# Trigonometry in a crank shaft diagram. Show that $\tan \alpha = \frac{r \sin\theta}{a-r \cos \theta}$

Let $OAB$ be a triangle with $\angle AOB = \theta, \angle OBA = \alpha$ and $|OA| = r$. Then show $$\tan(\alpha)= \frac{r\sin(\theta)}{a-r\cos(\theta)}.$$ I'm not sure how this is done. There's a similar question on Stack Exchange, but I don't understand converting the image to a parallelogram to find tan. I thought to maybe use the Sine Rule, but I'm not sure if that works out? I'm not the best at trigonometry.

## 2 Answers

Drop perpendicular from $A$ on $OB$. Name this point $P$. We will now get $OP = r \cos \theta$ and $AP = r \sin \theta$. Also $OB = a$

$$\tan\alpha = \frac{AP}{BP} = \frac{AP}{OB-OP} = \frac{r \sin \theta}{a-r \cos \theta}$$

Drop a perpendicular from $A$ to $OB$, let that intersect $OB$ at point $P$

In right $\triangle APB$, $\tan \alpha = \frac{AP}{PB}$

In right $\triangle OPA$, $AP = r\sin\theta$ and $OP = r\cos\theta$

$PB = OB - OP = a - r\cos\theta$

• Why the downvote? – Deepak Oct 29 '17 at 5:08