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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 ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at
The arcsine is a partial inverse of the sine function. These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since (+) = for every real x (and more generally sin(x + 2 π n) = sin(x) for every integer n).
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
3.4 Inverse functions. 3.5 Other properties. 4 Approximations. 5 Arbitrary curves and chords. 6 See also. 7 Notes. 8 References. ... In the form of sin 2 ...
The inverse function of sine is arcsine or inverse sine, denoted as "arcsin", ... The cosine double angle formula implies that sin 2 and cos 2 are, themselves, ...
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There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin −1, asin, or, as is used on this page, arcsin. For each inverse trigonometric integration formula below there is a corresponding formula in the list of integrals of inverse hyperbolic functions.