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Product to sum sine sine

Prerequisites

Angle addition for cosine | \(\cos(\theta + \phi) = \cos(\theta) \cdot \cos(\phi) - \sin(\theta) \cdot \sin(\phi)\)
Angle subtraction for cosine | \(\cos(\theta - \phi) = \cos(\theta) \cdot \cos(\phi) + \sin(\theta) \cdot \sin(\phi)\)

Description

Description coming soon...

\[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) = \frac{1}{2}(\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}))\]

Symbols Used:

This is a commonly used symbol to represent an angle in mathematics and physics.
\( \phi \)

This symbol means the same as Angle, which uses \(\htmlClass{sdt-0000000024}{\theta}\). It is a secondary symbol to use that represents an angle, when a different angle is already using \(\htmlClass{sdt-0000000024}{\theta}\).
\( \cos \)

This is the symbol for cosine, a trigonometric function that calculates the ratio of the adjacent side to the hypotenuse of a right-angled triangle.
\( \sin \)

This is the symbol for sine, is a trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Derivation

  1. Consider the angle addition for cosine identity:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
    and the angle subtraction for cosine identity:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
  2. We can now subtract the angle addition for cosine identity from the angle subtraction for cosine identity to get:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) - - \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
  3. From here, on the right hand side, the cosine terms are identical but of opposite sign so cancel each other out. The sine term has a double subtraction, so becomes an addition. The sine terms are also identical, so they aggregate, simplifying the equation to...
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = 2 \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
  4. Finally, we can divide both sides by \(2\), and swap around the left hand side and right hand side to get:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) = \frac{1}{2}(\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}))\]
    as required.

Example

Example coming soon...