Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
If sin(x) = 4cos(x), what is cos^-2(x)?
My goal is to post all Rapoport Geometry questions on the blog, but trig is always tricky. Some trig problems can be solved two different ways -- one using techniques from Chapter 14 of our Geometry text, and the other using methods not usually taught until Algebra II or Pre-calc.
Let's try the more advanced method first -- trig identities. We divide both sides of the equation:
sin(x) = 4cos(x)
sin(x)/cos(x) = 4
And there's our first identity: sin(x)/cos(x) = tan(x):
tan(x) = 4
Now let's go back to the original question: what is cos^-2(x)? That's a deep question -- what exactly does the notation cos^-2(x) mean? (The -2 here is a superscript which doesn't appear well in ASCII.) We already know that cos^2(x) means cos(x)^2, but cos^-1(x) does not mean cos(x)^-1 -- instead, it really means the inverse cosine of x -- arccos(x).
We might be tempted to think that cos^-2(x) could mean arccos(x)^2. But we look at the context -- the original equation mentions cos(x). Now cos(x) and arccos(x) rarely appear in the same problem -- after all, in cos(x), the input x would be an angle, but in arccos(x), the output would be an angle, not input. So it's much more logical to assume that cos^-2(x) means cos(x)^-2 and nothing involving arccos.
Of course, we avoid cos(x)^-1 by using the secant function instead. Thus the original question is really asking us to find sec^2(x). This is the second identity -- 1/cos(x) = sec(x).
So we know tan(x) and are asked to find sec^2(x). This leads to our third identity -- the Pythagorean identity in the form 1 + tan^2(x) = sec^2(x):
1 + 4^2 = sec^2(x)
17 = sec^2(x)
Thus the desired value is 17 -- and of course, today's date is the seventeenth.
So let's discuss that controversial notation cos^-2(x) in more detail. Only one person on Twitter has mentioned the questionable notation:
Z. Gershoff:
cos^{-2}: not to be confused with arccosine squared
The following website briefly mentions it:
https://math.vanderbilt.edu/schectex/commerrs/
Idiosyncratic inverses. We need to be sympathetic about the student's difficulty in learning the language of mathematicians. That language is a bit more consistent than English, but it is not entirely consistent -- it too has its idiosyncrasies, which (like those of English) are largely due to historical accidents, and not really anyone's fault. Here is one such idiosyncrasy: The expressions sinn and tann get interpreted in different ways, depending on what n is.
And of course, if n = 2 then it's an exponent, but if n = -1 then it's an arcfunction. Unfortunately, this website doesn't tell us about n = -2.
The following website discusses this a little as well:
https://tutorial.math.lamar.edu/extras/commonerrors/TrigErrors.aspx
It states that cos^n(x) = cos(x)^n only when n is a positive integer, which is why it doesn't apply in the case where n = -1. Again, this doesn't say anything about n = -2.
A few years ago, I thought I read some website that states that cos^n(x) = cos(x)^n for all real values of n other than -1, where it means arccos(x) instead. This is clearly how Rapoport is using it -- but, as usual, this was a website that I found years ago and can't find now.
If this is the case, then the equation y = cos^x(0) has an interesting graph -- it equals the constant function y = 1 everywhere except x = -1, where it takes on the value pi/2 instead. The line has a hole at (-1, 1) and a point at (-1, pi/2).
In the expression cos^n(x) = cos(x)^n, the implied operation is multiplication -- we are multiplying cos(x) by itself n times. But in the expression cos^-1(x) = arccos(x), the operation is composition -- the composite of the cos and arccos functions is the identity function (over a suitable domain).
Some mathematicians are interested in tetration and the composition of functions. These people use a special notation to indicate an implied operation of composition -- the subscript o. (It's the same circle we use with T o S to denote the composite of S and T, "T following S.") The following looks ugly in ASCII, where we have both a numerical superscript and the subscript o:
cos^2_o(x) = cos(cos(x))
cos^-1_o(x) = arccos(x)
I'd argue in favor of using cos^-1_o(x) instead of arccos(x), except that it doesn't work in ASCII. So we should just use arccos(x) instead. Then cos^-2(x) can unambiguously mean cos(x)^-2, just as Rapoport intended, since we would have cos^n(x) = cos(x)^n for any real number n.
I won't tweet anything to Rapoport this time, since it's not her error. She wasn't the one who invented the awkward notation cos^n(x).
I spent all of this time on trig notation that I still haven't given the purely geometric solution to today's Rapoport problem. So let me do so now:
We're given that sin(x) = 4cos(x). This means that in some right triangle, we have opposite/hypotenuse equal to four times adjacent/hypotenuse -- or, multiplying both sides by hypotenuse, we get that the opposite leg is four times the adjacent leg. So we draw a right triangle with adjacent leg = 1 and opposite leg = 4. By the Pythagorean Theorem, the hypotenuse is sqrt(17).
Notice that for this solution, it's good that Rapoport asked for cos^-2(x) rather than sec^2(x), since our Geometry students won't know what secant is. The cosine is adjacent/hypotenuse = 1/sqrt(17), and therefore our final answer is (1/sqrt(17))^-2 = 17.
There are a few more loose ends for me to discuss in this post. Let's start with Fawn Nguyen, who makes her latest post today:
https://www.fawnnguyen.com/teach/fractions-operations-using-rectangles
Nguyen begins by quoting a tweet:
@fawnpnguyen is the queen of this. Pretty sure she has a blog post on it.
