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trig identities and sum/product formulas
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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
x − x = ± 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
(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
sin cos
x x
cos sin
x x tanx = cot( o
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