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
