A star over Madrid
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 1.5
Adapted from IOAA 2013 (Volos), theory problem 9. We have restated it in our own words and kept the original data. The solution was last checked on 15 October 2026.
An observer in Madrid, at latitude 40° N, sees a star at a zenith distance of 30° and an azimuth of 50°, with the azimuth measured from the south point. Find the star's hour angle and declination.
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The problem does not say which way the azimuth is counted. Azimuths measured from the south were traditionally counted towards the west, and we assume that here, so 50° puts the star in the south-west, at azimuth
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The cosine rule for the side PX, as in (1.20), gives
so
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(1.19) rearranged gives
so the angle at P is 23.76°.
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The star is west of the meridian, so its hour angle is
AnswerThe star's declination is +18°05′ and its hour angle is 1h 35m, west of the meridian.
This problem also appears among the olympiad problems of Section 1.5, The astronomical triangle, which teaches the ideas it uses. Equation and figure numbers refer to the course.