The use of azimuth and elevation angles is prevalent across various fields and applications, each leveraging these angles to address specific challenges and achieve precise outcomes. Some of the systems that require an intricate use of these two angles are navigation systems, satellite communication systems, remote sensing systems, optical systems, etc. Azimuth and elevation angles can be used together to precisely specify the direction or position of an object in three-dimensional space, especially in relation to an observer or reference point. In this comprehensive study, we delve into the determination of an intricate mathematical relationship between azimuth and elevation angles. Through complex, yet basic trigonometry and limits, an attempt has been made to meticulously derive a mathematical relationship between azimuth and elevation angles. Along with the mathematical derivation, we also aim to mathematically prove why azimuth angle tends to become undefined or indeterminate at the singularity point, that is, when the elevation angle approaches 90°.