Mathematics reference
Trigonometric identities
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Various identities and properties essential in trigonometry.
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Legend.
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- x and y are independent variables,
- d is the differential operator,
- int is the integration operator,
- C is the constant of integration.
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Identities.
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tan x = sin x/cos x
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equation 1
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cot x = cos x/sin x
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equation 2
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sec x = 1/cos x
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equation 3
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csc x = 1/sin x
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equation 4
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cot x = 1/tan x
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equation 5
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sin2 x + cos2 x = 1
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equation 6
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tan2 x + 1 = sec2 x
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equation 7
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1 + cot2 x = csc2 x
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equation 8
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cos (x +- y) = cos x cos y -+ sin x sin y
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equation 9
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sin (x +- y) = sin x cos y +- cos x sin y
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equation 10
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cos (-x) = cos x
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equation 11
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cos (pi/2 - x) = sin x
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equation 12
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cos (2 x) = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x
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equation 13
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cos (x/2) = +-[(1/2) (1 + cos x)]1/2
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equation 14
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sin (-x) = -sin x
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equation 15
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sin (pi/2 - x) = cos x
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equation 16
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sin (2 x) = 2 sin x cos x
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equation 17
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tan (x +- y) = (tan x +- tan y)/(1 -+ tan x tan y)
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equation 18
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tan (-x) = -tan x
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equation 19
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tan (2 x) = 2 tan x/(1 - tan2 x)
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equation 20
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tan (x/2) = sin x/(1 + cos x)
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equation 21
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cos (pi +- x) = -cos x
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equation 22
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sin (pi +- x) = -+sin x
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equation 23
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tan (pi +- x) = +-tan x
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equation 24
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Derivatives.
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(d/dx) sin x = cos x
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equation 25
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(d/dx) cos x = -sin x
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equation 26
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(d/dx) tan x = sec2 x
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equation 27
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(d/dx) cot x = -csc2 x
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equation 28
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(d/dx) sec x = sec x tan x
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equation 29
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(d/dx) csc x = -csc x cot x
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equation 30
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Integrals.
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int sin x dx = -cos x + C
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equation 31
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int cos x dx = sin x + C
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equation 32
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int tan x dx = -ln |cos x| + C
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equation 33
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int cot x dx = ln |sin x| + C
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equation 34
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int sec x dx = ln |sec x + tan x| + C
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equation 35
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int csc x dx = -ln |csc x + cot x| + C
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equation 36
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Various identities essential in hyperbolic trigonometry.
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