Tangent is Sine divided by Cosine

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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:

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

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

$\blacksquare$


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