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Product to sum sine cosine

Prerequisites

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

Description

Description coming soon...

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

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 sine identity:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta})\]
    and the angle subtraction for sine identity:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta})\]
  2. We can now add these identities together to get:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta})\]
  3. From here, we have two \(\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi})\) terms that we can aggregate, and the \(\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta})\) terms cancel each other out. We can therefore simplify to:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = 2 \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi})\]
  4. Finally, we can divide both sides by \(2\), and swap around the order of the left hand side and the right hand side to get:
    \[\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) = \frac{1}{2}(\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}))\]
    as required.

Example

Example coming soon...