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TOPIC 1: USING DEGREES TO MEASURE ANGLES
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I am going to assume that everyone knows that you can measure an angle using the unit degrees. You measured angles with a protractor  See the images below in figure 1.

figure 1

Mathematicians also use a Greek letter, θ , to represent a degree measure. The symbol θ is spelled "theta" and pronounced "thay-ta." So let's agree, from now on, that we will try to use the greek letter theta, θ , to represent a degree measure. See some examples of theta usage below in figure 1.

ex. 1) In figure 1 above name all the degree measures using the Greek letter θ .

θ = 30°; θ = 180°; θ = 90°, θ = 270°; θ = 360°;

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TOPIC 2: ARC, CENTRAL ANGLE AND THE NUMBER OF DEGREES ALONG
THE ARC OF A CIRCLE
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Any part of the circumference of a circle is called an "arc." See figure 2 below.

If you connect the endpoints of the arc with 2 radii(plural for radius) connected to the center of the circle the angle created is called a "central angle." See figure 3 below.

In figure 4 below, I created an image of a circle showing that there are 360° in a circle - I am assuming that you know this geometrical fact.

figure 2	figure 3	figure 4

In figure 5 below, I placed a Circle Runner on the circle. The Circle Runner moved in a counter clockwise direction starting on the circumference of the circle where the circle intersected the positive x-axis.

figure 5

The Circle Walker first moved along the arc a degree measure of 90° , then to 180° , then to 270° , and finally to 360° . So it appears that there are 4 arcs of equal degree measure of 90°. Hence there is 360° around a circle. Also, and this is important, each of the 4 arcs in quadrant 1, 2, 3 and 4 are of equal length due to the fact that the circle is centered on the CCS.

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TOPIC 3 : THE CONCEPT OF A RADIAN:
AN ALTERNATIVE UNIT FOR MEASURING AN ANGLE
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A "radian" is an alternative way of measuring an angle. It was developed for a particular purpose that you will soon see is quite useful. So what is a radian?

If you place the radius of a circle on the outside of a circle, the central angle created with the arc is called "1 radian." See figure 6 below.

For each radius place on the outside of the circle you increase the angle of the central angle in radians. I created a central angle of 2 radians in figure 7 below.

In figure 8, I created an angle of 1/2 radian which is determined by 1/2 of a radius.

SO TO SUM UP: we have created an alternative way of measuring an angle using the radius of a circle. The unit of measurement is called a "radian."

figure 6	figure 7	figure 8

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TOPIC 4 : HOW MANY RADIANS IN A CIRCLE?
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So we know for each radius we place on the circumference of a circle we produce an angle called a "1 radian."
Can you suggest a way that we can determine how many radians there are in a circle? Well, we can place a bunch of radii(plural for radius) on the outside of a circle to get a sense of this. Let's try this. See figure 9 below.

figure 9

It appears that the angle in radians around the circle is a bit more than 6 radians. It might be a good idea to go to the RADIAN CREATOR MODULE which will give you an interactive experience with creating radians.

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TOPIC 5 : ANSWERING THE THE BIG QUESTION
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For some reasons many students have a difficult time understanding the connection between radians and degrees. It is important as you go through this section that you attempt to verbalize the logic that I develop here and make sure you can construct these ideas on a piece of paper.

Let's start by deconstructing multiplication.

I think you get the idea.

THE LOGIC OF THE TOTAL NUMBER OF RADIANS IN A CIRCLE

1. The circumference of a circle is 2π(length of the radius); hence, there are 2π radii on a circle or 6. 283 radii on a circle.
   ...I used the same reasoning that I did in the multiplication examples above. Please note that 6.283 is an approximation of π.
   
   
2. For each radius placed as an arc on the circumference of a circle an angle measure of 1 radian is created.
   ...Remember every time a radius of a circle is put on the outside of a circle we create an angle called 1 radian; 2 radii on the outside of the circle an angle of 2 radians; 3 radii on the outside of the circle an angle of 3 radians, etc.
   
   
3. Hence, since there are 2π radii ON a circle there MUST also be an angle measure of 2π radians in a circle - or 6. 283 radians - because for EVERY radius on a circle there is ALSO an angle of 1 radian.

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The total angle measure in radians in a circle
is
2π radians or approximately 6.283 radians
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So, just like we know there is an angle measure of 360° in a circle we now know there is an angle measure of 2π radians in a circle. PLEASE MAKE SURE YOU KNOW HOW TO DEVELOP THIS LOGIC OF RADIANS IN A CIRCLE!

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360°_{in a circle}  and  2π radians_{in a circle}

are two ways of stating the angle measure in a complete circle
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So just like we can say that 12 inches and 30.48 centimeters name the length 1 foot. We now know that 360° and 2π radians name the central angle produced in 1 revolution around a circle.