Fun lecture on geometric series (Part 3): The Fibonacci sequence

Every once in a while, I’ll give a “fun lecture” to my students. The rules of a “fun lecture” are that I talk about some advanced applications of classroom topics, but I won’t hold them responsible for these ideas on homework and on exams. In other words, they can just enjoy the lecture without being responsible for its content.

In the two previous posts, I introduced the idea of a generating function and then talked about its application to counting money. In this post, we’ll take on the famous Fibonacci sequence:

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$

The Fibonacci sequence starts with two $1$ s, and each subsequent term is defined as the sum of the two previous terms. Of course, the generating function for this sequence is

$f(x) = 1 + x + 2x + 3x^2 + 5x^3 + 8x^4 + 13x^5 + 21x^6 + 34x^7 + 55x^8 + \dots$

Slightly less famous than the Fibonacci sequence is (ahem) the Quintanilla sequence. It also begins with two $1$ s, but each subsequent term is defined as the sum of the previous term and twice the term that’s two back in the sequence. So,

The first term is $1$
The second term is $1$
The third term is $2(1) + 1 = 3$
The fourth term is $2(1) + 3 = 5$
The fifth term is $2(3) + 5 = 11$
The sixth term is $2(5) + 11 = 21$
The seventh term is $2(11) + 21 = 43$
The eighth term is $2(21) + 43 = 85$

And so on. The generating function for the Quintanilla sequence is

$Q(x) = 1 + x + 3x^2 + 5x^3 + 11x^4 + 21x^5 + 43x^6 + 85x^7 + \dots$

Both the Fibonacci and the Quintanilla sequences are examples of recursively defined sequences: we need to know the previous terms in order to get the next term. The major disadvantage of a recursively defined sequence is that, in order to get the 100th term, we need to know the 99th and 98th terms. To get the 98th term, we need the 96th and 97th terms. And so on. There’s no easy way to just plug in $100$ to get the answer.

When I mention this to students, they naturally start trying to figure it out on their own. Occasionally, a student will notice that each term is roughly double the previous term. With a little more time, they see that the even terms are one less than double the previous term, while the odd terms are one more than the previous term. That’s entirely correct. However, that’s another example of a recursively defined function. So if a student volunteers this, I’ll use this as an opportunity to note that it’s pretty easy to find a recursive definition for a function, but it’s a lot harder to come up with a closed-form definition.

In order to get a closed-form definition for the sequence — something that we could just plug in $100$ and get the answer — we will need to use the generating functions $f(x)$ and $Q(x)$ .

To simplify the infinite series $Q(x)$ , we pull something out of the patented Bag of Tricks. In case you’ve forgotten, Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.

Here’s the trick: let’s rewrite $Q(x)$ and also figure out $-xQ(x)$ and also $-2x^2 Q(x)$ :

$Q(x) = 1 + x + 3x^2 + 5x^3 + 11x^4 + 21x^5 + 43x^6 + 85x^7 + \dots$

$-xQ(x) = \, \, \, -x- x^2 - 3x^3 - 5x^4 -11x^5 - 21x^6 - 43x^7 - \dots$

$-2x^2Q(x) = \, \, \, \, \, \, -2x^2 -2 x^3 - 6x^4 - 10x^5 - 22x^6 - 42x^7 - \dots$

Let’s now add the last three green equations together. Notice that everything on the right-hand cancels except for $1$ ! Therefore,

$[1 - x - 2x^2] Q(x) = 1$

$Q(x) = \displaystyle \frac{1}{1-x-2x^2}$

The generating function for the Fibonacci sequence is similarly found. You can probably guess it since the Fibonacci sequence does not involve doubling a previous term.

It turns out that these generating functions can be used to find a closed-form definition for both the Quintanilla sequence and the Fibonacci sequence. More on this tomorrow.

Fun lecture on geometric series (Part 2): Ways of counting money

This series of posts describes a fun lecture that I’ve given to my Precalculus students after they’ve learned about partial fractions and geometric series.

