Trigonometry
      Trigonometric identities, examples
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 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:   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
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