Our well-regarded local junior college is the top destination for my high school’s graduates, a number of whom are more than bright enough to go to a four-year university but lack the money or the immediate desire to do so. Case in point: Jake, my best case for the hope that subsequent generations of Asian immigrants will adopt properly American values towards education, now at the local community college with a 4.0 GPA. He earned it entirely in math classes, having taken every course in the catalog–and nothing else. This from a kid who failed honors Algebra/Trig for not doing homework, and didn’t bother with any honors courses after that.
Jake visits four or five times a year, usually coming during class to see what’s up, working with other students as needed, then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition formula.
The day before, I started with the question: “cos(a+b) = cos(a) + cos(b)?” and let them chew on this for a bit before I
introduce remind them of proof by counterexample. A few test cases leads to the conclusion that no, they are not equal for all cases.
Then we went through this sketch that sets up the premise. I like the unit circle proof, because the right triangle proofs just hurt my head. So here we can see the original angle A, the original angle B, and the angle of the sum. Moreover, the unit circle proof includes a reminder of even and odd functions, a quick refresher as to why we know that cos(-B) = cos(B), but sin(-B) = -sin(b).
Math teachers often forget to point out and explain the seemingly random nature of some common proof steps. For example, proving that a triangle’s degrees sum up to 180 involves adding a parallel line to the top of the triangle and using transversal relationships and the straight angle.
Didn’t I make that sound obvious? You have this triangle, see, and you wonder geewhiz, how many degrees does it have? Hmm. Hey, I know! I’ll draw a parallel line through one vertex point! Who thinks like that? The illustration of a triangle’s 180 degrees is much more compelling than any proof.
So when introducing a proof, I try to make the transition from question to equation….observable. Answering the question requires that we define the question in known terms. What is the objective? How does the diagram and the lines drawn get us further to an answer?
Point 1 in the diagram defines the objective. Points 2 and 4 allow us to represent the same value in known terms–that is, cos(A) and cos(b). And thanks to some geometry that is intuitively obvious even if they’ve forgotten the theorem, we know that the distance between Point 1 and Point 3 [(1,0)] is equal to the distance between Point 2 and Point 4.
So I’d done this all the day before in first block, setting up the equation and doing the proof algebra myself, and the kids were lost. In my second block class, I turned the problem over to the kids at this point.
The solution involves coordinate geometry, algebra, and one Pythagorean identity. No new process, nothing to “discover”. Familiar math, unfamiliar objective. Perfect.
I grouped the second block kids by 5 or 6 instead of the usual 3 or 4 (always roughly by ability), giving each team one distance to simplify (P1P3 or P2P4). Once they were done, they joined up with kids who’d found the other distance, set the two expressions equal and solve for cos(A+B). The group with the strongest kids were tasked with solving the entire equation, no double teaming.
Block Two kids worked enthusiastically and quickly. I decided to retrace steps and do the same activity with block 1 the next day. Which is when Jake—remember Jake? This is a story about Jake—showed up.
“Hey, Jake! You here for the duration? Good. I’m giving you a group.”
Jake got those who had either been absent or were too weak at the math to be comfortable doing the work. I kept a watchful eye on the rest, who tussled with the algebra. I tried not to yell at them for thinking (cos(A) + cos(B))2 = cos(A)2 + cos(B)2, even though they all passed algebra 2 (often in my class), even though I’ve stressed binomial multiplication constantly throughout the year but no, I’m not bitter. Meanwhile, Jake carefully broke down the concept and made sure the other six understood, while they paid much more attention to him than they ever did to me but no, I’m not bitter.
Result: much better understanding of how and why cos(A+B) = cos(A)cos(B) – sin(A)sin(B). One of my most hostile students even thanked me for “making us do the math ourselves” because now, to her great surprise, she grasped how we had proved and thus derived the formula.
And then she went on to ask “But we have calculators now. Do we need to know this?” She looked at me warily, as I’m prone to snarl at this. But I decided to use my helper elf.
Jake, mind you, gave exactly the same answer I would have, but he’s just twenty years old, so they listened as he ran through the process for cosine 75 (degrees. 75 degrees. Jake’s a stickler for niceties.)
“But why is this better?” persisted my skeptic.
“It’s exact,” Jake explained. “Precise. When we use a calculator, it rounds numbers. Besides, who programs computers to make the calculations? You have to know the most accurate method to better understand the math.”
“Class, one thing I’d add to Jake’s answer is that depending on circumstances, you might want to factor the numerator, particularly if you are in the middle of a process.” and I added that in:
“Yeah, that’s right,” Jake confirmed. “like if you were multiplying this, I can think of all sorts of reasons a square root of two might be in the denominator. But other times you need to expand.”
I suddenly had another idea. “Hey. How about if we use right triangles?”
