Question

Two stars in the sky appear to be 18.5° apart. Star A is 16.5 ly from the Earth, and star B, appearing to the right of star A, is 23.5 ly from the Earth. To an inhabitant of a planet orbiting star A, what is the angle in the sky between star B and our Sun?Two stars in the sky appear to be 18.5° apart. Star A is 16.5 ly from the Earth, and star B, appearing to the right of star A, is 23.5 ly from the Earth. To an inhabitant of a planet orbiting star A, what is the angle in the sky between star B and our Sun?

Answer 1

Here is a drawing of the situation:

Answer 2

Are you familiar with the law of cosines:

\(a² = b² +c² - 2bc \cos \alpha \)?

Answer 3

Here, b=16.5, c=23.5 and alpha=18.5 degrees.
Put these numbers in the formula. This gives you a, or better, it gives you a².
You don't have to calculate a, because you need another angle of the triangle.

Now you can also use the formula in another way: if you know the values of all the vertices, you can put these in the formula to calculate the angles.

The formula I wrote down earlier, can also be written doen for other angles:

1. \(a^{2}=b^{2}+c^{2}-2bc \cos \alpha\) (this is the original formula)
2. \(b^{2}=c^{2}+a^{2}-2ca \cos \beta\) (take each "next letter")
3. \(c^{2}=a^{2}+b^{2}-2ab \cos \gamma\) (take each "next letter").

Answer 4

If we call the angle we are looking for gamma, we use formula 3. and solve for cos gamma:

\(\cos \gamma=\dfrac{a^{2}+b^{2}-c^{2}}{2ab}\).

Now substitute a, b and c into this formula (needing a² and a after all, in spite of what I said - sorry!).

The inverse cosine of this number is the angle you need.

It sounds much more difficult than it is. I trust you are already familiar with this law of cosines, and the varieties of it. So all you have to do is use it twice to find (using your calculator) the desired angle.