Trigonometry
      Trigonometric functions of double angles, double angle formulas
         Trigonometric functions expressed by the half 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 half angle
Substituting a/2 in the double angle formulas we obtain trigonometric functions expressed by the half angle,
and
and
and
or
or
Trigonometric identities examples
Example:   Using known values, sin 60 = 3/2 and sin 45 = 2/2 evaluate sin 105.
Solution:  Applying the sum formula for the sine function,  sin (a + b) = sin a cos b + cosa sin b
therefore,   sin 105 = sin (60 + 45) = sin 60 cos 45 + cos60 sin 45
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:   If  tan a = 3/4, find tan a/2.   
Solution:   Use formula  to express tan a/2  in terms of  tan a.
Example:   Prove the identity  
Solution:   Substitute  then
Example:   Express the given difference  sin 61 - sin 59  as a product.   
Solution:   Since 
Example:   Prove the identity  sin a + sin (a + 120) + sin (a + 240) = 0.   
Solution:   Applying the sum formula to the last two terms on the left side of the identity we get,
Example:   Prove the identity  
Solution:   Using the formula for the sum of the tangent 
Trigonometry contents A
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