3.+Given+an+angle+measure+in+degrees+or+radians,+draw+the+angle+in+standard+position

Mike Osipowicz

I chose the topic of drawing angles in their standard position when given an angle measurement in either degrees or radians for a few different reasons. To start, I really have a good understanding of this topic and have a few ideas to help teach it to everyone else. Also, drawing and understanding angle positions is very important in my favorite math subject, trigonometry.

The first step in drawing angles in their standard position correctly is actually understanding what standard position is. Standard position means one ray (or both) of the angle is located on the positive X axis, and the vertex of that angle is located on the origin. Example:
 * [[image:precalculusnwr7/standard.gif]]

The actual measurement of an angle by definition is the amount of rotation from one ray to the other. This measurement is taken from the X axis, and a clockwise rotation is a negative angle measure, while a counter-clockwise rotation is necessary for a positive angle measure.

An angle measurement has infinite possibilities because as soon as it goes over 360 degrees its second ray is located at X multiples of 360 minus the angle measure remaining. X is determined by finding how many full rotations around the origin the angle makes before coming to its end.

When an angle measure is given in degrees, one degree stands for 1/360 of a rotation. Meaning 9- degrees is 1/4 rotation, 180 degrees is 1/2 and so on. When an angle measure is given in radians it is a bit different. On the unit circle, an angle cuts off an arc and the length of that arc is the angles measurement in radians. Therefore, converting between degrees and radians requires the use of the formula: 1 degree= PI/180 radians.

There are two different commonly used ways of measuring an angle. The first, more common way is called the principal angle which is just an angle measurement from the initial to the terminal ray no matter how many revolutions or any fraction of revolutions it has. The second term used for recording the measure of an angle is called its reference angle. A reference angle is the measure of the angle closest to the X axis. With any angle inside the boundaries of the first quadrant, the reference angle is equivalent to the principal angle. For an angle with a terminal ray in the second or third quadrant, the reference angles measure is equal to 180-principal angle, and for the fourth quadrant, the reference angle is equal to 360-principal angle.

media type="file" key="Angles in standard position.avi" width="300" height="300"

Final Example Problem:

Imagine an angle measuring 240 degrees positive in the standard position. A: What is the NEGATIVE measure of this angle? B: What are all of the possible angle measures that would be co-terminal to 240 degrees? C: What is the reference angle measure of 240 degrees? D: What is the measure of this angle in radians?

E:Where would the initial ray of an angle measuring -240 degrees be located?

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