In the 1949 cartoon “Hare Do,” Bugs Bunny comes across the following sign when trying to buy candy (well, actually, a carrot) from a vending machine. The picture below can be seen at the 2:40 mark of this video: http://www.ulozto.net/live/xSG8zto/bugs-bunny-hare-do-1949-avi

How many ways are there of expressing 20 cents using pennies, nickels, dimes, and (though not applicable to this problem) quarters? Believe it or not, this is equivalent to the following very complicated multiplication problem:

$\left[1 + x + x^2 + x^3 + x^4 + x^5 + \dots \right]$

$\times \left[1 + x^5 + x^{10} + x^{15} + x^{20} + x^{25} + \dots \right]$

$\times \left[1 + x^{10} + x^{20} + x^{30} + x^{40} + x^{50} + \dots \right]$

$\times \left[1 + x^{25} + x^{50} + x^{75} + x^{100} + x^{125} + \dots \right]$

On the first line, the exponents are all multiples of 1. On the second line, the exponents are all multiples of 5. On the third line, the exponents are all multiples of 10. On the fourth line, the exponents are all multiples of 25.

How many ways are there of constructing a product of $x^{20}$ from the product of these four infinite series? I offer a thought bubble if you’d like to think about it before seeing the answer.

There are actually 9 ways. We could choose $1$ from the first, second, and fourth lines while choosing $x^{20}$ from the third line. So,

$1 \cdot 1 \cdot x^{20} \cdot 1 = x^{20}$

There are 8 other ways. For each of these lines, the first term comes from the first infinite series, the second term comes from the second infinite series, and so on.

$1 \cdot x^{10} \cdot x^{10} \cdot 1 = x^{20}$

$1 \cdot x^{20} \cdot 1 \cdot 1 = x^{20}$

$x^{10} \cdot 1 \cdot x^{10} \cdot 1 = x^{20}$

$x^5 \cdot x^{15} \cdot 1 \cdot 1 = x^{20}$

$x^{10} \cdot x^{10} \cdot 1 \cdot 1 = x^{20}$

$x^5 \cdot x^{15} \cdot 1 \cdot 1 = x^{20}$

$x^{20} \cdot 1 \cdot 1 \cdot 1 = x^{20}$

$x^5 \cdot x^5 \cdot x^{10} \cdot 1 = x^{20}$

The nice thing is that each of these expressions is conceptually equivalent to a way of expressing 20 cents using pennies, nickels, dimes, and quarters. In each case, the value in parentheses matches an exponent.

$1 \cdot 1 \cdot x^{20} \cdot 1 = x^{20}$ : 2 dimes (20 cents).
$1 \cdot x^{10} \cdot x^{10} \cdot 1 = x^{20}$ : 2 nickels (10 cents) and 1 dime (10 cents)
$1 \cdot x^{20} \cdot 1 \cdot 1 = x^{20}$ : 4 nickels (20 cents)
$x^{10} \cdot 1 \cdot x^{10} \cdot 1 = x^{20}$ : 10 pennies (10 cents) and 1 dime (10 cents)
$x^{15} \cdot x^5 \cdot 1 \cdot 1 = x^{20}$ : 15 pennies (15 cents) and 1 nickel (5 cents)
$x^{10} \cdot x^{10} \cdot 1 \cdot 1 = x^{20}$ : 10 pennies (10 cents) and 2 nickels (10 cents)
$x^5 \cdot x^{15} \cdot 1 \cdot 1 = x^{20}$ : 5 pennies (5 cents) and 3 nickels (15 cents)
$x^{20} \cdot 1 \cdot 1 \cdot 1 = x^{20}$ : 20 pennies (20 cents)
$x^5 \cdot x^5 \cdot x^{10} \cdot 1 = x^{20}$ : 5 pennies (5 cents), 1 nickel (5 cents), and 1 dime (10 cents)

Notice that the last line didn’t appear in the Bugs Bunny cartoon.

