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The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse A D ¯ {\displaystyle {\overline {AD}}} of the blue triangle has length 2 cos θ {\displaystyle 2\cos \theta } .
By means of integration by parts, a reduction formula can be obtained. Using the identity sin 2 x = 1 − cos 2 x {\displaystyle \sin ^{2}x=1-\cos ^{2}x} , we have for all n ≥ 2 {\displaystyle n\geq 2} ,
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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.
Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
List of trigonometric identities#Power-reduction formulae To a section : This is a redirect from a topic that does not have its own page to a section of a page on the subject. For redirects to embedded anchors on a page, use {{ R to anchor }} instead .
That cos nx is an n th-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre's formula: + = ( + ). The real part of the other side is a polynomial in cos x and sin x , in which all powers of sin x are even and thus replaceable through the identity cos 2 x + sin 2 x = 1 .