Notes for Unit Circle (2 Viewers)

od9895 · Jan 2, 2012

Hello
I need your help guys. Do any you guys have notes or past exams on the unit circle because apparantly my tutor said it is compulsory to know how to use the unit circle in trig. Please help!!!

Thanks

Carrotsticks · Jan 2, 2012

Perhaps you can use that tutor to help you get notes? That's what they're paid for after all....

Unless you want a tutor FOR your tutor, who tutors for your school.

deswa1 · Jan 2, 2012

The unit circle isn't a topic by itself so you won't find a whole exam on it. What exactly do you need help with and I'll try and come up with something

od9895 · Jan 2, 2012

I need help with everything on the unit circle because I am new to my tutor and I dont know anything about it

Carrotsticks · Jan 2, 2012

od9895 said:
I need help with everything on the unit circle because I am new to my tutor and I dont know anything about it

I presume this is a tutoring class?

RealiseNothing · Jan 2, 2012

Draw a unit circle (centre 0,0 on a number plane with radius 1).

Now draw 2 diagonal lines through the centre ( should look like y=x and y=-x ).

The angles these lines make with the x axis are called the related angle. The top right quadrant is the first, followed by the top left, then bottom left, and finally the bottom right being the 4th quadrant.

You should have 4 related angles, each in a different quadrant. Draw perpendicular lines straight down from the intersection of your diagonal lines and the unit circle, this will make 4 congruent right angled triangles. Let the angle made at the origin of the triangle be theta.

Now you can use sin/cos/tan to prove a few indentities for theta in each of the 4 different quadrants. Hint: Let the hypotenuse be 'r' as it is the radius of the unit circle, and then let the vertical side of the triangle be 'y' and the horizontal side be 'x'.

You will come up with a general case for sin, cos, and tan in each quadrant which you can use for any length 'r'. However since it's a unit circle and the radius is 1, let r=1 and see what happens to these identities. These new identities are special cases on a unit circle.

Using this diagram you can also derive other identities, so just play around with it a bit (see when sin/cos/tan are positive or negative).

You can also see things such as sin@ = cos(90-@) through the diagram.

Note: I did this instead of just telling you the identities as you will understand it more if you derive it yourself instead of accepting it as fact.

RivalryofTroll · Jan 2, 2012

Why can't you just ask your tutor on how to do it?
But, RealiseNothing just explained the whole thing in a nutshell!

such_such · Jan 2, 2012

What tutor do you go to? This might help.

SpiralFlex · Jan 2, 2012

*Reserved for posting later

$Introduction to the Unit Circle$

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$Before I start, there are a few things you must know (Derivation shall be done later via request.):$

$cosec \theta = \frac{1}{\sin \theta} \Rightarrow \sin \theta =\frac{1}{cosec \theta}$

$\sec \theta = \frac{1}{\cos \theta} \Rightarrow \cos \theta =\frac{1}{\sec \theta}$

$\cot \theta = \frac{1}{\tan \theta} \Rightarrow \tan \theta =\frac{1}{\cot \theta}$

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$\tan \theta = \frac{\sin \theta}{\cos \theta}$

$\cot \theta =\frac{\cos \theta}{\sin \theta}$

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$Throughout Kindy to Year 10, teachers have deliberately restricted trigonometry. They have purposely put a place-hold on the angle$

$\theta,$ where $0 \leq \theta \leq 90$ .

$What if we were to extend our confined definition of trigonometry, if we introduced angles larger than 90 degrees, what are the implications? What could we possibly do? The simple answer to that is a lot.$

$First, I will introduce you to a unit circle. What is it? Well it's exactly what the name says, a circle with radius of one unit.$

$Hence the equation of this circle is of course:$

$x^2+y^2=1$

$Here is what it looks like, of course we need to "place" it on the Cartesian plane for it to geometrically make sense to us.$

(I can't use image tags for some reason, we'll just have to deal with it.)

$First of all we will deal with TRIGONOMETRIC IDENTITIES.$

$As you can see, I have drawn a pretty circle on the Cartesian Plane. Nothing special right? But from this circle, we can derive an identity!$

$Our famous six star formula!$

★★★★★★

$\sin^2 \theta+\cos^2 \theta=1$

$So let's see how I got this!$

$We will use our diagram to aid us.$

$We know that the circle has a radius of one. Hence the name "unit circle". It is now noteworthy to use your basic trigonometry skills you have accumulated through Junior school!$

$So from our pretty diagram, our x, y coordinates have now been rewritten. Let us inscribe a little triangle in the first quadrant of the unit circle. Sure, we can use trigonometry,$

