Question

Guide: How to NOT memorize the unit circle.

Answer 1

Why do they call it a unit circle? Because it has a radius of 1, a single unit! Now, what do we know about triangles?
\[\ a^2+b^2=c^2\]and
all angles add up to 180 degrees

So using this bit of knowledge, here are all the tricks:



For the 45 degree triangle what do we know?

Well since we know all the angles add up to 180 degrees and that it's a right triangle, the other angle must also be 45 degrees! That must mean the two legs are the same length. Since it's on the unit circle, the hypotenuse must also be 1, so we have: \[a^2+a^2=1^2\] so here are the algebra steps to solve for the sides: \[2a^2=1 \\ a^2 = \frac{1}{2} \\ a=\sqrt{\frac{1}{2}} =\frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt2} \] Some people like to be fancy and rationalize it by multiplying by a fancy form of the number 1 since anything divided by itself is just 1. \[\ \frac{1}{\sqrt2}*1= \frac{1}{\sqrt2}*\frac{\sqrt2}{\sqrt2}=\frac{\sqrt2}{2}\]

Not difficult so far, the next one is just as easy too:

Answer 2

Similarly, if we want to find say sin(30) we can just think of this picture in our mind or draw it on the margin of our paper. Since the sum of angles must be 180, we have 30, 90 already used so the last angle must be 60. We know the hypotenuse must be 1 since it's no the unit circle but what about the other sides? Last time they were the same but now it's not so clear. But really a simple trick tells us immediately!

We want the "y" value so we reflect the triangle down. The lower corner has 60 degrees just like the top one and the other angle doubled from 30 to 60. And sure enough 60+60+60=180. So we just created a simple equilateral triangle, where the "y" value was 1/2 of the side. But that's just the same half of the hypotenuse. We can easily solve for the other side as well: \[a^2+(\frac{1}{2})^2=1^2\]\[a^2=1-\frac{1}{4}=\frac{3}{4} \\ a=\sqrt{\frac{3}{4}}=\frac{\sqrt3}{\sqrt4}=\frac{\sqrt3}{2}\]

Instead of wracking your brain every time, just simply think of this triangle and say, "ok if I reflect it to make the 30 degree angle double, will I be looking at the side that's half the hypotenuse?" So that way you know immediately if the side is 1/2 or if it's the slightly scarier older brother sqrt(3)/2.

Answer 3

I'm going to assume that 0 and 90 angles aren't a big deal to anyone here. Now what about angles that aren't in the first quadrant, then what? Don't worry too much, they're just the exact same angles as we've seen before except they've put signs on them. If you see: All you're really dealing with is the same triangle reflected across the y-axis. You can see that the y-components are in the same direction sin(135)=sin(45) because they both point up. However you can also see that cos(135)=-cos(45) since it's now in the opposite direction but the exact same magnitude. This is the unit circle, not the unit oval after all, so it is perfectly symmetrical.

In fact thinking like this hints you off to a lot of seemingly awful trig identities that you might have thought were impossible to memorize.

Answer 4

If you have any questions or there are any trig identities that seem weird or you don't understand them I guarantee you I have found some clever way of figuring it out without memorizing it, if you'd like to know that as well.

Answer 5

kai lets do some fun math :)

Answer 6

im high lol

Answer 7

:) im so ready lets do some programming or some math anything baby

Answer 8

i mised u man <3

Answer 9

HOW ARE YOU DOING BRO

Answer 10

Get on skype fool lol

Answer 11

my skyp is hacked

Answer 12

i can go on rachels skype wait

Answer 13

Get on rachel's skype

Answer 14

yeah lol

Answer 15

or come on google chat

Answer 16

wth is that

Answer 17

you have a google plug accn t we can tallk on that site

Answer 18

go to google plus risght now and u will see my msgs popping up

Answer 19

Hey Kai, Explain about sin and cos please! don't know what they are
really why do they make sense anyway

Answer 20

There is mnemonic ASTC (All Students Take Calculus) (See attachment)

I find it helpful. Just something I thought I should share.

Answer 21

@geerky42 I think things like FOIL and other mnemonics are an abomination though.

@xapproachesinfinity I'll explain that later, right now I'm a little busy sorry!

Answer 22

That's awesome, nicely done kai :P

Answer 23

Not all think like that toward mnemonics. Better shared something than doing nothing anyway.

Answer 24

I actually don't mind mnemonics, long as you understand what's going on behind the process it shouldn't be looked as anything terrible, just a short cut.

Answer 25

The only thing mnemonics do is allow you to "learn" something without actually learning it.

Answer 26

So how do you learn quadratic formula? Tell me please, as I am currently relying on mnemonic for it...

Answer 27

There's no mnemonic involved with quadratic formula

Answer 28

I developed my own mnemonic for it. How can I memorize or supposedly "learn" quadratic formula?

Answer 29

what do you need mnemonic for quadratic formula! you practiced it a lot tell it became second nature?

Answer 30

@xapproachesinfinity Here's my best attempt at an explanation for you, since I think a lot of people have trouble with accepting sine and cosine as functions.

What sine and cosine are really doing is looking at this specific triangle:

Do you remember doing triangle proofs in the past such as SAS or AAS things like this? This is exactly what we're using.

You see, above you automatically know one Side, the hypotenuse is 1 and one Angle, the right angle. So when you give sin() or cos() the specific angle such as 45 degrees or pi/3 radians it has now nailed down that triangle as being a specific triangle by AAS.

That means for that exact triangle, it has exactly one value for each x and y leg like in the picture above. Sine gives us the y value of that triangle and cosine gives us the x value. This is also why this identity holds true: \[\large \sin^2 \theta + \cos^2 \theta = 1\] since it's really just saying \[\large x^2+y^2=1\]

Now that's all well and good, but you will probably still feel unsatisfied at this point, right? The problem you're feeling is you don't know how to calculate it for any other values other than a couple fairly simple ones. Kind of frustrating I know. But remember because we're specifying two angles and a side this corresponds to an exact value whether we know what it is or not. It might seem weird to use a function that you can't exactly evaluate, but at least understand that it has a real meaning there.

For instance, what is the square root of 3? It's the same scenario, it's some ridiculous infinite decimal number slightly smaller than 2 that you don't quite know how to calculate. I may not have fully answered your question, but try your best to articulate what you feel uncomfortable with/don't understand and I'll try to make it more clear.