Pythagorean Identities
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cos 2 x + sin 2 x = 1
-
1 + tan 2 x = sec 2 x
-
cot 2 x + 1 = csc 2 x
Ratios of Trigonometric Functions
- tan x = sin x / cos x
- cot x = 1 / tan x = cos x / sin x
- sec x = 1 / cos x
- csc x = 1 / sin x
Sum/Difference of Angles Formulas
- cos(x + y)= cos x cos y - sin x sin y
- cos(x - y)= cos x cos y + sin x sin y
- sin(x + y)= sin x cos y + cos x sin y
- sin(x - y)= sin x cos y - cos x sin y
- 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/Identities
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cos(2 x)= cos 2 x - sin 2 x
- sin(2 x)= 2 sin x cos x
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tan (2 x)=(2 tan x)/(1 - tan 2 x)
Half Angle Formulas/Identities
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sin (x / 2) = + or - √ (1 - cos x)/ 2)
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cos (x / 2) = + or - √ (1 + cos x)/ 2)
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tan (x / 2) = + or - √ (1 - cos x)/ (1 + cos x)) = sin x / (1 + cos x) = (1 - cos x)/(sin x)
Power Reducing Formulas/Identities
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sin 2 x = 1 / 2 - (1 / 2) cos(2 x)
- $\cos^2 x = \dfrac{1}{2}+\dfrac{1}{2}\cos(2x)$
- $\sin^3 x = \dfrac{3}{4}\sin x -\dfrac{1}{4}\sin(3x)$
- $\cos^3 x = \dfrac{3}{4}\cos x +\dfrac{1}{4}\cos(3x)$
Negative Angle Identities
Odd Functions
- $\sin (-x) = -\sin x$
- $\tan (-x) = -\tan x$
- $\cot (-x) = -\cot x$
- $\csc (-x) = -\csc x$
Even Functions
- $\cos (-x) = \cos x$
- $\sec (-x) = \sec x$
Sum (to Product) of Trigonometric Functions
- $\sin x + \sin y = 2\sin \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$
- $\cos x + \cos y = 2\cos \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$
Difference (to Product) of Trigonometric Functions
- $\sin x - \sin y = 2\cos \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$
- $\cos x - \cos y = 2\sin \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$
Product (to Sum) of Trigonometric Functions
- $\sin x \sin y = \dfrac{1}{2}[\cos(x-y)-\cos(x+y)]$
- $\sin x \cos y = \dfrac{1}{2}[\sin(x-y)+\sin(x+y)]$
- $\cos x \cos y = \dfrac{1}{2}[\cos(x-y)+\cos(x+y)]$
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