This paper presents the findings of a research project on Sanhag Seubyu, a work by mid-19^th-century Joseon mathematician Jo Hui-sun. Joseon's efforts to adopt China's new calendrical system began in the mid-17^th century and continued for nearly two centuries, becoming fully documented by the 1860s. While mathematical records from this period are largely the collaborative work of Nam Byeong-gil and Lee Sang-hyeog, Jo Hui-sun's Sanhag Seubyu indicates that astronomical mathematics was of interest among scholars, suggesting that other scholars may have reached similar levels of expertise. Sanhag Seubyu comprises seven chapters, with all but the first and last focusing on spherical trigonometry. This paper presents our investigation into the content of the first two of the five chapters.