Hi Grace. To answer this question you first have to understand the directional bearing. N20 degrees E means 20 degrees east of north. So set up the coordinate axes to help you. The positive y direction is north. Measure off 20 degrees to the right of the y axes for east. (Eyeball it! These measurements don't have to be exact.) Draw a line from (0,0) with a length to represent 1 mile. At the endpoint, set up another (small) x,y axes. From the positive y line (which indicates North), estimate 70 degrees to the LEFT, for W (west). Draw a line in that direction 5 times longer than the first one, and label it 5 mi. The distance you are looking for is a side of a triangle! Draw in that third side so you can see the triangle.
Now the interesting thing is, the small y axis you just drew is parallel to the one you started with. So you have a set of alternate interior angles between these two y axes, that includes the original 20 degree angle you made. You can label that other angle 20 degrees, and it lies on the small y axis, along with the 70 degree angle you just drew. ----- You can't really know all that from just reading this question, nor would you readily understand what I am saying from what I am writing here. YOU HAVE TO DRAW IT OUT TO SEE IT. (That is the key to any geometry word problem!).
Hmmm. A 20 degree and 70 degree angle lying on a straight line (180 degrees). See it? What you have left in between is a 90 degree angle.
The distance you are looking for is the hypotenuse of a right triangle!
Pythagorean theorem. Don't forget to use units with your numerical answer.
Now you need a bearing - FROM the airport TO the aircraft. (We need to read carefully to know what they are asking!). From my rough drawing this will be WEST of North. You need that angle down by (0,0) - PART of which is your original 20 degrees. Since you now have all the SIDES of the triangle use trig to ge tthat whole angle that is part of the triangle down by your original (0,0):
sin(x) = 5/(hypotenuse).
Now the x you solve for INCLUDES the 20 degrees (EAST), and you don't want that in your bearing. Subtract 20, and you are left with the degrees WEST of North. Based on the explanation of directional bearings above, you have all you need to write it with the correct notation.
BOTTOM LINE: Draw out geometry word problems. It's the only way to see what is going on.
GOOD LUCK!
