Cosine Function is Even
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Theorem
For all $z \in \C$:
- $\map \cos {-z} = \cos z$
That is, the cosine function is even.
Proof 1
Recall the definition of the cosine function:
\(\ds \cos z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots\) |
From Even Power is Non-Negative:
- $\forall n \in \N: z^{2 n} = \paren {-z}^{2 n}$
The result follows.
$\blacksquare$
Proof 2
\(\ds \map \cos {-z}\) | \(=\) | \(\ds \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i z} + e^{-i z} } 2\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos z\) | Euler's Cosine Identity |
$\blacksquare$
Also see
- Sine Function is Odd
- Tangent Function is Odd
- Cotangent Function is Odd
- Secant Function is Even
- Cosecant Function is Odd
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.29$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosine
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Symmetry
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Symmetry