Sunday, February 10, 2008

The Great Almighty Unit Circle. Plus other stuff .. That can be almighty if you want it to. ^^

Hey everybody! Sorry for the two days and a half late post. Anyways, what I'm going to be discussing in my blog is what we've done that day. First, we started it off by finding that there is a super Pi on the Chinese coin. After that we discussed the points on the Unit Circle. What was Sinθ, Cosθ, and Tanθ were on each point of the Unit Circle. Then we discovered patterns and methods of remembering the whole unit circle.

The unit circle is called the unit circle because the radius is 1. The angles we used in the unit circle was 30°, 45° and 60°. We converted degrees into radians. 30° = Pi/6, 45° = Pi/4 and 60° = Pi/3.

**


- In a right angle triangle, we know that Sinθ = OPP / HYP, Cosθ = ADJ / HYP, and Tanθ = OPP / ADJ.
SOH CAH TOA
- We then find that Tanθ = Sinθ / Cosθ

** In the unit circle, Pi/6, if you drop straight down from P(θ) to form a right angle triangle, you can find the coordinates of P(θ). As you can see in my professional diagram below.


The coordinates of P(θ) is (Cosθ, Sinθ) .. Because we know the unit circle's radius is 1, and the hypotenuse is 1, then we can figure out Sinθ, and Cosθ, using Pi/6 (30 Degrees) ..

- Sin Pi/6 = 1/2
- Cos Pi/6 = v3/2

** Keep in mind that it will always be over 2
So P(θ) = P(v3/2, 1/2) Knowing that these are the points that represent P(θ) then that must mean that Cosθ is represented by the X axis and Sinθ is represented by the Y axis. -Tanθ = y/x

**
That is how we found the points for the unit circle.

- A short lesson we learned was complementary angles.

- It is the sum of 2 angles that add up to 90°

Ex. 30° + 60° = 90°

Sin30° = 1/2

Cos60° = 1/2

** The unit circle can be found on the last lesson's slide (The one with Happy New Year!!). Slide #6/8 .. It is a very good unit circle.
-The method of remembering each point is you have to remember 3 numbers.
1, 2, and 3.

Here is a link to my amazing, incredibly, outstanding paint skills for a clear example of one way of remembering the unit circle.

Click Here!

The explaining will be here.
-The black numbers represent Sin
θ
-The red numbers represent Cos
θ
-The green numbers represent Tan
θ

What do you notice?
-RIGHT! 1, 2 and 3. (You're probably thinking in your head .. "uh.. no")

**Anyways

The numbers you see will always be square rooted over 2.
Yes, Square root of 3, over 2
-Square root of 2, over 2
-Square root of 1, over 2.

**Now how do I know when the number is 1, 2 or 3?

The purple lines in the circle represent the Y axis, also Sin
θ.
Ditto for the blue lines, it represents the X axis, Cos
θ.

**What do you notice?
-RIGHT! That Pi/4 or 45° will always be square root of 2 over 2. (Not including Tangent)

As you see the longer the lines are extended, it is square root of 3 over 2.
The shorter it is, it is 1 half.
Between that is square root of 2 over 2.
Yea the clock dance becomes useful. Unfortunately.

**Now that we have figured out Sin
θ and Cosθ .. How about Tanθ?

Well .. You should have already picked up that the middle numbers are always 1.

**The closer tangent is to the X axis, it is under 1. Yes 1 over square root of 3.
The farther it is, (over the X axis) it is square root of 3 over 1. Or just square root of 3.

**I hope this helped. I was in a rush.

I'll skip to to the credits.

**The next scribe will be the person who sits the closest to me.

Yup, you got it right again! Joyce will be the next scribe =)

This is Agent Eleven, out.

Better days.

4 comments:

-zeph said...

Don't you mean Jamie, not Joyce? lol

benofschool said...

nice pic lol

Eleven said...

no, joyce hehe

JamieNeRd123C said...

don't get me into things benofschool.lol