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Product to Sum Trigonometry Identities Calculator

To use the product to sum identities calculator, enter the values of angles U & V, and hit calculate

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Product to Sum Identities Calculator

Product to sum identities calculator finds the result of two angles using the formulas of product to sum conversions.

What are Trigonometric Identities?

Trigonometric identities are equations that relate various trigonometric functions to each other. These identities are derived from the geometric properties of triangles and the unit circle. They play a crucial role in simplifying trigonometric expressions and solving trigonometric equations.

One such set of identities is the product to sum trigonometry identities.

What is Product to Sum Trigonometry Identities?

Product to sum trigonometry identities are formulas that express the product of two trigonometric functions as the sum or difference of two trigonometric functions. These identities are helpful when we need to simplify trigonometric expressions involving products.

By using these identities, we can rewrite a product of trigonometric functions as a sum or difference of trigonometric functions, which is often easier to work with.

Product to Sum Identities for Sine and Cosine:

The product to sum identities for sine and cosine are as follows:

  • sin(u) * sin(v) = (1/2) [cos (u - v) – cos (u + v)]
  • cos(u) * cos(v) = (1/2) [cos (u - v) + cos (u + v)]
  • sin(u) * cos(v) = (1/2) [sin (u - v) + sin (u + v)]
  • cos(u) * sin(v) = (1/2) [sin (u + v) - sin (u - v)]

Examples: Applying Product to Sum Trigonometry Identities

Example 1:

Simplify the expression: sin(3x) * sin(2x)

Solution:

Using Identity 1:  

sin(u) * sin(v) = (1/2) [cos (u - v) – cos (u + v)]

We have:

sin(3x) * sin(2x) = (1/2) [cos (3x - 2x) – cos (3x + 2x)]

sin(3x) * sin(2x) = (1/2) [cos(x) - cos(5x)]

Example 2:

Simplify the expression: cos(4x) * cos(3x)

Solution:

Using Identity 2:

cos(u) * cos(v) = (1/2) [cos (u - v) + cos (u + v)]

We have:

cos(4x) * cos(3x) = (1/2) [cos (4x - 3x) + cos (4x + 3x)]

cos(4x) * cos(3x) = (1/2) [cos (x) + cos (7x)]

Example 3:

Simplify the expression: sin(2x) * cos(2x)

Solution:

Using Identity 3:

sin(u) * cos(v) = (1/2) [sin (u - v) + sin (u + v)]

We have:

sin(2x) * cos(2x) = (1/2) [sin (2x - 2x) + sin (2x + 2x)]

sin(2x) * cos(2x) = (1/2) [sin (0) + sin(4x)]

sin(2x) * cos(2x) = (1/2) [0 + sin(4x)]

sin(2x) * cos(2x) = (1/2) sin(4x)

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