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Angle subtraction for cosine

Prerequisites

Cosine is an even function | \(\cos(\theta) = \cos(-\theta)\)
Sine is an odd function | \(\sin(-\theta) = -\sin(\theta)\)
Angle addition for cosine | \(\cos(\theta + \phi) = \cos(\theta) \cdot \cos(\phi) - \sin(\theta) \cdot \sin(\phi)\)

Description

The cosine angle subtraction formula is a fundamental identity in trigonometry that describes how the cosine of the difference between two angles can be expressed in terms of the sines and cosines of the individual angles. This formula is essential for solving problems involving trigonometric functions, where angles are combined or split, and for proving other trigonometric identities. It enables the simplification of expressions and the calculation of unknown angles or lengths in geometric figures. The formula also plays a crucial role in the study of wave interference, oscillations, and rotations in physics, illustrating how wave phases or orientations combine. Understanding and applying the cosine angle subtraction formula is a key skill in mathematics, providing a bridge to more complex concepts in calculus and analytical geometry.

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

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 equation for Angle addition for cosine:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} + \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) - \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
  2. We can now let (\( \htmlClass{sdt-0000000123}{\phi} \) = -\( \htmlClass{sdt-0000000123}{\phi} \)), and substitute it into the equation to get
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(-\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(-\htmlClass{sdt-0000000123}{\phi})\]
  3. We can now use the fact that cosine is an even function:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) = \htmlClass{sdt-0000000124}{\cos}(-\htmlClass{sdt-0000000024}{\theta})\]
    and that sine is an odd function:
    \[\htmlClass{sdt-0000000127}{\sin}(-\htmlClass{sdt-0000000024}{\theta}) = -\htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta})\]
    to simplify the equation in step (2) to:
    \[\htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta} - \htmlClass{sdt-0000000123}{\phi}) = \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000124}{\cos}(\htmlClass{sdt-0000000123}{\phi}) + \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000024}{\theta}) \cdot \htmlClass{sdt-0000000127}{\sin}(\htmlClass{sdt-0000000123}{\phi})\]
    as required.

Example

Coming soon...