Hello, fayeorwhatsoever!

We need a good sketch . . .

A magician standing on a platform subtends an angle of 15° at a point on the ground.

The angle of elevation of the platform from the same point is 45 degrees.

The magician is 180 cm high.

Find the height of the platform. Code:

* A
*|
* |
* | 180
* |
* |
* * B
* * |
*15°* |
* * | h
* * |
** 45° |
P * - - - - - * C
h

The magician is: $\displaystyle AB = 180$

The height of the platform is: $\displaystyle h = BC$

The observation point is $\displaystyle P\!:\;\;\angle APB = 15^o,\;\angle BPC = 45^o$

Since $\displaystyle \angle BPC = 45^o,\;\Delta BCP$ is an isosceles right triangle: .$\displaystyle PC = BC = h$

In $\displaystyle \Delta ACP\!:\;\;\tan60^o \:=\:\frac{h+180}{h} \quad\Rightarrow\quad \sqrt{3} \:=\:\frac{h+180}{h}\quad\Rightarrow\quad h \:=\:\frac{180}{\sqrt{3}-1}$

Therefore: .$\displaystyle h \;=\;245.8845... \;\approx\;246\text{ cm}$