## All posts tagged **periodic trig identities**

# Periodic

*Posted by John Quintanilla on June 29, 2017*

https://meangreenmath.com/2017/06/29/periodic/

# My Favorite One-Liners: Part 76

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that might arise in trigonometry:

Compute .

To begin, we observe that , so that

.

We then remember that is a periodic function with period . This means that we can add or subtract any multiple of to the angle, and the result of the function doesn’t change. In particular, is a multiple of , so that

.

Said another way, corresponds to complete rotations, and the value of cosine doesn’t change with a complete rotation. So it’s OK to just throw away any even multiple of when computing the sine or cosine of a very large angle. I then tell my class:

In mathematics, there’s a technical term for this idea; it’s called throwing.

*Posted by John Quintanilla on April 17, 2017*

https://meangreenmath.com/2017/04/17/my-favorite-one-liners-part-76/

# How I Impressed My Wife: Index

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

Yes, I married well indeed.

In this post, I collect the posts that I wrote last summer regarding various ways of computing this integral.

*Posted by John Quintanilla on January 6, 2016*

https://meangreenmath.com/2016/01/06/how-i-impressed-my-wife-index/

# How I Impressed My Wife: Part 5c

Earlier in this series, I gave three different methods of showing that

Since is independent of , I can substitute any convenient value of that I want without changing the value of . For example, let me substitute :

So that I can employ the magic substitution , I’ll divide the interval of integration into two pieces and then perform the substitution on the second piece:

*Posted by John Quintanilla on August 14, 2015*

https://meangreenmath.com/2015/08/14/how-i-impressed-my-wife-part-5c/

# How I Impressed My Wife: Part 3e

Previously in this series, I showed that

So far, I have shown that

.

where and (and is a certain angle that is now irrelevant at this point in the calculation).

I now write as a new sum by again dividing the region of integration:

,

.

For , I employ the substitution , so that and . Also, the interval of integration changes from to , so that

Next, I employ the trigonometric identity :

,

where I have changed the dummy variable from back to .

Therefore, becomes

.

Once again, the fact that the integrand is over an interval of length allows me to shift the interval of integration.

I’ll continue this different method of evaluating this integral in tomorrow’s post.

*Posted by John Quintanilla on July 30, 2015*

https://meangreenmath.com/2015/07/30/how-i-impressed-my-wife-part-3e-2/

# How I Impressed My Wife: Part 3d

Previously in this series, I showed that

So far, I have shown that

,

where , , and is a certain angle (that will soon become irrelevant).

I now write as a new sum by dividing the region of integration:

,

.

For , I employ the substitution , so that and . Also, the interval of integration changes from to , so that

Next, I employ the trigonometric identity :

,

where I have changed the dummy variable from back to .

Therefore, becomes

.

Next, I employ the substitution , so that and the interval of integration changes from to :

.

Almost by magic, the mysterious angle has completely disappeared, making the integral that much easier to compute.

I’ll continue this different method of evaluating this integral in tomorrow’s post.

*Posted by John Quintanilla on July 29, 2015*

https://meangreenmath.com/2015/07/29/how-i-impressed-my-wife-part-3d/

# How I Impressed My Wife: Part 2a

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

,

where

,

,

.

I’ll begin with and apply the substitution , or . Then , and the endpoints change from to . Therefore,

.

Next, we use the periodic property for both sine and cosine — and — to rewrite as

.

Changing the dummy variable from back to , we have

.

Therefore, we can combined into a single integral:

Next, we work on the middle integral . We use the substitution , or , so that . Then the interval of integration changes from to , so that

.

Next, we use the trigonometric identities

,

,

so that the last integral becomes

On the line above, I again replaced the dummy variable of integration from to . We see that , and so

I’ll continue with the evaluation of this integral in tomorrow’s post.

*Posted by John Quintanilla on July 20, 2015*

https://meangreenmath.com/2015/07/20/how-i-impressed-my-wife-part-2a/