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trignometry




Trigonometrics function of Sum and difference of angles

Sin and cos function

  1. cos(A+B)=cos(A)cos(B)-sin(A)sin(B)
  2. cos(A-B)=cos(A)cos(B)+sin(A)sin(B)
  3. cos(π/2 -A)=sin(A)
  4. sin(π/2 -A)=cos(A)
  5. sin(A+B)=sin(A)cos(B)+sin(B)cos(A)
  6. sin(A-B)=sin(A)cos(B)-sin(B)cos(A)
Similary we can have defined other sin and cos sum and differences

Tan and cot functions

  1. If none of the angles x, y and (x + y) is an odd multiple of π/2
    $tan(A+B)=\frac{tan(A)+tan(B)}{1-tan(A)tan(B)}$
    $tan(A-B)=\frac{tan(A)-tan(B)}{1+tan(A)tan(B)}$
  2. If none of the angles x, y and (x + y) is an multiple of π
    $cot(A+B)=\frac{cot(A)cot(B)-1}{cot(A)+cot(B)}$
    $cot(A-B)=\frac{cot(A)cot(B)+1}{cot(B)-cot(A)}$
    Now lets explore the multiple of x. These all can be proved from above equations
    Double of x
    $cos2x=cos^{^{2}}x-sin^{^{2}}x=2cos^{^{2}}x-1=1-2sin^{^{2}}x=\frac{1-tan^{^{2}}x}{1+tan^{^{2}}x}$
    $sin2x=2cos(x)sin(x)=\frac{2tan(x)}{1+tan^{^{2}}x}$
    $tan2x=\frac{2tan(x)}{1-tan^{^{2}}x}$
    Triple of x
    $sin3x=3sin(x)-4sin^{3}x$
    $cos3x=4cos^{3}x-3cos(x)$
    $tan(3x)=\frac{3tanx-tan^{^{3}}x}{1-3tan^{^{2}}x}$
Some other Important functions
  • $cos(A)+cos(B)=2cos\frac{A+B}{2}cos\frac{A-B}{2}$
  • $cos(A)-cos(B)=-2sin\frac{A+B}{2}sin\frac{A-B}{2}$
  • $sin(A)+sin(B)=2sin\frac{A+B}{2}cos\frac{A-B}{2}$
  • $sin(A)-sin(B)=2cos\frac{A+B}{2}sin\frac{A-B}{2}$





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