Trigonometric Functions

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The Six Functions

Sum & Difference Formulas

Reciprocal, Quotient, Pythagorean, and Even-odd identities.

Double & Half Angle Formulas

Pythagorean Identities

Tangent (tan)

Double Angle Formulas

Even Odd Identities

Quotient Identities

Reciprocal Identities

Cosecant (csc)

Cosine (cos)

Secant (sec)

Sine (sin)

Cotangent (cot)

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Examples

(Numbers are in Degrees)(sqrt=square root) cos(75)=cos(45+30)
=cos(45)cos(30)-sin(45)sin(30)
=((sqrt2)/2) x ((sqrt3)/2)-((sqrt2)/2)(1/2)
=((sqrt6)-(sqrt2))/4

tan(pi/12)=tan((pi/3)-(pi/4)
=(tan(pi/3)-tan(pi/4))/(1+tan(pi/3)(tan(pi/4))
((sqrt3)-1)/(1+(sqrt3)(1)
=2-(sqrt3)

tan(105)
=(tan(60)+tan(45))/(1-tan(60)tan(45)
=((sqrt3)+1)/(1-(sqrt3)(1)
=(1+(sqrt3))/(1-(sqrt3)
Multiply by reciprocal
(1+2(sqrt3)+3)/(1-3)
(4+2(sqrt3))/(-2)
=-2-(sqrt3)

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Sum Formulas
cos(a+B)=cos(a)cos(B)-sin(a)sin(B)
sin(a+B)=sin(a)cos(B)+cos(a)sin(B)
tan(a+B)=(tan(a)+tan(B))/(1-tan(a)tan(B)

Difference Formulas
cos(a-B)=cos(a)cos(B)+sin(a)sin(B)
sin(a-B)=sin(a)cos(B)-cos(a)sin(B)
tan(a-B)=(tan(a)-tan(B))/(1+tan(a)tan(B)

Examples

Numbers are in radians
cos(a)=(-4/5) and sin(B)=(5/13)

=((-3/5)(12/13))-((-4/5)(5/13))
=(-16/65)

cos(x-pi)=cos(x)cos(pi)+sin(x)sin(pi)
=(cos(x))(-1)+(sin(x))(0)
=-cos(x)

(in degrees) tan(330-45)
=(tan(330)-tan(45))/(1+tan(330)tan(45))
=(-((sqrt3)/3)-1))/(1-((sqrt3)/3)(1)
=(-3-(sqrt3))/(3-(sqrt3))
Multiply by Reciprocal
=(-9-6(sqrt3)-3)/(9-3)
=-2-(sqrt3)

sin(x)=1/csc(x)
cos(x)=1/sec(x)
tan(x)=1/cot(x)

csc(x)=1/sin(x)
sec(x)=1/cos(x)
cot(x)=1/tan(x)

tan(x)-(sin(x))/(cos(x))
cot(x)=(cos(x)/sin(x))

sin^2(x)+cos^2(x)=1
1+tan^2(x)=sec^2(x)
1+cot^2(x)=csc^2(x)

sin(-x)=-sin(x)
cos(-x)=cos(x)
tan(-x)=-tan(x)

csc(-x)=-csc(x)
sec(-x)=sec(x)
cot(-x)=-cot(x)

(y/r)

(x/r)

(y/x)

(r/y)

(r/x)

(x/y)

sin2x=2sin(x)cos(x)
cos2x=cos^2(x)-sin^2(x)
cos2x=2cos^2(x)-1
cos2x=1-2sin^2(x)
tan2x=(2tan(x))/(1-tan^2(x))

Half Angle Identities

sin(x/2)=(+/-)(sqrt((1-cosx)/2))
cos(x/2)=(+/-)(sqrt((1+cosx)/2))
tan(x/2)=(+/-)(sqrt((1-cos(x))/(1+cos(x)))
tan(x/2)=(1-cos(x))/(sin(x))
tan(x/2)=(sin(x))/1+cos(x)

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