삼각함수 곱, 합 차 공식과 문제 (Product and Sum Formulas)

Product and Sum Formulas

 

From the Addition Formulas, we derive the following trigonometric formulas (or identities)

 

      cos(a) cos(b) = ½ (cos(a + b) + cos(a - b))

 

      sin(a) sin(b) = ½ (cos(a - b) - cos(a + b))

 

      sin(a) cos(b) = ½ (sin(a + b) + sin(a - b))

 

      cos(a) sin(b) = ½ (sin(a + b) - sin(a - b))

 

 

 

Remark. It is clear that the third formula and the fourth are identical (use the propertysin(-x) = -sin(x) to see it).

 

The above formulas are important whenever need rises to transform the product of sine and cosine into a sum.

 

  

 

Example 01. Express the product cos(5x) sin(3x) as a sum of trigonometric functions.

 

  

Example 02. Express  cos(3x) + cos(7x) as a product.

 

  

Example 03. Find the real number x such that 0 ≤ x ≤π/2 and cos(4x) + cos(2x) = 0

 

  

 

Example 04. Verify the identity

 

        sin(5a) + 2sin(3a) + sin(a) = 4cos²(a) sin(3a)

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