Hello, magentarita!
A rectangle is inscribed in a semicircle of radius 1.
(a) Express the area $\displaystyle A$ of the rectangle as a function of $\displaystyle \theta.$
(b) Show that: .$\displaystyle A \:=\: \sin(2\theta)$ Code:
* * *
* | *
*-------+-------*
*| | * |*
| | 1* |y
* | | * θ | *
*-+-------+-------+-*
x
The area of the rectangle is: .$\displaystyle A \:=\:2xy$
We have: .$\displaystyle x \:=\:\cos\theta,\;y\:=\:\sin\theta$
Therefore: .$\displaystyle A \;=\;2\sin\theta\cos\theta \;=\;\sin2\theta$