The Use of Trigonometry in Indian Navigation

Aryabhata, an Indian mathematician and astronomer, is credited with developing the concept of sine in the 5th century CE.

The astrolabe was an instrument used by Indian sailors to measure the altitude of celestial bodies, such as the sun and stars.

The cosine function is used to calculate the distance between two points on a sphere, using the formula (d = R \cos^{-1}(\cos \phi_1 \cos \phi_2 + \sin \phi_1 \sin \phi_2 \cos \Delta \lambda)).

Altitude is the term used for the angle between the vertical and the line of sight to a celestial body.

The sine function is used to calculate the latitude of a place, using the formula (\phi = \sin^{-1}(\sin \delta \sin \phi_0 + \cos \delta \cos \phi_0 \cos t)).

Declination is the term used for the angle between the true north and the magnetic north.

The tangent function is used to calculate the longitude of a place, using the formula (\lambda = \tan^{-1}(\frac{\sin t}{\cos t \sin \phi_0 - \tan \delta \cos \phi_0})).

Latitude is the term used for the angle between the horizon and the celestial equator.

The Pythagorean theorem is used to calculate the distance between two points on a plane, using the formula (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

Altitude is the term used for the angle between the vertical and the line of sight to the horizon.

The tangent function is used to calculate the height of a celestial body above the horizon, using the formula (h = R \tan \alpha).

Variation is the term used for the angle between the true north and the compass north.

The tangent function is used to calculate the azimuth of a celestial body, using the formula (\alpha = \tan^{-1}(\frac{\sin A}{\cos A \sin \phi - \tan \delta \cos \phi})).

Obliquity is the term used for the angle between the celestial equator and the ecliptic.

The tangent function is used to calculate the time of sunrise or sunset, using the formula (t = \frac{1}{15} \tan^{-1}(\frac{\sin \delta}{\cos \delta \cos \phi - \sin \phi})).