9.04 Solve problems involving right-angled triangles

Exploration

Karen is standing $30$30 m from the base of a tower with an angle of elevation of $27^\circ$27° from the top of the tower. If Nick's angle of elevation is $44^\circ$44° from the top of the tower, what is his distance $x$x from the base of the tower?

In order to work out the distance from Nick to the base of the tower, we need to know either the height of the tower or Nick's distance from the top of the tower.

Since we are given the distance from Karen to the base of the tower, we can apply trigonometry once to find the height of the tower, then this value to find our desired value.

Using trigonometry once, we can calculate the height $h$h of the tower to be:

$h=30\tan27^\circ$h=30tan27°

Using this new value, we can find the distance from Nick to the base of the tower $x$x (rounding to two decimal places) to be:

$x$x $=$= $\frac{h}{\tan44^\circ}$htan44°​ $=$= $\frac{30\tan27^\circ}{\tan44^\circ}$30tan27°tan44°​ $=$= $15.83$15.83

Exploration

Joshua and Catherine are looking up at the top of a statue with angles of elevation of $29^\circ$29° and $43^\circ$43° respectively. If Catherine is standing $3$3 m closer to the statue than Joshua, what is the height $h$h of the statue?

Using both triangles, we can find the distance from both Catherine and Joshua to the base of the statue in terms of $h$h.

$\text{Joshua to the statue}=\frac{h}{\tan29^\circ}$Joshua to the statue=htan29°​

$\text{Catherine to the statue}=\frac{h}{\tan43^\circ}$Catherine to the statue=htan43°​

Since Catherine is $3$3 m closer to the base of the statue, we know that:

$\frac{h}{\tan29^\circ}-\frac{h}{\tan43^\circ}=3$htan29°​−htan43°​=3

Since this is now a simple equation involving only $h$h, we can solve for the height of the statue (rounding to two decimal places), which we find to be:

$h$h $=$= $\frac{3}{\frac{1}{\tan29^\circ}-\frac{1}{\tan43^\circ}}$31tan29°​−1tan43°​​ $=$= $4.10$4.10