Non-Right Triangles: Law of Cosines, Thank you!

Question

A steep mountain is inclined 74 degree to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 830 ft out in the plain from the base of the mountain. Find the shortest length of cable needed. Round your answer to the nearest foot.

The shortest length of cable needed is 

Answer 1

Since you mention the law of cosines in the your question title, I assume you're being asked to use that law. So here's how I would do it using the law of cosines.

The triangle you're considering is defined by three sides. The first is the straight cable from the ground to the mountain peak (straight because a straight line minimizes length); this side has length c. The second side is the mountain incline (plain up to 3400'); it has length b. These two meet at the summit. The last side connects the base of the mountain to the start of the cable on the ground (830' from the base of the mountain). It has length 830' (let's call it side a). We want c. Unfortunately we don't know b just yet.

The mountain's incline defines a right triangle. The hypothenuse goes from the plain to the peak at 3400'. One right angle side is horizontal, the other is vertical and length 3400'. The angle between the horizontal size and mountain incline is 74 degrees. This tells us that the hypothenuse has length:

3400/sin(74) = hypothenuse = b = 3537.018'

So now we know almost everything we need to get c. The last bit is the angle between the mountain incline and the valley floor (the angle opposite the side that follows the cable). This angle is 180 - 74 = 106 degrees.

So...

c = sqrt(a^2 + b^2 - 2abcos(106)) = sqrt(830^2 + (3537.018)^2 - 2(830)(3537.018)cos(106))

= 3849.388

Rounded to the nearest foot, the answer is 3849'