Notes - Trigonometric Identities

Angle Sum and Difference

Sine cross and same sign, cosine cross and opposite sign.

• sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b)

• sin(a-b)=sin(a)*cos(b)-cos(a)*sin(b)

• cos(a+b)=cos(a)*cos(b)-sin(a)*sin(b)

• cos(a-b)=cos(a)*cos(b)+sin(a)*sin(b)

• tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b))

• tan(a-b)=(tan(a)-tan(b))/(1+tan(a)*tan(b))

Product to Sum

Multiplying the same function results in cosine, with sine setting everything to negative. Multiplying different functions results in sine, with positive sign unless cosine is first.

• sin(a)*sin(b)=-1/2*[cos(a+b)-cos(a-b)]

• cos(a)*cos(b)=1/2*[cos(a+b)+cos(a-b)]

• sin(a)*cos(b)=1/2*[sin(a+b)+sin(a-b)]

• cos(a)*sin(b)=1/2*[sin(a+b)-sin(a-b)]

Sum to Product

Sine is crossing, while adding cosine is similar. Whatever that means. First function first.

• sin(a)+sin(b)=2*sin((a+b)/2)*cos((a-b)/2)

• sin(a)-sin(b)=2*cos((a+b)/2)*sin((a-b)/2)

• cos(a)+cos(b)=2*cos((a+b)/2)*cos((a-b)/2)

• cos(a)-cos(b)=-2*sin((a+b)/2)*sin((a-b)/2)

Double Angle

• sin(2*a)=2*sin(a)*cos(a)

• cos(2*a)=(cos^2)(a)-(sin^2)(a)

• cos(2*a)=2*(cos^2)(a)-1

• cos(2*a)=1-2*(sin^2)(a)

• cos(2*a)=(1-(tan^2)(a))/(1+(tan^2)(a))

• tan(2*a)=(2*tan(a))/(1-(tan^2)(a))

Half Angle

(sin^2)(x/2)=(1-cos(x))/2

(cos^2)(x/2)=(1+cos(x))/2

(tan^2)(x/2)=(1-cos(x))/(1+cos(x))

Limits Approaching 0

Generally, sin(x)=x for small x. However, automatically substituting that is often counterintuitive, so we use these identities to avoid that.

(lim_x→0)(sin(a*x)/(b*x))=(lim_x→0)((a*x)/sin(b*x))=a/b

(lim_x→0)(tan(a*x)/(b*x))=(lim_x→0)((a*x)/tan(b*x))=a/b

(lim_x→0)(sin(a*x)/tan(b*x))=(lim_x→0)(tan(a*x)/sin(b*x))=a/b

Harmonic Addition

a*cos(x)+b*sin(x)=k*cos(x-α)

a*cos(x)+b*sin(x)=k*sin(x+β)

k=√(,a2+b2)