Tangent Function is Odd
Theorem
For all $x \in \C$ where $\tan x$ is defined:
- $\map \tan {-x} = -\tan x$
That is, the tangent function is odd.
Proof
| \(\ds \map \tan {-x}\) | \(=\) | \(\ds \frac {\map \sin {-x} } {\map \cos {-x} }\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\sin x} {\cos x}\) | Sine Function is Odd; Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\tan x\) | Tangent is Sine divided by Cosine |
$\blacksquare$
Also see
- Sine Function is Odd
- Cosine Function is Even
- 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: $5.30$
- 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: $4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trigonometric function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trigonometric function
- 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