Trigonometry
      Sum to product and product to sum formulas or identities
         Sum to product formulas for the sine and the cosine functions
      The product to sum formulas for the sine and cosine functions
      Trigonometric identities, examples
Sum to product and product to sum formulas or identities
Sum to product formulas for the sine and the cosine functions
Adding the sum and difference formulas for the sine function,
sin (a + b) = sin a cos b + cosa sin b   (1)
sin (a - b) = sin a cos b - cosa sin b   (2)
yields sin (a + b) + sin (a - b) = 2sin a cos b  
and by subtracting the second from the first identity,
sin (a + b) - sin (a - b) = 2cosa sin b.
Then, substitute   a + b = x    and   a - b = y .
By adding and subtracting these equalities we get    
thus, and  
Using the same procedure for the cosine function,
cos (a + b) = cos a cos b - sin a sin b   (1)
cos (a - b) = cos a cos b + sin a sin b   (2)
by adding  (1) + (2) we get,         cos (a + b) + cos (a - b) = 2cos a cos b
and subtracting   (1) - (2)           cos (a + b) - cos (a - b) = -2sin a sin b
substitute,  a + b = x  and  a - b = y   so that,     
thus,   and    
The product to sum formulas for the sine and cosine functions
By adding and subtracting addition formulas derived are following product to sum formulas,
    and    
    and    
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:   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:   Prove the identity  
Solution:
Example:   Prove the identity  
Solution:
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 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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