Verified answer

first sketch it..

can't draw here but should look like

A--------->B---------->D

with another line from B up and to the right to point c. make C directly above the point D

the line from B to C has angle 33.5 above the horizontal.

now. let x = distance from B to D, y = distance from D to C

(a) distance from A to C

you know the distance from A to B = 200. if you can find x and y, then you would know that distance from A to C = [(200+x)^2 + y^2]^.5 right ? pythagorean theorem a^2 + b^2 = c^2

so find x and y.

you know the hypotenus = 360 and the angle = 33.5

sin 33.5 = y / 360

y = 360 * sin 33.5

cos 33.5 = x / 360

x = 360 * cos 33.5

then distance AC = [ (200 + 360 * cos 33.5)^2 + (360 * sin 33.5)^2] ^1/2

you calculate.....

(b)

sin angle = y / distance from AC

again. you calculate....

The best way to solve this, or any problem like it is to use right triangles. They are our friends, especially if you assume that the earth is flat. Now you can see that 200 km due west is not 200 km west on the surface of even a smooth ball or sphere, so you know why we make the assumption. On the earth in isn't very different, but on a small sphere like a moon of one of the giant planets it could be alot. I will make that assumption, which may not answer your question since you ask for straight-line distance, and then again maybe it will. I will have to think about it.

But, if you know the angle is 33.5 North and the distance is 360 km, then this 360 km is the hypotenuse (sp?) of the right triangle, the distance that you will have to travel east, through city A is the adjacent side, so this will be the long side right?

You must always draw a diagram of the problem and if you do and you realize that the angle is less than 45 degrees the the adjacent side has to be the long side. Then even if you forget if it is the sin or cos of 33.5, which ever gives you the longest length has to be right. So you can check your work.

In this case it is 360 km * cos 33.5 =300 km east. The distance north must then be 360 km * sin 33.5 =199 km.

To find the straight-line distance from C to A, you can not use the distance north because that is the same for both cities, you need the difference in distances travelled east (300 - 200 = 100 km east). The straight-line distance is then (100^2+199^2)^(1/2) which is the hypotenuse.

the adjactent divided by the hyp. gives you the cosine of the angle of C north from east. (100/AnswerA) = angle of city C north of east, BUT you want the angle north of west, so subtract that number of degrees from 180 and you get Answer b. (kind of a wierd way to ask for an answer if you are asking for things relative so you probably want to note that in your answer.) How you go from cosine to degrees really depends on your calc. but if you have a cos^-1 sign on yours that will convert it to degrees.

I hope that helps

Draw it out. AB is a 200km long line due east-west.

BC is a 360km line that runs towards the NW at an angle of 33.5 degrees from the first line.

Now you have a obtuse triangle. turn it so the hypotenuse is on the bottom, and drop a vertical line from the point and solve using Trig function (sin, cos, tan) to solve the remaining angles and lengths.

Here's a handy mnemonic for remembering those pesky Trig identities. I still use these, 28 years after high school, and people wonder how I can do trig so fast.

Sin(angle) = length of opposite side (from the angle) / hypotenuse

Cos(angle) = length of adjacent side (from the angle) / hypotenuse

Tan(angle) = length of opposite side (from the angle) / adjacent side (from the angle)

Sin = O/H

Cos = A/H

Tan = O/A

Sin = Oh / Heck

Cos = Another / Hour

Tan = Of / Algebra