How would you solve this geological trig question?

Question

Sally is a surveyor in the badlands in South Dakota. At point A on the south side of a deep canyon, she sights across to a fire tower on the north side of the canyon at point B. Then she moves a measured 500 feet along the south rim of the canyon to a point C on a line perpendicular to her original sight line AB. Setting up her transit again, she finds that her line of sight, CB, to the tower is 83 degrees. Correct to the nearest whole foot, how far is point C from the tower?

Answer 1

The south rim is parallel to the north rim. So moving along the south rim is the same as moving perpendicular to AB. That means B is exactly opposite A across the canyon. ie AB is perpendicular to the rims of the canyon.

AC = 500.
Her line of sight is 83 degrees. That's very vague. Line of sight from what?
If you assume that to be 83 deg from the rim of the canyon, then ABC is a right angled triangle with ACB = 83 deg.
So BC = AC / cos83 = 4103 feet.

AB = 4072 feet which would be the width of the canyon.

Answer 2

Draw two parallel lines representing the N and S side of the canyon. Create a triangle. Point A and C are on the S side and point B is directly across from Point A creating a right angle at Angle A. Angle C is 83 degrees.
Remember SohCahToa: Sin=opposite/hyp, cos=adj/hyp, tan=opp/adj.

You have an angle(83 deg), the length of the side adjacent to the angle (500ft) and the length you are looking for (hypotenuse).

cos(ang) = adj/hyp

Plug in and solve.

Answer 3

using cosine it will give you adjacent/hyp
Calculate cos83 from that formula and you will get the answer

Answer 4

x = 500/cos83 = ~4100 ft