Trigonometry
      Trigonometric functions of double angles, double angle formulas
         Trigonometric functions expressed by the half angle
         Trigonometric functions of double angles expressed by the tangent function
      Trigonometric functions expressed by the cosine of the double angle
      Trigonometric identities, examples
Trigonometric functions of double angles, double angle formulas
By substituting b with a in the sum formulas,
sin (a + b) = sin a cos b + cosa sin b      and     cos (a + b) = cos a cos b - sin a sin b,
  and
thus for example,   sin 2a = sin (a + a) = sin a cos a + cosa sin a = 2sin a cos a   so we get,
  sin 2a = 2 sin a cos a and cos 2a = cos2 a - sin2 a  
and
The double angle formula for the cosine function can be expressed by sine or cosine function using the identity  sin2 a + cos2 a = 1,
  cos 2a = 2cos2 a - 1 or cos 2a = 1 - 2sin2 a  
and  1 + cos 2a = 2 cos2 a or 1 - cos 2a = 2sin2 a  
Trigonometric functions expressed by the cosine of the double angle
Replacing a/2 by a in the above identities, we get
         
Trigonometric identities, examples
Example:   Express sin 3x in terms of sin x.
Solution:   Using the sum formula and the double angle formula for the sine function,
sin 3x = sin (2x + x) = sin 2x cos x + cos 2x sin x = 2sin x cos x cos x + (cos2 x - sin2 x) sin x
            = 2sin x (1 - sin2 x) + (1 - 2sin2 x) sin x = 3sin x - 4sin3 x.
Example:   Express tan 3x in terms of tan x.
Solution:   Using the sum formula and the double angle formula for the tangent function,
Example:   Prove the identity  
Solution:
Example:   Prove the identity  
Solution:
Example:   Prove the identity  
Solution:   Using the formula for the sum of the tangent 
Pre-calculus contents F
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