Trigonometry
      Trigonometric functions of double angles, double angle formulas
         Trigonometric functions of double angles expressed by the tangent function
         Trigonometric functions expressed by the tangent of the half angle
      Trigonometric identities, examples
Trigonometric functions of double angles expressed by the tangent function
The double angle formulas can be expressed in terms of a single function, so using the identity
and dividing the numerator and denominator by cos2 a we get  
and from  
dividing the right side by cos2 a gives  
  then    
Trigonometric functions expressed by the tangent of the half angle
Replacing a by a/2 in the above identities, we get
         
Trigonometric identities examples
Example:   Using known values, sin 60° = Ö3/2 and sin 45° = Ö2/2 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:   Use,  tan 45° = 1 and  tan 60° = Ö3,  to prove that  tan 15° = 2 - Ö3.
Solution:
Example:   Prove the identity
Solution:   Using the addition formula
Example:   Verify the identity  
Solution:   We divide the numerator and denominator on the left side by sin a and to the right side we use the cotangent formula for the difference of two angles, thus
Example:   Express sin 3x in terms of sin x.
Solution:   Using the addition 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 addition 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 
Example:   Prove that  
Solution:  Replace sin a by cos (p/2 - a) and cos a by sin (p/2 - a) and use the sum to product formula
Trigonometry contents A
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