Trigonometric Identities

Trigonometric Identities
Reciprocal Identities
Complement Formulas
1
csc x
csc x =
cos x =
1
sec x
sec x =
1
cosx
tan x =
1
cot x
tan x =
sin x
cosx
1
tan x
cot x =
sin x =
cot x =
1
sin x
cosx
sin x
sinx = cos (90–x)
cosx = sin (90–x)
tanx = cot (90–x)
cotx = tan (90–x)
secx = csc (90–x)
cscx = sec (90–x)
Supplement Formulas
Pythagorean Identities
sin(π-x) = sinx
sin 2 x + cos2 x = 1
cos(π-x) = -cosx
sin2 x = 1− cos2 x
tan(π-x) = -tanx
cos2 x = 1− sin2 x
csc(π-x) = cscx
sec(π-x) = -secx
sec 2 x − tan2 x = 1
cot(π-x) = -cotx
sec 2 x = tan2 x + 1
tan2 x = sec 2 x −1
Addition & Subtraction Formulas
csc 2 x − cot 2 x = 1
cos(x + y) = cos x cos y − sin x sin y
csc 2 x = cot 2 x + 1
cos(x − y) = cos x cos y + sin x sin y
cot 2 x = csc 2 x − 1
sin(x + y) = sin x cos y + cos xsin y
Symmetry Identities
sin(x − y) = sin x cos y − cosx sin y
sin(−x) = −sin x
cot(− x) = − cot x
cos(−x) = cos x
sec(−x) = sec x
tan(− x) = − tan x
csc(− x) = − csc x
tan( x + y) =
tan x + tan y
1 − tan x tan y
tan( x − y) =
tan x − tan y
1 + tan x tan y
Double-Angle Formulas
Sum Formulas
sin2x = 2sin x cos x
1
1
sin x + sin y = 2sin (x + y)cos (x − y)
2
2
2
2
cos2x = cos x − sin x
1
1
sin x − sin y = 2sin (x − y)cos (x + y)
2
2
2
cos2x = 2cos x − 1
cos2x = 1− 2sin2 x
tan 2x =
1
1
cos x + cos y = 2 cos (x + y) cos (x − y)
2
2
2tan x
1− tan 2 x
1
1
cos x − cos y = −2sin (x + y)sin (x − y)
2
2
Half-Angle Formulas
1
1− cos x
sin x = ±
2
2
Inverse Functions
−1
if and only if x = sin y
π
π
and − ≤ y ≤
2
2
y = sin x
1
1+ cos x
cos x = ±
2
2
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1
sin x
tan x =
2
1 + cos x
−1
if and only if x = cos y
y = cos x
and 0 ≤ y ≤ π
1
1− cos x
tan x =
2
sin x
y = tan x
−1
if and only if x = tan y
π
π
and − ≤ y ≤
2
2
1
1− cos x
tan x =
2
1+ cos x
Product Formulas
sin x sin y =
cos x cos y =
sin x cos y =
SOH-CAH-TOA
1
[cos(x − y) − cos(x + y)]
2
1
[cos(x + y) + cos(x − y)]
2
1
[sin(x + y) + sin(x − y)]
2
sin x =
opposite
hypotenuse
cos x =
adjacent
hypotenuse
tan x =
opposite
adjacent
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