# A plane is 120 miles north and 85 miles east of an airport. If the pilot wants to fly directly back to the airport, what angle should be taken?

May 21, 2017

The plane has to turn 125.31° right to fly directly back towards the airport.

This would be on a bearing of 215°

#### Explanation:

We can firstly map out how far north and east the pilot was from the airport, to find the angle to get back to the Airport.

(not to scale)

$x = {\tan}^{-} 1 \left(\frac{120}{85}\right) \text{ }$see note below

x = 54.69º

Because the plane is flying due east, it has to turn 90º to face south, before it turns the extra 35.31º to fly towards the airport.

90° + 35.31° = color(blue)(125.31º)

It actually turns through the exterior angle of the triangle :
180-54.69° = 125.31°

The question would be better if it asked for the bearing.
(Which is what a pilot would use)

Note: ${\tan}^{-} 1$ is read as "arctan: (to find an angle from the ratio)

and is not the same as $\frac{1}{\tan}$ which would be $\cot$