Question

Cosine law help! Really appreciate your guys hslp

Answer 1

Answer: Either 25.0 or 25 depending on how your teacher wants you to format the answer

===========================================================

Explanation:

To start off, it probably helps to translate what the question wants. 
It states "For the pilot of airplane B, calculate the angle between the lines of sight to the airplane at C and Jenny's airplane [at point A]". 
This fairly long, and possibly complex, sentence boils down to "find angle B"

To find angle B, we need to find the length of side 'a' first

Let,
a = x
b = 4.2
c = 5.7

Note how the lowercase letters (a,b,c) are opposite their uppercase counterparts (A,B,C). This is often the conventional way to label triangles. The lowercase letters are usually for the side lengths while the upper case is for the angles.

We have angle A = 120 degrees

Plug these values into the law of cosines formula below. Then solve for x
a^2 = b^2 + c^2 - 2*b*c*cos(A)
x^2 = 4.2^2 + 5.7^2 - 2*4.2*5.7*cos(120)
x^2 = 17.64 + 32.49 - 47.88*cos(120)
x^2 = 17.64 + 32.49 - 47.88*(-0.5)
x^2 = 17.64 + 32.49 + 23.94
x^2 = 74.07
x = sqrt(74.07)
x = 8.60639297266863
x = 8.6064

So side 'a' is roughly 8.6064 kilometers when we round to four decimal places

Now we'll use this to find angle B
Use the law of cosines again, but this time, the formula is slightly altered so that angle B is the focus instead of angle A

Plug in the side lengths (a,b,c). Solve for angle B
b^2 = a^2 + c^2 - 2*a*c*cos(B)
(4.2)^2 = (8.6064)^2 + (5.7)^2 - 2*(8.6064)*(5.7)*cos(B)
17.64 = 74.07012096 + 32.49 - 98.11296*cos(B)
17.64 = 106.56012096 - 98.11296*cos(B)
17.64 - 106.56012096 = 106.56012096 - 98.11296*cos(B)-106.56012096
-88.92012096  = -98.11296*cos(B)
(-88.92012096)/(-98.11296)  = (-98.11296*cos(B))/(-98.11296)
0.906303519535034 = cos(B)
cos(B) = 0.906303519535034
arccos(cos(B)) = arccos(0.906303519535034)
B = 25.0005785532867

It's a bit messier this time around, but we get the approximate angle
B = 25.0005785532867
which rounds to
B = 25.0 degrees
when we round to the nearest tenth. We can write "25.0" as simply "25"