Dear Uncle Colin,

When I'm 8 metres away from a flagpole, the angle of elevation to its top is exactly 40º, according to this angle-measurey-majigger I have here. What will it be when I'm only five metres away?

The Hypotenuse Eludes One; Does One Look Into Trigonometric Expressions?

Hi, THEODOLITE, and thanks for your message!

This is a classical two-step trigonometry problem: the first step is to figure out whatever you can, and the second is to find the thing you need. (In principle - and certainly in the new GCSE - the first step might be several steps.)

Here, you can work out the height of the flagpole: using right-angled trigonometry on the bigger triangle (with a base angle of 40º and an adjacent side of 8m), it's $8\tan(40º) \approx 6.713$ metres tall.

Now move on to the smaller triangle, with unknown base angle, but an adjacent side of 5m and a height of 6.713 . You need to work out $\tan^{-1}\br{\frac{Ans}{5}} \approx 53.3º$.

### Whoosh

"A ratio of 5:6.7 is about 15:20, so it's close to a 3-4-5 triangle. The larger base angle there is 0.927 radians, or about 53º."

"Thank you, sensei!"

Hope that helps!

* Edited 2018-08-22 to add a comment from Adam Atkinson. Thanks, Adam!

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.