$\sin \theta = \frac{y}{1}$

$y=\sin \theta$

$Similarly,$

$\cos \theta = \frac{x}{1}$

$x=\cos \theta$

$Our coordinates have been rewritten into "Polar Coordinates"$

$x\Rightarrow \cos \theta, y\Rightarrow\sin \theta$

$Something pretty has just occurred! Now what happens if we apply Pythagoras?$

$x^2+y^2=1$

$\cos^2 \theta+\sin^2 \theta=1$

$Or more conventionally,$

★★★★★★

$\fbox{\sin^2 \theta + \cos^2 \theta =1}$

$Please understand this is ONE of the MOST IMPORTANT formulae you will ever see! So it is worth knowing how to derive it!$

$There are two more identities you will need to learn! But if you learn the first one, you will never need to remember the second one.$

$Here's how many students start off with learning the second identity off by heart. We can derive it algebraically,$

$\sin^2 \theta + \cos^2 \theta =1$

$Now divide all sides by \cos^2 \theta,$

$\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} =\frac{1}{\cos^2 \theta}$

$\tan^2 \theta+1=\sec^2 \theta$

$Another IMPORTANT identity$

$\fbox{\tan^2 \theta+1=\sec^2 \theta}$

$Thirdly,$

$\sin^2 \theta + \cos^2 \theta =1$

$Now divide all sides by \sin^2 \theta,$

$\frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} =\frac{1}{\sin^2 \theta}$

$1+\cot^2 \theta = cosec^2 \theta$

$Third formula,$

$\fbox{\cot^2 \theta +1= cosec^2 \theta}$

$Excellent! Let's begin some examples involving Trigonometric identities$

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$Example 1$

$I will start VERY basic!$

$Prove the identity: (\cos \theta + \sin \theta)^2=1+2 \cos \theta \sin \theta$

$We conventionally start with the LHS (Left Hand Side), then see if we can work our way to making it equal to the RHS (Right Hand Side)$

$Our initial thoughts are to expand!$

$LHS=(\cos \theta + \sin \theta)^2$

$=\cos ^2 \theta + 2\cos \theta \sin \theta +\sin^2 \theta$

$Ah we notice the six star formula can be used here, \sin^2 \theta + \cos^2 \theta = 1$

$$=1+ 2\cos \theta \sin \theta$

$=RHS$

$DONE! (Though once you learn double angles you can advanced further. But this is an introduction so I will keep it simple!)$

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$Example 2$

$There is nothing hard with these questions, it's just manipulation.$

$Sometimes they will ask you to simplify, instead of show the following identity.$

$Simplify:$

$\frac{\cos \theta}{\sin^2 \theta -1}$

$Almost immediately you should release we're dealing with a square of sine. So instantly, our formulae pops into mind. But there's a slight altercation, there is no \sin^2 \theta + \cos^2 \theta to eliminate.$

What we can do is re-arrange the formula like so,

$\sin^2 \theta + \cos^2 \theta=1$

$\sin^2 \theta=1- \cos^2 \theta$

So now,

$\frac{\cos \theta}{\sin^2 \theta -1}$

$=\frac{\cos \theta}{(1-\cos^2 \theta) -1}$

$=\frac{\cos \theta}{-\cos^2 \theta}$

$=-\frac{1}{\cos \theta}$

$=-\sec \theta$

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$Example 3$

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Writing (Estimated finish time -1 am)

Currently at ice-cream break

Sanical · Jan 2, 2012

All...Stations...To...Central.
^ That's enough to last you all of maths.

I was surprised it came up in 2011 3U - tricked a lot of people.

Peeik · Jan 2, 2012

Hey what textbook do you have? I'm sure most will have the unit circle in them.

SpiralFlex · Jan 2, 2012

Note, I will probably be done by tomorrow so check then OP!

wildchild666 · Jan 2, 2012

Sanical said:
All...Stations...To...Central.
^ That's enough to last you all of maths.

I was surprised it came up in 2011 3U - tricked a lot of people.

That's how I remember it..

SpiralFlex · Jan 2, 2012

Other variations I have seen over my time are,

Add Sugar To Coffee

All Silly Tom Cats

All Surfies To Cronulla

Bored_of_HSC · Jan 2, 2012

All Stations To China

SpiralFlex · Jan 2, 2012

Bored_of_HSC said:
All Stations To China

That too...

asdfqwerty · Jan 2, 2012

All sluts take cock.

such_such · Jan 3, 2012

ASTC doesn't apply to Dr. Truong. So if OP goes Truongs, forget about learning it the textbook way

SpiralFlex · Jan 3, 2012

such_such said:
ASTC doesn't apply to Dr. Truong. So if OP goes Truongs, forget about learning it the textbook way

Explain. Which method does he teach? Graphically?

RealiseNothing · Jan 3, 2012

Graphically should be how it's learnt, and ASTC should be how it's remembered, in my opinion.

Notes for Unit Circle (2 Viewers)

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