Trigonometry satellite question

Question

The figure below shows a satellite orbiting Earth. The satellite passes directly over two tracking stations A and B, which are 82 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 83.9 degrees and 86.2 degrees respectively. How far is the satellite from station A and how high is the satellite above the ground? Round answers to the nearest whole mile.

Answer 1

Find the angle from the satellite (angle C). Add 83.9^o and 86.2^o to get 170.1^o. Then subtract from 180^o to get 9.9^o.

Use the law of sines to find the 2 missing side lengths.

sin(83.9)/a = sin(86.2)/b = sin(9.9)/82

a = [82*sin(83.9)/sin(9.9)]

b = [82*sin(86.2)/sin(9.9)]

You should get a = 474 miles and b = 476 miles.

Answer 2

I drew a pattern. Side a=82; angle α=9.9°; β=83.9°; γ=86.2°.

You want side "c"; and side "b" of a right triangle.

So a/(sinα)=b/(sinβ)=c/(sinγ)

82/(sin9.9)=476.94=c/(sin86.2) so c=475.89.

Now the rt triangle is used.

So a^2+b^2=c^2. In this case it's got to be rearranged.

475.89^2-82^2=b^2; b=468.77