trigonometry application question

Question

I have a trigonometry question from my textbook that I am not sure how to approach.

View from a Satellite: The figures on the next page indicate that the higher the orbit of a satellite, the more of the earth the satellite can “see.” Let u, s, and h be as in the figure, and assume that the earth is a sphere of radius 3960 mi.

(a) Express the angle $\theta$ as a function of h.

I know the dotted lines measure 3960 since that is the radius but there is no right angle in the triangles. I am not sure how to approach this question.

Answer 1

There is, in fact, a right angle in this question, but it isn't drawn in for you. The black lines in the diagram are tangent to the circle, so they are perpendicular to its radius, forming a right triangle with sides of the circle's radius, the tangent to the circle, and the line from the satellite to the center of the earth. In this case, since you now have a right triangle, you can say that $$\cos\theta=\frac{3960}{3960+h}$$ $$\frac{1}{\cos\theta}=\frac{3960+h}{3960}$$ $$\frac{3960}{\cos\theta}=3960+h$$ $$h=\frac{3960}{\cos\theta}-3960$$ Which should be your final answer.

Answer 2

There is, actually a right angle. The lines that form a big triangle are nothing but tangents to the surface. Thus that is actually nothing but a right triangle.

Therefore, $$\sec\theta=\frac{r+h}{r}$$ where $r=360 miles$