Using the formula for an infinite geometric series (and assuming $-1 < x < 1$ ), we may write the infinite product as

$f(x) = \displaystyle \frac{1}{(1-x)(1-x^5)(1-x^{10})(1-x^{25})}$

When written as an infinite series — that is, as a Taylor series about $x =0$ — the coefficients provide the number of ways of expressing that many cents using pennies, nickels, dimes and quarters. This Taylor series can be computed with Mathematica:

Looking at the coefficient of $x^{20}$ , we see that there are indeed 9 ways of expressing 20 cents with pennies, nickels, dimes, and quarters. We also see that there are 242 of expressing 1 dollar and 1463 ways of expressing 2 dollars.

The United States also has 50-cent coins and dollar coins, although they are rarely used in circulation. Our answers become slightly different if we permit the use of these larger coins:

Finally, just for the fun of it, the coins in the United Kingdom are worth 1 pence, 2 pence, 5 pence, 10 pence, 20 pence, 50 pence, 100 pence (1 pound), and 200 pence (2 pounds). With these different coins, there are 41 ways of expressing 20 pence, 4563 ways of expressing 1 pound, and 73,682 ways of expressing 2 pounds.

For more discussion about this application of generating functions — including ways of determining the above coefficients without Mathematica — I’ll refer to the 1000+ results of the following Google search:

https://www.google.com/search?q=pennies+nickles+dimes+quarters+%22generating+function%22

FYI, previous posts on an infinite geometric series:

https://meangreenmath.com/2013/09/16/formula-for-an-infinite-geometric-series-part-9

https://meangreenmath.com/2013/09/17/formula-for-an-infinite-geometric-series-part-10

https://meangreenmath.com/2013/09/18/formula-for-an-infinite-geometric-series-part-11

Previous posts on Taylor series:

https://meangreenmath.com/2013/07/01/reminding-students-about-taylor-series-part-1/

https://meangreenmath.com/2013/07/02/reminding-students-about-taylor-series-part-2/

https://meangreenmath.com/2013/07/03/giving-students-a-refresher-about-taylor-series-part-3/

https://meangreenmath.com/2013/07/04/giving-students-a-refresher-about-taylor-series-part-4/

https://meangreenmath.com/2013/07/05/reminding-students-about-taylor-series-part-5/

https://meangreenmath.com/2013/07/06/reminding-students-about-taylor-series-part-6/

Fun lecture on geometric series (Part 1): Generating functions

This series of posts describes a 50-minute fun lecture — on the topic of generating functions — that I’ve given to my Precalculus students after they’ve learned about partial fractions and geometric series.

To launch the topic: In the 1949 cartoon “Hare Do,” Bugs Bunny comes across the following sign when trying to buy candy (well, actually, a carrot) from a vending machine. The picture below can be seen at the 2:40 mark of this video: http://www.ulozto.net/live/xSG8zto/bugs-bunny-hare-do-1949-avi

Notice that the price of candy from vending machines have increased somewhat since 1949. (Elsewhere in the cartoon, the price of a ticket to the movies was listed as 55 cents for adults, 20 cents for children, and 10 cents for rabbits.)

I wasn’t alive in 1949, but I vividly remember seeing this essentially mathematical problem while watching cartoons after school in the late 1970s. Now that I’m a little older — and can freeze-frame the above sign — I can see that the animators actually missed one way of expressing 20 cents. More on that later.

Definition. The generating function of a sequence is defined to be an infinite series whose coefficients match the sequence.

Example #1. Consider the (boring) sequence $1, 1, 1, 1, \dots$ . The generating function for this sequence is

$f(x) = 1 + 1x + 1x^2 + 1x^3 + \dots$

If $-1 < x < 1$ , then $f(x) = \displaystyle \frac{1}{1-x}$ , using the formula for an infinite geometric series.

Example #2. For the slightly less boring sequence of $1, -1, 1, -1, \dots$ , the generating function is

$f(x) = 1 - x + x^2 - x^3 + \dots$ ,

which (if $-1 < x < 1$ ) is $f(x) = \displaystyle \frac{1}{1+x}$ .

Example #3. Suppose $a_n = \displaystyle {10 \choose n}$ if $0 \le n \le 10$ and $a_n = 0$ for $n>10$ . Then the generating function is

$f(x) = \displaystyle \sum_{n=0}^{10} {10 \choose n} x^n = (x+1)^{10}$ .

