Nilakantha Somayaji lived in the 15th century.

Nilakantha Somayaji is best known for his work on trigonometry.

Nilakantha Somayaji developed the sine series for the sine function.

Nilakantha Somayaji developed the cosine series for the cosine function.

The general formula for the sine series is $sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$.

The general formula for the cosine series is $cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$.

Using the sine series, we have $sin(\frac{\pi}{2}) = \frac{\pi}{2} - \frac{(\frac{\pi}{2})^3}{3!} + \frac{(\frac{\pi}{2})^5}{5!} - \frac{(\frac{\pi}{2})^7}{7!} + \cdots = 1$.

Using the cosine series, we have $cos(\frac{\pi}{2}) = 1 - \frac{(\frac{\pi}{2})^2}{2!} + \frac{(\frac{\pi}{2})^4}{4!} - \frac{(\frac{\pi}{2})^6}{6!} + \cdots = 0$.

Using the sine series, we have $sin(\frac{\pi}{3}) = \frac{\pi}{3} - \frac{(\frac{\pi}{3})^3}{3!} + \frac{(\frac{\pi}{3})^5}{5!} - \frac{(\frac{\pi}{3})^7}{7!} + \cdots = \frac{\sqrt{3}}{2}$.

Using the cosine series, we have $cos(\frac{\pi}{3}) = 1 - \frac{(\frac{\pi}{3})^2}{2!} + \frac{(\frac{\pi}{3})^4}{4!} - \frac{(\frac{\pi}{3})^6}{6!} + \cdots = \frac{1}{2}$.

Using the sine series, we have $sin(\frac{\pi}{4}) = \frac{\pi}{4} - \frac{(\frac{\pi}{4})^3}{3!} + \frac{(\frac{\pi}{4})^5}{5!} - \frac{(\frac{\pi}{4})^7}{7!} + \cdots = \frac{1}{\sqrt{2}}$.

Using the cosine series, we have $cos(\frac{\pi}{4}) = 1 - \frac{(\frac{\pi}{4})^2}{2!} + \frac{(\frac{\pi}{4})^4}{4!} - \frac{(\frac{\pi}{4})^6}{6!} + \cdots = \frac{1}{\sqrt{2}}$.

Using the sine series, we have $sin(\frac{\pi}{6}) = \frac{\pi}{6} - \frac{(\frac{\pi}{6})^3}{3!} + \frac{(\frac{\pi}{6})^5}{5!} - \frac{(\frac{\pi}{6})^7}{7!} + \cdots = \frac{1}{2}$.

Using the cosine series, we have $cos(\frac{\pi}{6}) = 1 - \frac{(\frac{\pi}{6})^2}{2!} + \frac{(\frac{\pi}{6})^4}{4!} - \frac{(\frac{\pi}{6})^6}{6!} + \cdots = \frac{\sqrt{3}}{2}$.