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APEX Calculus

Section B.3 Trigonometry Reference

The Unit Circle.

Subsection B.3.1 Definitions of the Trigonometric Functions

Unit Circle Definition.

sin(θ)=y cos(θ)=x
csc(θ)=1y sec(θ)=1x
tan(θ)=yx cot(θ)=xy

Right Triangle Definition.

sin(θ)=OH csc(θ)=HO
cos(θ)=AH sec(θ)=HA
tan(θ)=OA cot(θ)=AO

Subsection B.3.2 Common Trigonometric Identities

  1. sin2(x)+cos2(x)=1
  2. tan2(x)+1=sec2(x)
  3. 1+cot2(x)=csc2(x)
List B.3.1. Pythagorean Identities
  1. sin(2x)=2sin(x)cos(x)
  2. cos(2x)=cos2(x)sin2(x)=2cos2(x)1=12sin2(x)
  3. tan(2x)=2tan(x)1tan2(x)
List B.3.2. Double Angle Formulas
  1. sin(π2x)=cos(x)
  2. cos(π2x)=sin(x)
  3. tan(π2x)=cot(x)
  4. csc(π2x)=sec(x)
  5. sec(π2x)=csc(x)
  6. cot(π2x)=tan(x)
List B.3.3. Cofunction Identities
  1. sin(x)=sin(x)
  2. cos(x)=cos(x)
  3. tan(x)=tan(x)
  4. csc(x)=csc(x)
  5. sec(x)=sec(x)
  6. cot(x)=cot(x)
List B.3.4. Even/Odd Identities
  1. sin2(x)=1cos(2x)2
  2. cos2(x)=1+cos(2x)2
  3. tan2(x)=1cos(2x)1+cos(2x)
List B.3.5. Power-Reducing Formulas
  1. sin(x)+sin(y)=2sin(x+y2)cos(xy2)
  2. sin(x)sin(y)=2sin(xy2)cos(x+y2)
  3. cos(x)+cos(y)=2cos(x+y2)cos(xy2)
  4. cos(x)cos(y)=2sin(x+y2)sin(xy2)
List B.3.6. Sum to Product Formulas
List B.3.7. Product to Sum Formulas
  1. sin(x)sin(y)=12(cos(xy)cos(x+y))
  2. cos(x)cos(y)=12(cos(xy)+cos(x+y))
  3. sin(x)cos(y)=12(sin(x+y)+sin(xy))
List B.3.8. Angle Sum/Difference Formulas
  1. sin(x±y)=sin(x)cos(y)±cos(x)sin(y)
  2. cos(x±y)=cos(x)cos(y)sin(x)sin(y)
  3. tan(x±y)=tan(x)±tan(y)1tan(x)tan(y)
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