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For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin( x ) and cos( x ) by means of the quotient rule applied to functions such ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
Excel at using Excel with these keyboard hotkeys that will save you minutes of time—and hours of aggravation. The post 80 of the Most Useful Excel Shortcuts appeared first on Reader's Digest.
3.1 Functions of the form a g(x) ... This is the definition of the derivative. ... These limits both follow from the continuity of sin and cos.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/ π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175.
The most general power rule is the functional power rule: for any functions and , ′ = () ′ = (′ + ′ ), wherever both sides are well defined. Special cases: If f ( x ) = x a {\textstyle f(x)=x^{a}} , then f ′ ( x ) = a x a − 1 {\textstyle f'(x)=ax^{a-1}} when a {\textstyle a} is any nonzero real number and x {\textstyle x} is ...
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