It turns out that the above problem from the Bugs Bunny cartoon can be viewed as a generating function. Let $a_n$ denote the number of ways that $n$ cents can be formed using pennies, nickels, dimes, and quarters? The Bugs Bunny cartoon is related to the value of $a_{20}$ . What about one dollar? Two dollars? I’ll provide the answer in tomorrow’s post.

FYI, previous posts on an infinite geometric series:

https://meangreenmath.com/2013/09/16/formula-for-an-infinite-geometric-series-part-9

https://meangreenmath.com/2013/09/17/formula-for-an-infinite-geometric-series-part-10

https://meangreenmath.com/2013/09/18/formula-for-an-infinite-geometric-series-part-11

Ramanujan and Futurama

From a recent article that appeared in the BBC:

The year 1913 marked the beginning of an extraordinary relationship between an impoverished Indian clerk and a Cambridge don. A century later, their remarkable friendship has left its mark in the strangest of places, namely in Futurama, the animated series from The Simpsons creator Matt Groening and physics graduate David X Cohen…

For example, in order to pay homage to Ramanujan, Keeler has repeatedly inserted 1,729 into Futurama, because this particular number cropped up in a famous conversation between Hardy and Ramanujan.

According to Hardy, he visited Ramanujan in a nursing home in 1918: “I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavourable omen. ‘No,’ he replied. ‘It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’ “…

It is in recognition of Ramanujan’s comment that Bender, Futurama’s cantankerous robot, has the unit number 1729.

The number also appears in an episode titled “The Farnsworth Parabox”. The plot involves Futurama characters hopping between multiple universes, and one of them is labelled “Universe 1729”.

Moreover, the starship Nimbus has the hull registration number BP-1729.

Engaging students: Finding points in the coordinate plane

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Derek Skipworth. His topic, from Pre-Algebra: finding points in the coordinate plane.

A. Applications – How could you as a teacher create an activity or project that involves your topic?

When I think of the coordinate plane, one of the first things that come to mind is mapping. When I think of my teenage years, I think of how I always wanted more money. By using these two ideas together, an activity could easily be created to get the students involved in the lesson: a treasure map!

The first part of the activity would be providing the students with a larger grid. Then provide them with a list of landmarks/items at different locations (i.e. skull cave at $(3,2)$ ) that would then be mapped onto the grid. By starting out with one landmark, you could also build off previously identified landmarks, such as “move 3 units East and 4 units North to find the shipwreck. The shipwreck is located at what coordinates?” These steps could also be based off generic formulas with solutions for $x$ and $y$ . After all landmarks were identified, there would be a guide below that would trace out a path to find the treasure, which is only discovered after the full path is completed.

Courtesy of paleochick.blogspot.com

B. Curriculum: How can this topic be used in your students’ future courses in mathematics or science?

One of the good things about the exercise above is that it integrates several different ideas into one. A big one that stands out to me is following procedures. This is vital once you get into high school sciences. By building the map step-by-step, which each one building off the previous step, you cannot find the treasure without replicating the map exactly if you miss/misinterpret a step along the way.

As far as the coordinate plane, finding locations on the plane is important when graphing functions. Being able to find the intercepts and any asymptotes gives you starting points to work with. From there you generally only need a few more points to create a line of the function based off plotted points. This also has applications in science/math when creating bar graphs/line graphs and similar graphs.

D. How was this topic adopted by the mathematical community?

As discussed in my Geometry class this semester (Krueger), the Cartesian plane opened up a lot of doors in the world of Geometry. Euclid had already established a great working knowledge of a vast amount of Geometric ideas and figures. One thing he did not establish was length. In his teachings, there were relative terms such as “smaller than” or “larger than”. No values were ever assigned to his figures though. By introducing the Cartesian plane (and in effect, being able to plot points on said plane), we were able to actually assign values to these figures and advance our mathematical knowledge. The Cartesian plane acts as a bridge between Algebra and Geometry that did not exist before. Because of this, we can know solve problems based in Geometry without ever even needing to draw the figure in the first place (example: Pythagorean Theorem).

Engaging students: Solving linear systems of equations with matrices

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alyssa Dalling. Her topic, from Algebra II: finding the area of a square or rectangle.

A. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

A fun way to engage students on the topic of solving systems of equations using matrices is by using real world problems they can actually understand. Below are some such problems that students can relate to and understand a purpose in finding the result.

The owner of Campbell Florist is assembling flower arrangements for Valentine’s Day. This morning, she assembled one large flower arrangement and found it took her 8 minutes. After lunch, she arranged 2 small arrangements and 15 large arrangements which took 130 minutes. She wants to know how long it takes her to complete each type of arrangement.

(Idea and solution on http://www.ixl.com/math/algebra-1/solve-a-system-of-equations-using-augmented-matrices-word-problems )

The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make?

(Idea and solution on http://www.algebra-class.com/system-of-equations-word-problems.html )

A. How could you as a teacher create an activity or project that involves your topic?

For this topic, creating a fun activity would be one of the best ways to help students learn and explore solving systems of equations using matrices. One way in which this could be done is by creating a fun engaging activity that allows the students to use matrices while completing a fun task. The type of activity I would create would be a sort of “treasure hunt.” Students would have a question they are trying to find the solution for using matrices. They would solve the system of equations and use that solution to count to the letter in the alphabet that corresponds to the number they found. In the end, the solution would create different blocks of letters that the student would have to unscramble.

For Example: The top of the page would start a joke such as “What did the Zero say to the Eight?…

Solve $x+y=26$ and $4x+12y=90$ using matrices.

To solve this, the student would put this information into a matrix and find the solution came out to be $x=12$ and $y=14$ . They would count in the alphabet and see that the 12^th letter was L and the 14^th letter was N. Then at the bottom of their page, they would find where it said to write the letters for x and y such as below-

N __ __ __ __ __ L __! (Nice Belt!)

x a c z d z y w

E. How can technology be used to effectively engage students with this topic?

This activity would be used after students have learned the basics of putting a matrix into their calculator to solve. The class would be separated into small groups (>5 or more if possible with 2-3 kids per group) The rules are as follows: a group can work together to set up the equation, but each individual in the group had to come up to the board and write out their groups matrices and solution. The teacher would hand out a paper of 8-12 problems and tell the students they can begin. The first group to finish all the problems correctly on the board wins. There would be problems ranging from 2 variables to 4.

Ex: One of the problems could be and . The groups would have to first solve this on their paper using their calculator then the first person would come up to the board to write how they solved it-

Written on the board:

The technology of calculators allows this to be a fun and fast paced game. It will allow students to understand how to use their calculator better while allowing them to have fun while learning.

Engaging students: Dividing fractions

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Dale Montgomery. His topic, from Pre-Algebra: dividing fractions.

Applications

A Short Play On Numbers

By: Dale Montgomery

You see two brothers talking in the playground.

Timmy: (little brother) Gee Jonny, it sure was a good idea to sell Joe our old Pokémon deck. Now he finally has some cards to play with and we have some money to buy some new cards.

Jonny: (older brother) Yeah, I am glad we could help him get started. He has been wanting some cards for so long. Ok, you have the money so give me half.

Timmy: Ummm… (puzzling) Jonny I don’t know how to make half of 6 dollars and fifty cents, can you help?

Jonny: Of course Timmy, I learned how to divide fractions last week… lets see. (Jonny writes on the board 6 and ½ divided by 1/2 and does the division)

Timmy: How is half more than what we started with?

Jonny: I don’t know, this is the way my teacher taught me to do it. I guess you just have to find 13 dollars to give me so I can have half.

End Scene

Teacher: So class, what did Jonny do?

I came up with this idea thinking about the student asked question regarding dividing pie in half. I feel this could be a common misconception that would be addressed if we could teach students to think about math in context, rather than just a process. Dividing fractions is not the easiest thing to conceive. This short skit could be presented in any number of formats. I like the idea of having some sort of recorded show, just because it would make the intro to class go much faster. This skit introduces a situation that is very similar to word problems that children do. Also, the content can easily be modified to fit the majority class interest. For example it could have been an old Nintendo DS game that the brothers no longer play. This puts a problem that could be very real for the students right in front of them to figure out the correct process. It could lead to good discussion and make for a good lesson on dividing fractions.

