Tangent is Sine divided by Cosine

Theorem

Let $\theta$ be an angle such that $\cos \theta \ne 0$.

Then:

$\tan \theta = \dfrac {\sin \theta} {\cos \theta}$

where $\tan$, $\sin$ and $\cos$ mean tangent, sine and cosine respectively.

Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

$\ds \frac {\sin \theta} {\cos \theta}$

$=$

$\ds \frac {y / r} {x / r}$

$\ds $

$=$

$\ds \frac y r \frac r x$

$\ds $

$=$

$\ds \frac y x$

$\ds $

$=$

$\ds \tan \theta$

When $\cos \theta = 0$ the expression $\dfrac {\sin \theta} {\cos \theta}$ is not defined.

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Definitions of the ratios
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.15$
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
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae