i need serious help with trigonometric values in the four quadrants of a circle.

- katherinesmith

i need serious help with trigonometric values in the four quadrants of a circle.

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- katieb

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- katherinesmith

i have read my lesson over again 2 times and still can not figure out the process to solving something like this:
what is the exact value of sin \[\frac{ \pi }{ 2}\] as found on the unit circle?

- katherinesmith

sin goes in front of the fraction

- katherinesmith

any idea @zepdrix ?

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## More answers

- zepdrix

Ok let's start really basic first.
Do you know your orientation within the unit circle?
Like do you know where an angle of \(\large \theta=0\) starts?
And do you know which way to rotate and how far to go to get to \(\large \pi/2\) ?

- katherinesmith

i know pi/2 is equivalent to 90 degrees

- zepdrix

ok good.

- zepdrix

|dw:1377747637549:dw|

- zepdrix

|dw:1377747711955:dw|Before we figure this problem out ~ I just wanna make 2 really quick notes that I think might help.
Let's start with some arbitrary angle like this.

- zepdrix

|dw:1377747788638:dw|There are a couple ways to get to this point on the unit circle.
By spinning theta radians around the circle
OR
by moving a distance in the x direction,
and a distance in the y direction.

- zepdrix

|dw:1377747862223:dw|Our trig functions show us how x and y relate to the angle.
In the unit circle:
`x corresponds to cosine of our angle`
`y corresponds to sine of our angle`

- zepdrix

|dw:1377748018329:dw|So when our angle is up here at pi/2....

- zepdrix

When they ask us `What is the SINE of pi/2?`
They're asking, `what is the y-component of your angle`.

- zepdrix

|dw:1377748102085:dw|

- zepdrix

So what would that be? :x
The y-value of that particular point?

- katherinesmith

|dw:1377748216852:dw|
oh my god i don't know i kinda just made it a triangle. how do i know where to make it!

- katherinesmith

you're so good at explaining honestly i'm just stupid

- zepdrix

To make a triangle only draw a line `straight down`.|dw:1377748252926:dw| But don't worry about doing the triangle thing right now. That might be a little bit too tricky at this point.

- katherinesmith

i understand that but the line is at 90 degrees which is straight up so there's no way to make a triangle

- zepdrix

Yess good observation! :)
We can't make a triangle.

- zepdrix

The question I'm asking you is a lot more simple than it seems.
It's just hard to make the connection.
Think for a sec :)
What is the `y-coordinate` of the point (0,1)?

- katherinesmith

1

- zepdrix

Yay good good.
Noooo you're not stupid! :D
You're like one of those special cookies.. we just have to... leave it in the oven a little longer than the others. :x

- katherinesmith

HAHAHA that is true.

- zepdrix

Ok so here is the connection.\[\Large \color{royalblue}{y=\sin \theta}\]
And we determined that,\[\Large \color{royalblue}{y}=1, \qquad\text{when }\theta=\frac{\pi}{2}\]

- katherinesmith

alright.

- zepdrix

\[\Large \color{royalblue}{\sin\theta}=1, \qquad\text{when }\theta=\frac{\pi}{2}\]\[\Large \color{royalblue}{\sin\frac{\pi}{2}}=1\]

- zepdrix

Let's try one more real quick!! D:
What does this give us?\[\Large \cos \pi=?\]

- katherinesmith

what?!

- zepdrix

|dw:1377748722656:dw|

- zepdrix

So pi is a distance of half circle right?

- katherinesmith

wait a minute. so the answer is 1?

- zepdrix

yes sin(pi/2)=1
:3

- katherinesmith

okay well i have another one of my own, can i ask you?

- zepdrix

fine fine fine :3

- katherinesmith

What is the exact value of
\[\cos \frac{ \pi }{ 6 }\]
as found on the unit circle

- zepdrix

Hmm this one is tough to answer without going into some detail.
It really depends how much you want to commit to memory.
Trig is hard because you have to remember a ton of stuff.
So I always feel like the less you have to memorize, the better.
There are some good tricks for memorizing the special angles but I think that might be a bit tough for you at this point. :[
http://mathematica.stackexchange.com/questions/2456/generate-a-unit-circle-trigonometry
Check out the unit circle in this link.

- zepdrix

Remember what I told you earlier. Try to burn it into your brain!!
x = cos
y = sin

- katherinesmith

well i have that chart on a piece of paper sitting in front of me

- zepdrix

Oh good hehe.

- katherinesmith

pi/6 is 30 degrees, now i just dont know what to do

- zepdrix

So if you go to the angle pi/6, what is your x-coordinate?
THAT is the value that corresponds to cosine.

- zepdrix

Do you have the coordinates all written down?
If you don't then you'll want to use that link :o

- katherinesmith

are you talking about \[\frac{ \sqrt{3} }{ 2 }, \frac{ 1 }{ }\]

- katherinesmith

1/2 ^

- zepdrix

yes. Those are the coordinates of the point along the unit circle that correspond to our angle.
cosine pi/6 is given by the `x coordinate`

- katherinesmith

okay.

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