1. ## Trig Problem

The ‘London Eye’ can be considered to be a circular frame of radius 67.5m on the
circumference of which are ‘capsules’ carrying a number of people round the circle. Take
a co-ordinate system where O is the base of the circle and Oy is a diameter. At any time
after starting off round the frame, the capsule will be at height h metres when it has
rotated .°θ

Thanks

2. Hello, p4pri!

Did you make a sketch?

The ‘London Eye’ can be considered to be a circular frame of radius 67.5m
on the circumference of which are ‘capsules’ carrying a number of people round the circle.
Take a coordinate system where O is the base of the circle and OY is a diameter.
At any time after starting off round the frame, the capsule will be at height h metres
when it has rotated θ°.

Apparently answer is: .$\displaystyle 67.5(1-\cos\theta)$ . . . why?
Code:
              * * *
*           *
*               *
*                 *

*         C         *
-   *         o         *
:   *    R  * |         *
:         * θ | Rcosθ
R    *  *     |        *
:   A o - - - + B     *
:     : *     |     *
-   --+-----*-*-*------
D       O

The center of the circle is $\displaystyle C.$
The capsule is at $\displaystyle A.$
$\displaystyle \angle ACO = \theta$

The radius is: .$\displaystyle CA = CO = R$

In right triangle $\displaystyle CBA\!:\;\cos\theta \:=\:\frac{CB}{R} \quad\Rightarrow\quad CB = R\cos\theta$

The height of the capsule is: .$\displaystyle h \;=\;AD \;=\;BO \;=\;CO - CB \;=\;R - R\cos\theta$

Therefore: .$\displaystyle h \;=\;R(1-\cos\theta)$