1.
A mountain is due north and the peak is 20°
above the horizon (the angle between a level line drawn straight north and a
line drawn to point at the peak is 20°). You walk 1000 feet due east and the
mountain is now 10° west of north (you have to rotate 10° counterclockwise from
facing north to be facing the peak). How tall is the mountain? Note that the
ground is level as you walk east and we are assuming the earth is flat.

Thank you for the opportunity to help you with your question!

The picture above is your problem. (not in scale)

If you walk 1000ft east and you need to turn your head 10 degree to see the peak when you are facing north that means you have an angle of 80 degree with the West-East. Since you moved 1000 ft you can calculate the distance between you and the the peak before you walk 1000 ft (d).

tg 80 =d/1000 => d=5671.3 ft

so peak height (h) can be calculated

tg 20= h/d=h/5671.3

so h=2064.2 ft

Please let me know if you need any clarification. I'm always happy to answer your questions.