Manipulative

Fraction bars are great tools to help students visualize dividing fractions. For example, if you wanted to divide 2/3 by 1/6 you would line up two of the third bars alongside one of the sixth bar and find out how many times that fraction goes into 2 thirds. In this case it would be four. Fractions themselves are extremely difficult to visualize, and dividing by fractions seems conceptually ridiculous. It can be difficult to adjust student’s thinking to this area. A manipulative like fraction bars are a good starting point in helping kids understand just how fractions work.

Curriculum, future uses

The topic of dividing fractions has many uses in future courses. Primarily these will be in algebra 1 and 2 for most students. Having a good conceptual knowledge of fractions will help students tremendously in these courses. As an algebra student you would be required to use your knowledge of fractions on an almost daily basis. Being introduced to the concept of multiple variables and canceling them out as you divide polynomials is a very complicated process that gets even more complicated if you do not understand fractions. Laying this conceptual framework is important when you consider all that students must use these concepts for at the higher level math classes. As you consider this in the lessons don’t forget the previous concepts held here such as grouping into equal parts and counting by intervals (3,6,9).

Why do we still require students to rationalize denominators?

Which answer is simplified: $\displaystyle \frac{1}{2 \sqrt{2}}$ or $\displaystyle \frac{ \sqrt{2} }{4}$ ? From example, here’s a simple problem from trigonometry:

Suppose $\theta$ is an acute angle so that $\sin \theta = \displaystyle \frac{1}{3}$ . Find $\tan \theta$ .

To solve, we make a right triangle whose side opposite of $\theta$ has length $1$ and hypotenuse with length $3$ . The adjacent side has length $\sqrt{3^2 - 1^2} = \sqrt{8} = 2\sqrt{2}$ . Therefore,

$\tan \theta = \displaystyle \frac{ \hbox{Opposite} }{ \hbox{Adjacent} } = \displaystyle \frac{1}{2 \sqrt{2}}$

This is the correct answer, and it could be plugged into a calculator to obtain a decimal approximation. However, in my experience, it seems that most students are taught that this answer is not yet simplified, and that they must rationalize the denominator to get the “correct” answer:

$\tan \theta = \displaystyle \frac{1}{2 \sqrt{2}} \cdot \frac{ \sqrt{2} }{ \sqrt{2} } = \displaystyle \frac{ \sqrt{2} }{4}$

Of course, this is equivalent to the first answer. So my question is philosophical: why are students taught that the first answer isn’t simplified but the second is? Stated another way, why is a square root in the numerator so much more preferable than a square root in the denominator?

Feel free to correct me if I’m wrong, but it seems to me that rationalizing denominators is a vestige of an era before cheap pocket calculators. Let’s go back in time to an era before pocket calculators… say, 1927, when The Jazz Singer was just released and stars of silent films, like Don Lockwood, were trying to figure out how to act in a talking movie.

Before cheap pocket calculators, how would someone find $\displaystyle \frac{1}{2 \sqrt{2}} ~~$ or $~~ \displaystyle \frac{ \sqrt{2} }{4}$ to nine decimal places? Clearly, the first step is finding $\sqrt{2}$ by hand, which I discussed in a previous post. So these expressions reduce to

$\displaystyle \frac{1}{2 (1.41421356\dots)}$ or $\displaystyle \frac{1.41421356\dots}{4}$

Next comes the step of dividing. If you don’t have a calculator and had to use long division, which would rather do: divide by $4$ or divide by $2.82842712\dots$ ?

Clearly, long division with $4$ is easier.

It seems to me that ease of computation was the reason that rationalizing denominators was required of students in previous generations. So I’m a little bemused why rationalizing denominators is still required of students now that cheap calculators are so prevalent.

Lest I be misunderstood, I absolutely believe that all students should be able to convert $\displaystyle \frac{1}{2 \sqrt{2}}$ into $\displaystyle \frac{ \sqrt{2} }{4}$ . But I see no compelling reason why the “simplified” answer to the above trigonometry problem should be the second answer and not the first.