A laser beam is aimed...

Question

A laser beam is aimed 15.95° above the horizontal at a mirror 11,648 away. It glances off the mirror and continues for an additional 8570. m at 11.44° above the horizon until it hits its target. Calculate the angle of elevation from the initial position to the final position. Please answer in 4 decimal places :)

Answer 1

I am ignoring the curvature of Earth's surface for this problem. My sketch has an initial angle of 15.95° above the x axis, hypotenuse 11,648m, opposite side h_a, and adjacent side x_a. From the mirror, I have another triangle at 11.44° above the x axis, hypotenuse 8,570.m, opposite side h_b, and adjacent side x_b. So the height of the target above the x axis is h = h_a + h_b, and total horizontal distance is x = x_a + x_b.

For the first triangle, sin 15.95 = h_a/11,648 and h_a = 3200.₈ [I carry 5 figures to get 4 sig fig precision]; cos 15.95 = x_a/11,648 and x_a = 11,200.

For the second triangle, sin 11.44 = h_b/8,570 and h_b = 1699.₈; cos 11.44 = x_b/8,570 and x_b = 8,400.

Now h = 4,900.₆ and x = 19,600. For the big triangle from the origin to the target, tan θ = h/x = 0.2500₃ and θ = 14.04°.