Srinivasa Ramanujan's Contributions to Trigonometry

Ramanujan's Pi Formula is a series that provides an approximation of the value of (\pi). It is given by the formula (\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{(4n)! (1103 + 26390n)}{(n!)^4 396^{4n}}).

Ramanujan's Sum of Two Squares Theorem states that every positive integer can be expressed as the sum of two squares in four different ways. This theorem is also known as the Four Squares Theorem.

Ramanujan's Sine-Cosine Identity is a trigonometric identity that relates the sine and cosine functions. It is given by the formula (\sin^2 \theta + \cos^2 \theta = 1).

Ramanujan's Tangent Approximation Formula is a series that provides an approximation of the value of the trigonometric function (\tan \theta). It is given by the formula (\tan \theta = \frac{\theta}{1 + \frac{\theta^2}{3} + \frac{2\theta^4}{15} + \frac{17\theta^6}{315} + \cdots}).

Ramanujan's Modular Equation Theorem states that the modular equation has a unique solution for every value of (\tau) in the upper half-plane. This theorem is also known as the Ramanujan-Petersson Conjecture.

Ramanujan's Modular Function is a function that is related to the modular equation. It is defined by the formula (\Delta(\tau) = \eta(\tau)^{24}), where (\eta(\tau)) is the Dedekind eta function.

Ramanujan's Modular Function Identity is an identity that relates the modular function and the Dedekind eta function. It is given by the formula (\Delta(\tau) = \eta(\tau)^{24}).

Ramanujan's Modular Function Zero Theorem states that the modular function has a simple zero at (\tau = 0) and a simple pole at (\tau = \infty).

Ramanujan's Modular Function Approximation Formula is a series that provides an approximation of the value of the modular function. It is given by the formula (\Delta(\tau) = \sum_{n=1}^{\infty} \tau^{n(n+1)/2}).

Ramanujan's Modular Form Theorem states that the modular form (\phi(\tau)) has a simple zero at (\tau = 0) and a simple pole at (\tau = \infty).

Ramanujan's Modular Form Approximation Formula is a series that provides an approximation of the value of the modular form (\phi(\tau)). It is given by the formula (\phi(\tau) = \sum_{n=1}^{\infty} \tau^{n^2}).

Ramanujan's Modular Form Identity is an identity that relates the modular form (\phi(\tau)) and the Dedekind eta function. It is given by the formula (\phi(\tau) = \eta(\tau)^{24}).

Ramanujan's Modular Form Zero Theorem states that the modular form (\phi(\tau)) has a simple zero at (\tau = 0) and a simple pole at (\tau = \infty).

Ramanujan's Modular Form Approximation Formula is a series that provides an approximation of the value of the modular form (\psi(\tau)). It is given by the formula (\psi(\tau) = \sum_{n=1}^{\infty} \tau^{n(n+1)/2}).