Symmetry (Meta) Math Example 4

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Example 4

medium
Use the symmetry of sin⁡\sin (odd function) and cos⁡\cos (even function) to simplify: sin⁡(−θ)+cos⁡(−θ)\sin(-\theta) + \cos(-\theta).

Solution

  1. 1
    sin⁡\sin is odd: sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta.
  2. 2
    cos⁡\cos is even: cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta.
  3. 3
    So sin⁡(−θ)+cos⁡(−θ)=−sin⁡θ+cos⁡θ=cos⁡θ−sin⁡θ\sin(-\theta)+\cos(-\theta) = -\sin\theta + \cos\theta = \cos\theta - \sin\theta.

Answer

sin⁡(−θ)+cos⁡(−θ)=cos⁡θ−sin⁡θ\sin(-\theta)+\cos(-\theta) = \cos\theta - \sin\theta
Knowing the symmetry type (odd vs even) of trigonometric functions allows immediate simplification of expressions with negative arguments.

About Symmetry (Meta)

A property of a mathematical object that remains unchanged under a specified transformation — reflection, rotation, translation, or algebraic substitution.

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More Symmetry (Meta) Examples

Example 1 easy

Show that the equation [formula] is symmetric about both coordinate axes and the origin. Verify by s

Example 2 medium

Use symmetry to evaluate [formula] for even [formula].

Example 3 easy

Determine whether [formula] is odd, even, or neither, by testing the symmetry condition.

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