9b)++Angles+on+the+unit+circle

[[image:unitcircle.gif align="center" caption="The Unit Circle"]]
====Our next step in our exploration of trigonometry is to evaluate trigonometric functions of any angle. Before we can take this step, we must first review the amazing unit circle! The unit circle, a circle with a radius of 1 and a center at (0,0) connects sine and cosine to x- and y-values in the coordinate plane. This section will set the foundation for the next two sections. You will learn the basics of drawing angles in the coordinate plane, as well as identifying and finding critical values pertaining to these angles.==== ====Throughout this section it will be necessary to be familiar with the following terminology: //terminal sides, standard position, reference angles,// and //coterminal angles.// In addition to having a familiarity with the terminology, we will also convert between //radians// and degrees, sketch angles, and find reference and coterminal angles.====

Example Problems:

1. Draw 470° in standard position. Then find  two coterminal angles (one positive and one negative) and the reference angle. Lastly, convert 470° to radians.
 * Standard Position:
 * First, when we draw an angle in standard position, we put the initial side on the positive x-axis.
 * Next, notice that 470° is greater than 360°. This is why we must draw the arrow representing this angle starting from the initial side and making more than one revolution. Additionally, it is positive. So we move in the counterclockwise direction.
 * Since 470° is 110° more than 360°, we have our terminal side in Quadrant II, where the terminal side for 110° lies.
 * Coterminal Angles:
 * Recall: We must add or subtract multiples of 360°. Also recall, coterminal angles share terminal sides!
 * 470° - 360° = 110° (this angle is drawn on the figure below)
 * 470° - 720° = -250°
 * Reference Angle:
 * Recall: The reference angle is the //**acute**// angle formed between the **//terminal side//** and the //**x-axis**//
 * So, our reference angle is in Quadrant II, which means we want to take 180° and subtract 110°. This gives us our reference angle of 70°, which is the red angle below.
 * Degrees to Radians:

2. Draw -130° in standard position. Then find  two coterminal angles (one positive and one negative) and the reference angle. Lastly, convert -130° to radians.
 * Standard Position:
 * First, when we draw an angle in standard position, we put the initial side on the positive x-axis.
 * Next, notice that -130° negative. So we move in the clockwise direction.
 * The terminal side of -130° will lie in Quadrant III since it is an obtuse angle, but a smaller than -180°
 * <span style="color: rgb(0, 0, 255);">Coterminal Angles:
 * <span style="color: rgb(0, 0, 255);">Recall: We must add or subtract multiples of 360°. Also recall, coterminal angles share terminal sides!
 * <span style="color: rgb(0, 0, 255);">-130° - 360° = -490°
 * <span style="color: rgb(0, 0, 255);">-130° + 360° = 230°
 * <span style="color: rgb(255, 0, 0);">Reference Angle:
 * <span style="color: rgb(255, 0, 0);">Recall: The reference angle is the //**acute**// angle formed between the **//terminal side//** and the //**x-axis**//
 * <span style="color: rgb(255, 0, 0);">So, our reference angle is in Quadrant III, which means we want to take 180° and subtract 130°. This gives us our reference angle of 50°, which is the red angle below.
 * <span style="color: rgb(0, 255, 0);">Degrees to Radians:

3. Draw in standard position. <span style="color: rgb(0, 0, 0);">Then find <span style="color: rgb(0, 0, 255);"> two coterminal angles (one positive and one negative) and the <span style="color: rgb(255, 0, 0);">reference angle. Lastly, <span style="color: rgb(0, 255, 0);">convert <span style="color: rgb(0, 255, 0);"> to degrees. >> <span style="color: rgb(0, 0, 255);">
 * Standard Position:
 * First, when we draw an angle in standard position, we put the initial side on the positive x-axis.
 * Next, we know this is a positive angle. So we move in the counterclockwise direction.
 * The terminal side of this angle will lie in Quadrant III since it greater than [[image:curatola8.gif]], but a smaller than [[image:curatola9.gif]].
 * //*If it helps... you may want to convert this to degrees first!//
 * <span style="color: rgb(0, 0, 255);">Coterminal Angles:
 * <span style="color: rgb(0, 0, 255);">Recall: We must add or subtract multiples of 360°, which is [[image:curatola10.gif]]. Also recall, coterminal angles share terminal sides!
 * <span style="color: rgb(0, 0, 255);"> [[image:curatola11.gif]]
 * <span style="color: rgb(0, 0, 255);"> [[image:curatola12.gif]]
 * <span style="color: rgb(255, 0, 0);">Reference Angle:
 * <span style="color: rgb(255, 0, 0);">Recall: The reference angle is the //**acute**// angle formed between the **//terminal side//** and the //**x-axis**//
 * <span style="color: rgb(255, 0, 0);">So, our reference angle is in Quadrant II, which means we want to take [[image:curatola9.gif]]<span style="color: rgb(255, 0, 0);"> and subtract [[image:curatola1.gif]]<span style="color: rgb(255, 0, 0);">. This gives us our reference angle of [[image:curatola13.gif]], which is the red angle below.


 * <span style="color: rgb(0, 255, 0);">Degrees to Radians:

[|Coterminal Angles: The Math Page]

 * =====Scroll to bottom for a review and a problem set on coterminal angles=====