Useful Mathematical Equations, Definitions and Identities

1. Trigonometry

The Unit Circle

Quad II x<0, y>0 cosθ<0 secθ<0 sinθ>0 cscθ>0 tanθ<0 cotθ<0			Quad I x>0, y>0 cosθ>0 secθ>0 sinθ>0 cscθ>0 tanθ>0 cotθ>0
Function Definitions cosθ = x secθ = x-1 sinθ = y cscθ = y-1 tanθ = y/x cotθ = x/y			Polar Definitions x = rcosθ y = rsinθ x2 + y2 = r2 tanθ = y/x
Quad III x<0, y<0 cosθ<0 secθ<0 sinθ<0 cscθ<0 tanθ<0 cotθ<0			Quad IV x>0, y<0 cosθ>0 secθ>0 sinθ<0 cscθ<0 tanθ<0 cotθ<0

The Unit Triangle

cosθ = a/h secθ = h/a sinθ = o/h cscθ = h/o tanθ = o/a cotθ = a/o

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Identities

General

sin2θ + cos2θ = 1	tan2θ + 1 = sec2θ
sinθ/cosθ = tanθ	cot2θ + 1 = csc2θ
secθ = 1/cosθ	cos(-θ) = cosθ
cscθ = 1/sinθ	sin(-θ) = -sinθ
cotθ = 1/tanθ	tan(-θ) = -tanθ
sin(π/2-θ)=cosθ	cos(π/2-θ)=sinθ
sin(90°-θ)=cosθ	cos(90°-θ)=sinθ
sin(θ+2πk)=sinθ	cos(θ+2πk)=cosθ

Double and Half angle

sin2θ = 2sinθcosθ	sin2θ = (1-cos2θ)/2
cos2θ = cos2θ-sin2θ	cos2θ=2cos2θ-1
cos2θ=1-2sin2θ	cos2θ=(1+cos2θ)/2
tan2θ=(2tanθ)/(1-tan2θ)	cos(s/2)=+or-[(1+cos(s))/2]1/2
tan(s/2)=sin(s)/(1+cos(s))	sin(s/2)=+or-[(1-cos(s))/2]1/2

Addition and Sum to Product

sin(s+t)=sin(s)cos(t)+cos(s)sin(t)	sin(s-t)=sin(s)cos(t)-cos(s)sin(t)
cos(s+t)=cos(s)cos(t)-sin(s)sin(t)	cos(s-t)=cos(s)cos(t)+sin(s)sin(t)
tan(s+t)=(tan(s)+tan(t))/(1-tan(s)tan(t))	tan(s-t)=(tan(s)-tan(t))/(1+tan(s)tan(t))
sinα+sinβ=2sin((α+β)/2)cos((α-β)/2)	sinα-sinβ= 2cos((α+β)/2)sin((α-β)/2)
cosα+cosβ=2cos((α+β)/2)cos((α-β)/2)	cosα-cosβ=-2sin((α+β)/2)sin((α-β)/2)

Product to Sum

sinαsinβ=(1/2)[cos(α-β)-cos(α+β)]

sinαcosβ=(1/2)[sin(α+β)+sin(α+β)]

cosαcosβ=(1/2)[-2sin(α+β+cos(α-β)]

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