I sketched out two triangles.
“Oh, good idea. Except you forgot the right triangle mark.”
I sighed. “Class, you see how Jake is insanely nitpicky? Like he’s always making me write in degrees? He’s right. I’m wrong. I’ve told you that before; I’m not a real mathematician and they have conniptions at my sloppiness. But…” I’m struck by an idea. “I don’t need to mark it here! These have to be right triangles. Neener.” (I nonetheless added them in, although I left them off here out of defiance.)
“This is good. So suppose you want to add the two angles here. These right triangles have integer sides, but their angle measures are approximations. Let’s find those values using the inverse.”
Ahmed has his calculator out already. “Angle A is…53 degrees, rounded down. Angle B is 67.38 degrees.”
Me: “Just checking–does everyone understand what Ahhmed did?” I wrote out cos-1(3⁄5). “He used the inverse function on the calculator; it’s just a reverse lookup.”
” Let’s keep them rounded to integers. So 53 + 67 is 120 degrees, which has a cosine of ….what?” Jake paused, waiting for a response. Born teacher, he is.
By golly, my efforts on memorization have paid off. Several kids chimed in with “negative one half.”
“Meanwhile, if we multiply all these values using the cosine addition formula…” he worked through the math with the students, “we get -33⁄65“.
Dewayne punched some numbers and snorted. “-0.507692307692. That’s practically the same thing!” .
I had another idea. “You know how I said you should look at things graphically? Let’s graph this out on the unit circle.”
Jake was pleased. “This is excellent. So where would cosine(A+B) show up? We need to find the sine of each to plot it on the circle.” We worked through that and I entered the points.
Isaac: “Yeah, Dewayne is right. The two points are the same on the graph!”
“But this is a unit circle,” Jake said. “Just a single unit. As the values get bigger….I wish we could show it on this graph. Could we make a bigger circle? Or that probably wouldn’t scale.”
“How about if we just show all the values for every x? We could plot the line through that point? From the origin?”
“What would the slope be?” Gianna asked.
“Yeah, what would the slope be? Rise over run. And in the unit circle, the rise is sine, the run is cosine, so…”
“Tangent!” everyone chorused.
Jake was impressed. “See, this is why I should have taken trigonometry. I never thought about that.”
“OK, so I’m going to graph two lines. One’s slope is the tangent of 120, the other’s is the tan(cos-1(-33⁄65))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam.”
We then looked more closely at different points on the graph and agreed that yes, this piddling difference became visible over time.
“So the lines show how far apart the points would be for 120 and the addition formula number if you made the circle to that radius?” Katie asked.
“Yep. And that’s just what we can see,” Jake added. “The difference matters long before that point.”
When second block started, after brunch, Abdul rushed in, “Ahmed said we had a genius guest lecturer? Where is he?”
I faced a cranky crowd when I told them the genius had to go to class, so Jake will have to come back sometime soon.
Two months ago, Jake stopped by for a chat and I asked him about his transfer plans.
“Oh, I don’t know. Four year universities, I’ll have to take other classes, instead of what interests me.”
“You can’t be serious.”
“Well, maybe in a few years. But I have to wait a while for the computer programming classes I need to take, and the math classes are more fun.”
“Yeah. That’s what I want to….what. Why are you laughing.”
“Do you know anything about computers?”
“No, but it’s a good field, right?”
“I think you’re one of the most gifted math students I’ve bumped into, and you’ve never shown the slightest interest in technology or programming.”
Jake sat up. “My professor told me that, too. He said I should think about applied math. Is that what you mean?”
“Eventually, probably, but let’s go back to why the hell you don’t have a transfer plan.”
“Well, should I go to [name of a local decent state university]?”
I brought up his school website, keyed in “transfer to [name of elite state university system]”.
Jake looked on. “Wait. There’s a procedure to apply to [schools much better than local decent state university]?”
“You will go to your counselor, tell her or him you want to put together a transfer plan. Report back to me with the results in no less than 2 weeks. Is that clear?”
Just five days later, Jake’s cousin, Joey, my best algebra 2 student, reported that Jake had a transfer plan started and was getting the paperwork ready.
So after this class, I asked him about transfer plans.
“Oh, yeah. I’m scheduled to transfer to [extremely elite public university] in fall of 2017. I’ve been taking all math classes, so I have a bunch of GE to take. But it’s all in place.” He grinned wryly. “I didn’t think I’d be eligible for a school that good.”
“And that’s just the guarantee, right?”
“Yes, I want to look at [another very highly regarded public]. Do you think that’s a good idea?”
“I do. You should also apply to a few private universities, just for the experience. It’s worth learning if they give transfer students money.” I named a few possibilities. “And ask your professors, too.”
“Okay. And you don’t think I should major in computer programming?”
