# Area of Triangle with Unit Circle

On the image below line DB is tangent to the circle. The problem is to try and express both $\Delta OBC$ and $\Delta OBD$ in terms of $\sin\theta$ and $\cos\theta$.

For $\Delta OBC$ $$A=\frac{1}{2}OB(CA)$$ I found out all the trig functions for it$$\sin\theta=\frac{CA}{OC}$$ $$\cos\theta=\frac{OA}{OC}$$ $$\tan\theta=\frac{CA}{OA}$$ I am having trouble using the trig functions to express area in $\sin\theta$ and $\cos\theta$ • Area of triangle $OBD$ is $\frac{1}{2} \cdot OB \cdot DB = \frac{1}{2} \tan \theta$ – TheSimpliFire Oct 30 '17 at 20:29
• @TheSimpliFire That clears things up, thanks – RedRobin Oct 30 '17 at 21:36
• $\tan\theta=DB$, then use similar triangles. – amd Oct 30 '17 at 22:36

## 1 Answer

$\Delta OBC$ and $\Delta OBD$ are not similar triangles. $\Delta OBC$ is not a right triangle.

Also, by "$\theta$", I assume you mean $x$? If so, $\tan\theta\neq\frac{CA}{OC}$, it is $\tan\theta=\frac{CA}{OA}$. Because $OC = 1$, the trig functions simplify to $$\sin \theta = CA$$ $$\cos \theta = OA$$

$$A_{OBC}= \frac{1}{2}(OB)(CA) = \frac{1}{2}(1)(\sin \theta) = \frac{1}{2}\sin \theta$$