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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 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 of the ...
The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as ...
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
A 2020 study indicates that the global mean loss of life expectancy (LLE) from air pollution in 2015 was 2.9 years, substantially more than, for example, 0.3 years from all forms of direct violence, albeit a significant fraction of the LLE is considered to be unavoidable. [39]
Taxation as a percentage of production of final goods may have reached 15–20% during the 17th century in places such as France, the Netherlands, and Scandinavia. During the war-filled years of the eighteenth and early nineteenth century, tax rates in Europe increased dramatically as war became more expensive and governments became more ...
According to Roderick Harrison "wealth is a measure of cumulative advantage or disadvantage" and "the fact that black and Hispanic wealth is a fraction of white wealth also reflects a history of discrimination". [114] Whites have historically had more opportunities to accumulate wealth. [115]
Machin-like formulas for π can be constructed by finding a set of integers , =, where all the prime factorisations of + , taken together, use a number of distinct primes , and then using either linear algebra or the LLL basis-reduction algorithm to construct linear combinations of arctangents of . For example, in the Størmer formula ...