find the length of cable

Question

A communication tower (the side CB) is located at the top (the point C) of a steep hill. The angle of inclination of the hill is 58 degrees. A guy wire is to be attached to the top (the point B) of the tower and to the ground (the point A), 105 m downhill from the base of the tower (the side AC). The angle <BAC in the figure is 12 degrees.

Find the length of cable (the side AB) required for the guy wire

answer is BLANK m

Answer 1

I'll do this with trigonometry for right angled triangles.

First, sketch the problem as described, to show the triangle ABC where A is ground, B is top of tower, C is base of tower.

Add one more point, Z, which vertically below the tower (and both C and B), and horizontal from A. This forms a right angled triangle AZC.

We have the angle of inclination (angle CAZ), and the hypotenuse (AC).

Hence we can use cos (angle CAZ) = adjacent (AZ) / hypotenuse (AC)

=> cos (58 deg) = AZ / 105

=> AZ = 105 cos (58 deg)

Now we can figure out the hypotenuse (required side AB) of larger right angled triangle AZB.

cos (angle BAZ) = adjacent (AZ) / hypotenuse (AB)

=> cos (12 deg + 58 deg) = 105 cos (58 deg) / AB

=> AB = 105 cos (58 deg) / cos (70 deg)

=> AB = 162.7 meters (to 1 dp)