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The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.
In that case, a and b are π / 2 − φ 1,2 (that is, the, co-latitudes), C is the longitude separation λ 2 − λ 1, and c is the desired d / R . Noting that sin( π / 2 − φ) = cos(φ), the haversine formula immediately follows. To derive the law of haversines, one starts with the spherical law of cosines:
The formula is given in verses 17–19, chapter VII, Mahabhaskariya of Bhāskara I. A translation of the verses is given below: [3] (Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees).
When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.
A simple way to calculate the mean of a series of angles (in the interval [0°, 360°)) is to calculate the mean of the cosines and sines of each angle, and obtain the angle by calculating the inverse tangent. Consider the following three angles as an example: 10, 20, and 30 degrees.
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
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Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [51] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [52]