Aryabhata's Contributions to Trigonometry

Aryabhata used a table of sines to calculate the values of (\sin \theta) for (\theta = 0^\circ, 1^\circ, 2^\circ, ..., 90^\circ). According to his table, (\sin 30^\circ) is equal to (\frac{1}{2}).

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 30^\circ) is equal to (\frac{\sqrt{3}}{2}).

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 30^\circ) is equal to (\frac{1}{\sqrt{3}}).

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\sin \theta). According to his table, (\sin 45^\circ) is equal to (\frac{1}{\sqrt{2}}).

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 45^\circ) is equal to (\frac{1}{\sqrt{2}}).

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 45^\circ) is equal to (1).

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\sin \theta). According to his table, (\sin 60^\circ) is equal to (\frac{\sqrt{3}}{2}).

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 60^\circ) is equal to (\frac{1}{2}).

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 60^\circ) is equal to (\sqrt{3}).

Aryabhata used the half-angle formula for (\sin \theta) to calculate the values of (\sin \theta) for angles greater than (45^\circ). According to his table, (\sin 75^\circ) is equal to (\frac{\sqrt{6 + \sqrt{3}}}{4}).

Aryabhata used the half-angle formula for (\cos \theta) to calculate the values of (\cos \theta) for angles greater than (45^\circ). According to his table, (\cos 75^\circ) is equal to (\frac{\sqrt{6 - \sqrt{3}}}{4}).

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta) for angles greater than (45^\circ). According to his table, (\tan 75^\circ) is equal to (\frac{\sqrt{6 + \sqrt{3}}}{\sqrt{6 - \sqrt{3}}}).

Aryabhata defined (\sin 90^\circ) to be equal to (1). This is because (\sin \theta) is the ratio of the opposite side to the hypotenuse, and in a right triangle with an angle of (90^\circ), the opposite side is equal to the hypotenuse.

Aryabhata defined (\cos 90^\circ) to be equal to (0). This is because (\cos \theta) is the ratio of the adjacent side to the hypotenuse, and in a right triangle with an angle of (90^\circ), the adjacent side is equal to (0).

Aryabhata did not define the value of (\tan 90^\circ). This is because (\tan \theta) is the ratio of the opposite side to the adjacent side, and in a right triangle with an angle of (90^\circ), the adjacent side is equal to (0). Therefore, (\tan 90^\circ) is undefined.