Jump to Power-reduction formulae - [edit]. Obtained by solving the second and third versions of the cosine double-angle formula. More in this series available at the Math by Video channel. The trigonometric power reduction identities allow us to rewrite expressions involving trigonometric terms with trigonometric terms of smaller powers.


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Calculus II - Integrals Involving Trig Functions

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Integrals Involving Trig Functions In this section we are going to look at quite a few integrals involving trig functions and some of the techniques we can use to help us evaluate them. Example 1 Evaluate the following integral.

Reduction Formula | Trigonometry | Siyavula

Notice that we were able to do reduction formula trigonometry rewrite that we did in the previous example because the exponent on the sine was odd.

In these cases all that we need to do is strip out one of the sines.

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We could strip out a sine, but the remaining sines would then have an odd exponent and while we could convert them to cosines the resulting integral would often be even more difficult than the original integral in most cases.

Example 2 Evaluate the following integral. So, reduction formula trigonometry can use a similar technique in this integral.

reduction formula trigonometry

List of trigonometric identities

Of course, if both exponents are odd then we can use either method. Each integral is different and in some cases there will be more than one way reduction formula trigonometry do the integral. With that being said most, if not all, of integrals involving products of sines reduction formula trigonometry cosines in which both exponents are even can be done using one or more of the following formulas to rewrite the integrand.


The last is the standard double angle formula for sine, again with a small rewrite. Example 3 Evaluate the following integral.


This integral is an example of that. There are at least two solution techniques for this problem.

Solution 1 In this solution we will use the two half angle formulas above and just substitute them into the integral. In fact to eliminate the remaining problem term all that we need to do is reuse the first half angle formula given above.

Solution 2 In this solution we will use reduction formula trigonometry half angle formula to help simplify the integral as follows. In the previous example reduction formula trigonometry saw two different solution methods that gave the same answer.


Note that this will not always happen. In fact, more often than not we will get different answers.

Trigonometric Power-Reduction Formulas -- from Wolfram MathWorld

However, as we discussed in the Integration by Parts section, the two answers will differ by no more than a constant. Sometimes in the process of reducing reduction formula trigonometry in which both reduction formula trigonometry are even we will run across products of sine and cosine in which the arguments are different.

These will require one of the following formulas to reduce the products to integrals that we can do.