A pilot flies her plane on a bearing of 35 degrees minutes from point X to point Y, which is 400mi from X. Then she turns and flies on a bearing of 145 degrees to point Z, which is 400 mi from her starting point X. What is the bearing of Z from X, and what is the distance YZ?

400 sin 145 degree/sin 35 degree = this gave me 400mi which I already know.

I'm stuck on the steps to find what I need the bearing and distance.

Any insight is appreciated, thank you.

To find the bearing of point Z from point X, we need to determine the angle between the line XZ and the reference line (north, for example). We can do this by finding the difference in bearings between the flights XY and YZ.

Given that the bearing of XY is 35 degrees and the bearing of YZ is 145 degrees, we can subtract the bearing of XY from the bearing of YZ to find the angle between the legs of the trip:

145 degrees - 35 degrees = 110 degrees

Thus, the bearing of Z from X is 110 degrees.

To find the distance YZ, we can use the Law of Cosines. Let's denote the distance YZ as d.

Using the Law of Cosines, we have:

d^2 = 400^2 + 400^2 - 2 * 400 * 400 * cos(110 degrees)

Simplifying the equation:

d^2 = 400^2 + 400^2 - 2 * 400 * 400 * cos(110 degrees)

d^2 = 160,000 + 160,000 - 2 * 400 * 400 * cos(110 degrees)

d^2 = 320,000 - 2 * 400 * 400 * cos(110 degrees)

Calculating the value:

d^2 = 320,000 - 2 * 400 * 400 * (-0.3420)

d^2 = 320,000 - 2 * 400 * 400 * (-0.3420)

d^2 = 320,000 - 2 * 400 * 400 * (-0.3420)

d^2 ≈ 320,000 - 275,520

d^2 ≈ 44,480

Taking the square root of both sides, we get:

d ≈ sqrt(44,480)

d ≈ 210.93

Therefore, the distance YZ is approximately 210.93 miles.

To find the bearing of Z from X, you'll need to add the initial bearing of 35 degrees and the turn of 145 degrees. Adding them together will give you a bearing of 180 degrees, since the total sum is greater than 180 degrees, you need to subtract 180 from the total. Therefore, the bearing of Z from X is 180 degrees - 145 degrees = 35 degrees.

Now, to find the distance YZ, we can use the Law of Cosines. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle.

In this case, we have a triangle with side XY of 400 miles, side XZ of 400 miles, and the included angle YXZ of 180 degrees - 35 degrees = 145 degrees.

Applying the Law of Cosines, we have:

YZ^2 = XY^2 + XZ^2 - 2 * XY * XZ * cos(YXZ)

Plugging in the values we know:

YZ^2 = (400mi)^2 + (400mi)^2 - 2 * (400mi) * (400mi) * cos(145 degrees)

Now, calculating this equation will give you the square of the distance YZ. Then, taking the square root of that value will give you the actual distance YZ.

I hope this clears up the steps for finding the bearing of Z from X and the distance YZ. Let me know if you have any further questions!

There are about 20 ways to do this, wondering what your instructor wanted.

1) Easiest way: sketch the triangle, you have an isosceles triangle, and you can easily deduce from your sketch the included angle. Law of cosines to find the unknown side, then law of sines to get the other angles, and from the sketch, you can deduce the bearing from those angles.
2) add the given vectors:
R is resultant, so you are looking for -R
R=vector1 + vector 2
= 400cos35 N + 400sin35 E + 400cos145 N + 400sin145 E
combine the N, E components to get a single N, E componsent.
-R is the negative of those.
bearing= arctan(Eastcomponent/Ncomponent)
distance^2=Ncompon^2 + E component^2