# Rectangle Inscribed in Semicircle...Part 1

• Aug 29th 2008, 12:04 PM
magentarita
Rectangle Inscribed in Semicircle...Part 1
A rectangle is inscribed in a semicircle of radius 1.

(a) Express the area A of the rectangle as a function of the angle theta.

(b) Show that A = sin(2theta)
• Aug 29th 2008, 02:27 PM
Jhevon
Quote:

Originally Posted by magentarita
A rectangle is inscribed in a semicircle of radius 1.

(a) Express the area A of the rectangle as a function of the angle theta.

(b) Show that A = sin(2theta)

um, what angle is theta?

in any case, i think it would be a good idea for you do draw this on a pair of axis. let the rectangle sit on the x-axis with its top vertices touching the upper-half circle of radius 1 centered at the origin. try to center the rectangle also. see if that gives you any ideas
• Aug 29th 2008, 02:32 PM
Soroban
Hello, magentarita!

Quote:

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: .$\displaystyle A \;=\;2\sin\theta\cos\theta \;=\;\sin2\theta$

• Aug 29th 2008, 06:30 PM
magentarita
Quote:

Originally Posted by Soroban
Hello, magentarita!

Code:

              * * *           *    |    *         *-------+-------*       *|      |    * |*         |      |  1*  |y       * |      | * θ  | *       *-+-------+-------+-*                     x
The area of the rectangle is: .$\displaystyle A \:=\:2xy$

We have: .$\displaystyle x \:=\:\cos\theta,\;y\:=\:\sin\theta$

Therefore: .$\displaystyle A \;=\;2\sin\theta\cos\theta \;=\;\sin2\theta$

Good job Soroban. Nice try Jhevon.