A cross country skier observes the top of a radio tower, due south of her, at an angle elevation of 32 degrees. She then skis on a bearing of 130 degrees for 560m and finds herself due east of the tower.

a. Calculate the height of the radio tower
b. Calculate the distance from the tower to the skier
c. Calculate the angle of elevation of the top of the tower from the new position

This is solved by trigonometry.

A line from the top of the radio tower to the ground makes a right-angle with the ground. You connect that line to the position of the observer to get a triangle. This triangle has a 32 degree angle from the ground. The side opposite the 32 degrees is the height of the tower. If you knew the distance from the tower to the observer, you could calculate the height from the tangent of the angle.

A second triangle is formed on the ground by the tower, the original location of the skiier, and the final location of the skiier. Note that "South" is a bearing of 180 degrees. You know the length of one of these legs.

The two triangles that share an edge.

Because you are due south of the tower, the triangle is a right triangle lying on the ground.

The hypotenuse of the ground-based triangle is 560m and the angle can be determined from the bearing you she traveled at, and the fact that she is now south of the tower.

Knowing the angle and the length of one side is sufficient to calculate the length of the common side from the sine of the angle that you calculated.

Since the side is common to the two triangles, you now know one side of the vertical triangle and you know the angle. Using that, you can calculate the height of the side of the vertical triangle that represents the tower's height.