“Do you know anything about programming right now? If not, why commit?”
“I don’t know. I never knew about applied math possibilities. It sounds interesting.”
“Or pure math, even. So you’ve got some research to do, right? And keep your GPA excellent with all that GE.”
“And at some point, you’re going to think wow, I never would have done any of this without my teacher’s fabulous support and advice.”
“I already think that. Really. Thanks.”
Just in case you think his visits pay dividends in only one direction.
December 20th, 2015 at 10:35 pm
[…] Source: Education Realist […]
December 21st, 2015 at 1:12 am
The sine and cosine formulas are the analytic expression of the fact that a rotation of angle theta1 followed by a rotation of angle theta2 is equivalent to a rotation of angle theta1 + theta2. That is to say the use of sine and cosine reduce composition of rotations to addition of real numbers.
The actual formulas are the result of composing the formula for a rotation of angle theta1 with the formula for a rotation of angle theta2. Just plug the formulas for a rotation of angle theta1 into the formulas for a rotation of angle theta2 and you get the formulas for a rotation of angle theta1 + theta2.
x’ = cos(theta1) x – sin(theta1) y
y’ = sin(theta1) x + cos(theta1) y
x” = cos(theta2) x’ – sin(theta2) y’
y” = sin(theta2) x’ + cos(theta2) y’
Plug the first set of equations into the second. The result is the equations for a rotation of angle theta1 + theta2.
December 21st, 2015 at 4:15 am
Cool. I didn’t know that. I used the proof in the book.
December 21st, 2015 at 3:15 pm
Yes it’s kind of interesting. The sine-cosine addition formulas don’t seem very intuitive but they are making in analytical language exactly the same statement as that the angles of rotations add up when the rotations follow one another which is a very intuitive geometric statement.
December 25th, 2015 at 8:17 am
I am so glad to see teacher like you who really cares students’ “understanding”. Following is my idea about the sin-cosine addition law:
Let us see things in a triangle (the starting point of trigonometry).
sinA, sinB are actually the lengths of the opposite sides of the angle A and B in the triangle ABC (just take 2R=1 in the law of sine).
Therefore, sinA cosB = length of side b times cos(adjacent angle) = projection of side b on side c.
sinAcosB+sinBcosA = sum of projections = length of side c = sinC
C= pi -A-B, we get sin(A+B)=sinAcosB+sinBcosA. QED.
I think this is the “real geometrical” proof of the addition formula
Remark: sinA originally is the opposite side of A, in a right triangle; in view of the law of sin, sinA IS the opposite side of A in any triangle, as long as we see things geometrically—-taking 2R=1 means an expansion of the figure and does NOT change the GEOMETRY of the whole picture.
Hope that you like the geometric proof (which I invented some years ago)
December 25th, 2015 at 4:09 pm
I’m going to have to look at both of these. From a teaching standpoint, I really like the unit circle, as it lets me beat some more algebra into their heads. Thanks for the additional ideas.
December 25th, 2015 at 7:38 pm
That is a nice proof. But aside from the details of any proof of the sine-cosine addition formulas these formulas have a meaning and significance in themselves. And this meaning and significance is exactly that the assignment of real numbers to rotations as angles makes the addition of real numbers correspond to the composition of rotations thus serving to reduce the study of the geometry of the plane to the study of the addition of real numbers.
The fundamental law of trigonometry is
Rotation(theta1 + theta2) = Rotation(theta2) o Rotation(theta1)
in words – “A rotation of angle theta1 followed by a rotation of angle theta2 is equal to a rotation of angle theta1 + theta2”.
The sine-cosine addition formulas are a somewhat disguised version of the above fundamental law. They are just the analytic equivalent of the fundamental law using the linearity of rotations.
December 25th, 2015 at 7:59 pm
Trig has a fundamental law? Really?
December 25th, 2015 at 8:21 pm
That’s what I call it. Alternatively the expression “Fundamental Law of Trigonometry” might be applied to the statement –
SO(|R,2) = R/Z
January 2nd, 2016 at 7:30 am
[…] some tremendous discussions throughout the math community. I couldn’t have been more pleased. Jake’s Guest Lecture and The Test that made them go Hmmmm is an accurate representation of my classroom discussions. The […]
January 4th, 2016 at 6:36 pm
Great success story, warms my heart!
January 4th, 2016 at 6:41 pm
A lot of companies hire math majors. Jake can check with the placement office at the university he attends, and the profs, job fairs.
April 30th, 2016 at 5:01 pm
[…] to many students of all races and ethnicities who couldn’t otherwise afford it–even after graduation. There are kids in top colleges today who once never had a thought of attending, because I had the […]
October 29th, 2016 at 9:29 pm
[…] course. Every year, a few students make coffee dates, just to chat. Still others just stop by my classroom and say […]