Identities and Formulas of Trigonometric Functions

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The following is a review of the Six Trigonometric Functions and the Trigonometric Identities. One should note that the right triangle definitions referer to the angle A where 0 < A < pi/2.

Six Trigonometric Functions

     Right triangle definitions.
      sin A = opp./hyp.       cos A = adj./hyp.       tan A = opp./adj.
      csc A = hyp./opp.       sec A = hyp./adj.       cot A = adj./opp.
     Circular function definitions, where A is any angle
      sin A = y/r       cos A = x/r       tan A = y/x
      csc A = r/y       sec A = r/x       cot A = x/y

Trigonometric Identities

     Pathagorean identities
      (sin A)² + (cos A)² = 1       (tan A)² + 1 = (sec A)²       (cot A)² + 1 = (csc A)²

     Sum and difference of two angles note that -(±) means the opposite of ±
      sin(A ± B) = sin A cos B ± cos 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}

     Half-angle formulas
      (sin A)² = 1/2(1- cos 2A)       (cos A)² = 1/2(1+ cos 2A)

     Double angle formulas
      sin 2A = 2 sin A cos A
      cos 2A = 2(cos A)² - 1
           = 1- 2 (sin A)²
           = (cos A)² - (sin A)²

     Law of Cosines and Sines note that A is the opposite angle of side a
      a² = b² + c² - 2bc cos A
         _{a    =    b    =    c}   
      ^{sin A   sin B   sin C}