# Correct longitude from longitude over 180°

Last week, one of my exercise tasks was about determining the an observer's position on Earth, when in their zenith is a certain star with known right ascension and declination. We were supposed to use this relation for the hour angle $$\tau$$ to determine the longitude $$\lambda$$: $$\tau = \text{GST} - \lambda - \alpha$$, which would transform to $$\lambda = \text{GST}-\alpha-\tau$$. Everything to here is clear to me, including calculating $$\lambda$$.

My problem is that in the case of our exercise, the calculated longitude is bigger than 180°. For simplicity, let's say $$\lambda=230°\text{E}$$. My naive thought process last week was, that this would mean that the longitude would simply continue going an additional 50° into the Western hemisphere, such one could say the true $$\lambda$$ on the Western side is 360° minus the calculated value. In my example: $$\lambda=130° \text{W}$$.
However, this week, I was told this is wrong and that we should still calculated 360° minus the calculated value, but instead on the Western hemisphere, we still would be on the Eastern one. Thus, $$\lambda=130°\text{E}$$. Can someone explain to me, why this is the case? I've asked our tutor, but they couldn't explain this, without confusing me (and my fellow students) even more.

After an emergency landing, you find yourself on an unknown beach. Directly above you, you can locate the star $$\Psi$$ Leo. You know that for this star $$\alpha=9^\text{h} 43^\text{m} 54^\text{s}$$ and $$\delta=14^\circ1'15"$$. The time is $$17^\text{h} 17^\text{m} 12^\text{s}$$ UT. In addition, you know that for the current day at 0 UT, we have $$8^\text{h} 19^\text{m} 50^\text{s}$$ GMST. Where are you located? Determine the longitude and latitude, as well as the name of the location.
Relation between hour angle and right ascension is given by $$\tau = \theta_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \lambda - \alpha.$$ Since the star is directly above us, it's in the zenith and thus, $$\tau=0$$. It follows for the longitude that $$\lambda = \theta_\text{G}(t=0) + t\left(\frac{366.24}{365.24}\right) - \alpha.$$ It is given from the exercise that $$\theta_\text{G}(t=0) = 8^\text{h} 19^\text{m} 50^\text{s} = 8.33056\text{h},$$ $$t = 17^\text{h} 17^\text{m} 12^\text{s} = 17.28667\text{h},$$ $$\alpha = 9^\text{h} 43^\text{m} 54^\text{s} = 9.73167\text{h}.$$ Plugging everything into the equation for $$\lambda$$ leads to $$\lambda \approx 15.93289\text{h}=238.99335^\circ E.$$ Until this point everything was marked as correct (and matches with the sample solutions of the lecturer).