# Unit Circle Essays

1.explain what a radian measure represents using the unit circle as a reference.

-A radian measure on a unit circle is the measure of the length of the arc at certain points on the unit circle.

2.how do special right triangles directly relate to a unit circle.

-On the unit circle the radius of the circle is also the hypotenuse. therefore if you set the hypotenuse to be a value such as 1. the side (x,y) of the triangle will be the sine and cosine values on the unit circle.

3. Suppose that you did not have the Unit Circle on Circle A, but rather a circle of radius

5 Will the angle measures in degrees and/or radians change? Why or why not?

- If I did not have a unit circle of circle A but instead had a circle with a radius of 5, the angle measures in degrees and/or radians would not change. Because no matter the radius one revolution around the circle will always be 2π therefore the sin and cos values of the unit circle would have a correlation with the radius in the x and y direction and the method in which we use to get the values of trigonometric functions would all still apply to the circle.

4. Suppose that you did not have the Unit Circle on Circle A, but rather a circle of radius 5. What do you suppose the x- and y-coordinates will be for that circle in Quadrant I?

- The coordinates for quadrant I would be (5,0). This is if you take a basic unit circle with a radius of 1 the coordinates for quadrant I would be (1,0) therefore if you have a circle with a radius of 5 the coordinate values would be (5,0) because the circle only expanded in size.

5.Consider the two points in Quadrant I on Circle B. What is the special relationship between them? (Consider the relationship between the angles whose terminal sides pass through these points.

- The special relationship between the two points in quadrant I on circle be is that when a line a drawn through those points it will form an angle that is in standard position.

6.Consider the point in Quadrant...

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The unit circle is the circle whose center is at the origin and whose radius is one. The circumfrence of the unit circle is 2*Π*. An arc of the unit circle has the same length as the measure of the central angle that intercepts that arc. Also, because the radius of the unit circle is one, the trigonometric functions sine and cosine have special relevance for the unit circle. If a point on the circle is on the terminal side of an angle in standard position, then the sine of such an angle is simply the y-coordinate of the point, and the cosine of the angle is the x-coordinate of the point.

Figure %: The unit circle

This relationship has practical uses concerning the length of an arc on the unit circle. If an arc has one endpoint at (1,0) and extends in the counterclockwise direction, the other endpoint of the arc can be determined if the arc length is known. Given an arc length s, the other endpoint of the arc is provided by the coordinates (cos(*s*), sin(*s*)). This is a common alternative way to plot the unit circle. Most often, the unit circle can be drawn according to the equation *x*^{2} + *y*^{2} = 1. As we have seen here, though, it can also be drawn according to the equations *x* = cos(*s*), *y* = sin(*s*), where s is the length of the arc starting at (1,0).

Figure %: The unit circle

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