And yes, I've called Nguyen the Queen many times on this blog before. Here, she's the Queen of using rectangles to motivate the rules for fraction operations:
I am queen only in my own head, and no, I did not have a blog post on adding fractions. I thought surely there must be a wide assortment of videos on adding fractions using rectangles. But the very first two that I’d clicked on - this and this - really astonished me. They both used grid papers without using the grids. Like, what the heck.
Say we want to add 2/3 and 4/5, same two fractions that popped in my head when I replied to Mary.
Draw two same-size rectangles using the denominators as dimensions.
I'll stop the quote here, since the rest of this is visual. Read the rest of our Queen's post to see the rest of this ingenious method.
The next thing for me to mention is quick note on the school calendar in my districts. In both of my Orange County districts, today is Day 168. But in my LA County district, last Friday was Day 168 while today is the last of the floating days. Notice that both the Monday before the AP exams and today were floating days -- presumably to avoid having no school on the day of an exam. But unexpectedly, the AP Stats exam was delayed to today due to the Eid al-Fitr holiday last Thursday. Thus AP Stats students had to attend school on a day when all others had the day off.
Tomorrow will be Day 169 in all three districts. For the first time ever on the blog, all of my districts agree on the day count, and they stay in agreement for the last twelve days of school (Days 169-180).
In recent posts, I've been writing a COVID-97 What If? -- that is, what if the pandemic had started, not in December 2019, but in December 1997, when I was in high school? I would have lost most of my junior Track season in 1998, and started my senior Cross County season in February 1999. By May 1999, I would have moved on to my final Track season.
I created this COVID-97 What If? by taking my alma mater's current 2021 Track season and mapping it back to 1999. Last Saturday -- May 15th, 2021 -- my school participated in an invitational, and so let's say that I'd have participated in an invite on Saturday, May 15th, 1999 in COVID-97 world.
Unlike the previous weekend's Irvine Distance Carnival, I don't have much information on this meet -- who hosted it, how many school participated, and so on. It doesn't matter, since this invite likely didn't exist in 1999 anyway. Just assume that someone would have put a meet together for us. And no, it wasn't the Mt. SAC Relays -- that race I would have recognized. It's another prestigious meet for which I wouldn't have qualified anyway.
But like the Mt. SAC Relays, most events are contested as relays. Earlier, I heard that there would be a 4 * 1600 relay race, which is rarely contested. But there's no evidence that any 4 * 1600 relays occurred last weekend -- instead, there appears to be only an open 1600.
Now that we're past the date when my 1999 season ended in the original timeline, I have to guess what my times would have been. Lately, I've been following a certain junior runner whose times are very close to my PR. On Saturday, he ran in the 1600 and improved slightly, so it's logical that I could have matched his time in 1999.
He also ran in 800 meters that day. While he's doubled several times over the season (usually it's the 800/3200 double), I've preferred to focus on 1600 on the blog instead. But I assume that this 800 was part of a 4 * 800 relay. It's logical to assume that someone would have needed me as a fourth runner in a 4 * 800 relay, if such a race were contested in 1999. So let me give myself this same double:
On May 15th, 1999 (a weekend invitational), my 1600 time would have been 5:05, and my 800 time would have been 2:26 (as part of a 4 * 800 relay).
This is what I wrote two years ago about today's lesson:
Question 15 of the SBAC Practice Exam is on systems of equations:
A store sells new and used video games. New video games cost more than old video games. All used video games cost the same. All new video games also cost the same.
Omar spent a total of $84 on 4 used video games and 2 new video games. Sally spent a total of $78 on 6 used video games and 1 new video game. Janet has $120 to spend.
Enter the number of used video games Janet can purchase after she purchases 3 new video games.
Here is the system to solve:
4u + 2n = 84
6u + n = 78
Using the substitution method:
n = 78 - 6u
4u + 2(78 - 6u) = 84
4u + 156 - 12u = 84
-8u = -72
(By the way, if we use elimination instead, multiply the 2nd equation by 2 to obtain -12u - 2n = -156 which also leads to -8u = -72.)
u = 9
6(9) + n = 78
54 + n = 78
n = 24
So Janet purchases three video games for $72, leaving her with $48, enough for five used games. And therefore, the number students must answer with is 5.
Both the girl and the guy from the Pre-Calc class correctly answer 5 for this question. It helps that this is one of the questions I specifically show them in class.
Question 16 of the SBAC Practice Exam is on systems of inequalities:
Click on the region of the graph that contains the solution set of the system of linear inequalities.
y < (-1/2)x + 3
y > 2x - 2
The correct graph must be below the (-1/2)x + 3 (or the downward-sloping) line and above the 2x - 2 (upward-sloping) line. These intersect in the region to the left -- the correct answer to shade.
The girl from the Pre-Calc class correctly answers this question. But the guy only marks the point where the lines intersect -- which is the solution to the system of equations, not the system of inequalities as directed. Well, at least he knows that the solution to a system of graphed equations is the point where they intersect (as many students don't know this). On the actual SBAC, I'm not sure whether it's possible to plot a point, since the software might be automatically set up to shade one of the four regions instead. That's fortunate for this guy, since once he realizes that the correct answer is a region, he'll probably figure out the inequalities.
SBAC Practice Exam Question 16
Commentary: In the U of Chicago Algebra I text, systems appear late -- not until Chapter 11, with systems of inequalities in Lesson 11-10 as well as systems of equations in the earlier parts of the chapter. In the Glencoe text, however, systems appear in Chapter 6. Nonetheless, systems are considered a part of second semester Algebra I. Make sure that the students know the difference between a system of equations and a system of inequalities.
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