Double Trig Questions

Question

As modeled below, a movie is projected onto a large outdoor screen. The bottom of the 60-foot-tall screen is 12 feet off the ground. The projector sits on the ground at a horizontal distance of 75 feet from the screen. Determine and state, to the nearest tenth of a degree, the measure of θ, the projection angle.

Answer 1

It is unclear what the theta is. So I drew three angles here alpha, beta and gamma.

So we have

tan(gamma) = 12/75,

tan(beta) = (60+12)/75.

So

beta = arctan(72/75) = 43.8 deg,

gamma = arctan(12/75) = 9.1 deg.

and

alpha = beta - gamma = 34.7 deg.

Answer 2

1. Use this right triangle to solve for the hypotenuse leg (either one would work actually). I will just use the 12, 75 to solve for the hypotenuse of that triangle.
2. 12² + 75² = C² and solve for C, C ≅ 75.95 feet
3. Use that base leg (C) and use the tan θ = 60/75.95, take the inverse tan of both sides to solve for θ
4. θ = tan^-1 (60/75.95) = 38.3°