cos 2 tan

2

cos2x = cos 2 x – sin 2 x sin2x = 2sinx cosx 2

1 cos

sin

xx = ± 2

1 cos

cos

x + x

x

x x 2 1 tan

2 tan tan 2 −

x

x

x

x

x

x x

sin

1 cos

1 cos

sin

1 cos

1 cos

tan

Reduction FormulasReduction FormulasReduction FormulasReduction Formulas Law of cosinesLaw of cosinesLaw of cosinesLaw of cosines aaaa 2222 = b= b= b= b 2222

  • c+ c+ c+ c 2222
  • ––– 2bc cos A2bc cos A2bc cos A2bc cos A

(where A is the angle of a scalene  opposite side a)

sin(-x) = -sinx cos(-x) = cosx

sin(x) = -sin(x – π) cos(x) = -cos(x – π) CoCoCoCo----FunctionsFunctionsFunctionsFunctions

tan(-x) = -tanx tan(x) = tan(x – π) sinx = cos( o

  • x) cosx = sin( o - x)

sin cos

x x  

cos sin

x x tanx = cot( o

  • x) secx = csc( o - x)

Sum and Product FormulasSum and Product FormulasSum and Product FormulasSum and Product Formulas

sina cosb = ½[sin(a + b) + sin(a - b)] cosa cosb = ½[cos(a + b) + cos(a - b)]

cosa sinb = ½[sin(a + b) – sin(a - b)] sina sinb = ½[cos(a – b) – cos(a + b)]

cos 2

sin sin 2 sin

x y x y x y  

sin 2

sin sin 2 cos

x y x y x y

cos 2

cos cos 2 cos

x y x y x y  

sin 2

cos cos 2 sin

x y x y x y