What is the observed wavelength of the Hα line?

[Nat1234123]

Homework Statement

A star moves away from us at a speed of 10 km / s. the angle between the speed direction and the line of sight are 30 °. What is the observed wavelength of the Hα line?
Then we suppose the observer is also moving at a speed of 20 km / away from the star.
What is the observed wavelength of the Hα line?

Homework Equations

v_r = v_s cos(Θ) - v_o cos(Φ)
Δλ = λ - λ_o
Δλ/λ = v_r/c

The Attempt at a Solution

I know the answer should be = 656,30nm
And for the second, : = 656,343 nm
My attempt is:
Δλ = 10000/3*10^8 * 656.28 = 0.02
--> λ = 656.30
Same for the second one but instead of 10000 is 20000. And I got the right answer.
However, I haven't used the angel, which is what worries me that my answer is wrong

 

[DrClaude]

You have to post a full attempt at a solution.

Also, clearly state what you are after. Do you simply want a confirmation that the result is correct?

 

[Nat1234123]

You have to post a full attempt at a solution.

Also, clearly state what you are after. Do you simply want a confirmation that the result is correct?

More as, how to get there. I have the solution manual but I can't get to the solution. Hence I said the answer 'should' be.

 

[haruspex]

More as, how to get there. I have the solution manual but I can't get to the solution. Hence I said the answer 'should' be.

I am confused about a couple of things in your posts.

In post #1, you appear to be saying that you got the right answers but only by ignoring the information about the angle. In post #3, you say you can't get the "right solution". Do you mean that in the sense that the right solution ought to involve the angle?

In post#1 you show the details of how you got 656.30. If I follow the same method for the 20000km/s case I get 656.32, but you quote the correct answer as 656.34 and write that you got that. How did you get 656.34?

The question statement strikes me as ambiguous. It says the star is "moving away from us" at 10km/s. That could mean it is moving at 10km/s in a direction that is predominantly away from us, but sounds more like the rate at which it is getting further from us is 10km/s.
In the first interpretation you should be applying cos(30) to the speed (not that it makes much difference to the answer), whereas in the second the angle is irrelevant.