# 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.

## Proof

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

Then:

\(\displaystyle \frac {\sin \theta} {\cos \theta}\) | \(=\) | \(\displaystyle \frac {y / r} {x / r}\) | Sine of Angle in Cartesian Plane and Cosine of Angle in Cartesian Plane | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac y r \frac r x\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac y x\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tan \theta\) | Tangent of Angle in Cartesian Plane |

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

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.15$