Question

In the context of the mathematical identities found in Siddhantic trigonometry, choose the correct set of definitions:
A. \(\frac{\text{Trijya} \times \text{Kotijya}}{\text{Bhujajya}} = \text{Kotisparsharekha} \) (Cotangent)
B. \(\text{Bhujajya} - \text{Trijya} = \text{Natamshajya} \)
C. \(\frac{\text{Trijya} \times \text{Kanchha}}{\text{Tri}} = \text{Vijya} \)
D. \(\text{Tri} - \text{Bhujajya} = \text{Padujya} \)
E. \(\frac{\text{Bhujajya} \times \text{Jizjya}}{\text{Trijya}} = \text{Kranjya} \) (Sine of Declination)

• A. Only A, B, C
• B. Only A, C, E
• C. Only A, D, E
• D. Only B, C, D

Hint:

Formula for Sun's Declination (Kranjya) is the most repeated formula in this exam. It is always: (Sine of Longitude $\times$ Sine of 24) Radius.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Indian trigonometry is arc-based rather than angle-based. The standard radius is called 'Trijya'. The primary functions are 'Jya' (Sine), 'Kotijya' (Cosine), and derived ratios like 'Sparsharekha' (Tangent/Cotangent). These are used to calculate the 'Kranti' (Declination) of the Sun based on its position in the zodiac.

Step 2: Detailed Explanation of Statements:
1. Cotangent Definition (A): In modern terms, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). In Siddhantic terms, this is \(\frac{R \cdot \text{Kotijya}}{R \cdot \text{Jya}}\). Multiplying by the radius constant (Trijya) for scaling, we get \(\frac{\text{Trijya} \times \text{Kotijya}}{\text{Jya}}\). This is exactly the definition of 'Kotisparsharekha'. Statement A is TRUE.
2. Kranjya/Declination (E): The most famous formula in the 'Tripra-shna' chapter is to find the declination (Kranti). It is given by:
\[ \sin(\delta) = \sin(\lambda) \times \sin(\epsilon) \]
In Sanskrit terms: \(\text{Kranjya} = \frac{\text{Bhujajya (Sine of Longitude)} \times \text{Jina-jya (Sine of max declination } 24^\circ\text{)}}{\text{Trijya}}\). The 'Jizjya' in the OCR represents the 'Jina-jya' ($24^\circ$). Statement E is TRUE.
3. Statement C: This represents a proportionality used in similar triangles for planetary distance or refraction, following the 'Rule of Three' (Trairashika). In the set provided, A and E are mathematically standard, making (2) the correct grouping.

Step 3: Verification:
- Statement B is false because subtracting a Sine from the Radius gives the Versine (Utkramajya), not Natamshajya.
- Statement D is also a variation of the Versine definition and is incorrectly labeled as 'Padujya'.

Step 4: Final Answer:
The correct mathematical identities are A, C, and E.