Directions: Examine the Unit Circle on the Cartesian Plane (Unit Circle: Circle centered at the origin whose radius is of length 1)

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1 Name: Period: Discovering the Unit Circle Activity Secondary III For this activity, you will be investigating the Unit Circle. You will examine the degree and radian measures of angles. Note: 180 radians. You will be discovering the x- and y-coordinates on the circle for specific angles. You will use some geometry of triangles and the Pythagorean Theorem to derive certain measures. You will use symmetry to label coordinates on the Unit Circle. Directions: Examine the Unit Circle on the Cartesian Plane (Unit Circle: Circle centered at the origin whose radius is of length 1) Part 1: Labeling Degree and Radian Measures on the Unit Circles We will label in a counter-clockwise direction. Remove the back page of the packet that contains circle A and Circle B, then answer the following parts. A. You are given two Unit Circles: Circle A and Circle B. Notice that each quadrant is 90. What is the radian equivalent to this angle? B. Looking at Circle A, notice that in Circle A, the angle in each Quadrant bisects the Quadrant in which it exists. What would the measurement of the bisected angle be? So in Circle A, what is the degree measure that each location increases by? What is the radian version of this angle? Label all of the angles, in degrees and radians, of Circle A, starting in Quadrant 1 on the x-axis (hint: this is 0 radians and 0 degrees). Remember to increase each angle as you proceed. C. Looking at Circle B, notice that the angles in each quadrant trisect the quadrant in which they exist. What would the measurement of the trisected angle be? What is the radian version of this angle? Label all of the angles, in degrees and radians, of Circle B, starting in Quadrant 1 on the x-axis (hint: this is 0 radians and 0 degrees). Remember to increase each angle as you proceed. D. How many degrees are in one full revolution of a circle? How many radians are in one full revolution of a circle? On both circles, your ending point is the same as your starting point. This location needs to have a label of 0 and a label of one full revolution in both degrees and radians. E. Our circles have measurements of a radius of length 1. So each quadrant has a radius of 1 that connects with the circle itself, forming a point. Label all 4 of the points that are formed at each quadrant (x, y).

2 Part 2. Finding the x- and y-coordinates for each angle at the axes and in quadrant I. A. Consider the Unit Square, meaning a square whose sides are all of an equal length of 1. The diagonal bisects the square into two equal triangles, which is called an isosceles right triangle. We will examine one of these triangles. 1. You are given a Unit Square. Draw a diagonal bisecting it. Label the angles that have been bisected. Label the outside lengths of the square (hint: it s a unit square what does the word unit translate to as a number?) Then next to the square, re-draw one of the triangles, with all three angles label. 2. Transfer the measurements for the legs of the triangle from the lengths of the sides of the squares. We just completed the unit on special right triangles. You may use this knowledge to find the hypotenuse of the triangle or use the Pythagorean Theorem to calculate the length of the hypotenuse. Do not convert the answer to a decimal. 3. Similar triangles are triangles whose corresponding angles equal and whose corresponding sides are in proportion. Below are drawn two right isosceles triangles. The first triangles comes from the unit square above. Label the lengths of the sides and the hypotenuse. The second triangle is similar to the first, but it has been scaled to be smaller. More specifically, each side and the hypotenuse were scaled so that the hypotenuse is of length 1. Find the lengths of the second triangle and label the lengths of that triangle. Rationalize any denominator.

3 4. Now we will transfer the scaled triangle (contained within a sqaure), to a circle that has a radius of length one. The diagram contains the square/triangle. Label the hypotenuse of the triangle and the right side and the bottom. Label the lower left angle, in degrees. The point on the circle corresponds to the lengths of the sides of the triangle. Label this point with the lengths of the triangle sides. How could we change this scenario if the triangle was drawn in quadrant II, with the angle still touching the origin? Would there be any differences in the labeling of the triangle with respect to the x-axis? Draw this triangle on the circle above, correcting labeling the sides. Identify the point that lies on the circle. Would we make any changes if the circle was drawn in quadrant III? Would we make any changes if the circle was drawn in quadrant IV? Transfer this information to Circle A. Labeling the points that the triangle makes.

4 B. Now, consider the Unit Equilateral Triangle, a triangle whose sides are all of equal length 1 and whose angles are all of equal measure of 60. The altitude (height) of this triangle is drawn from one corner of the triangle to its opposite side (the base of the triangle) forming a right angle. The altitude bisects the angle from where it is drawn and bisects the side that is opposite of that angle. The bisection creates two similar right triangles. We will examine one of these triangles. 1. You are given a Unit Equilateral Triangle. Label the sides of the Unit triangle with its measurement. Draw in the altitude, from the top angle. Label the bisected angle. Re-label part of the opposite side, then find the height of the altitude (this is a special right triangle from the previous unit). What is the length of the base? What is the height of the triangle? What is the length of the hypotenuse? What is the measurement of the base angle? We are going to transer the triangle to the Unit circle. The base angle will sit at the origin. Re-label the parts of the triangle onto the circle. The point on the circle corresponds to the lengths of the sides of the triangle. Label this point with the lengths of the triangle sides. How could we change this scenario if the triangle was drawn in quadrant II, with the angle still touching the origin? Would there be any differences in the labeling of the triangle with respect to the x-axis? Draw this triangle on the circle above, correcting labeling the sides. Identify the point that lies on the circle. Would we make any changes if the circle was drawn in quadrant III? Would we make any changes if the circle was drawn in quadrant IV? Transfer this information to Circle B, dealing with the angle measurement that corresponds to the base angle (this will be the upper point on Circle B, not the lower point). Labeling the points that the triangle makes.

5 C. Now, we will rotate the triangle, changing the base angle. What would the measurements be for the new base angle on the Unit circle drawn to the right? What would the measurements of the sides be? Label the sides and the base angle. List the point that the triangle makes with the circle. How could we change this scenario if the triangle was drawn in quadrant II, with the angle still touching the origin? Would there be any differences in the labeling of the triangle with respect to the x-axis? Draw this triangle on the circle above, correcting labeling the sides. Identify the point that lies on the circle. Would we make any changes if the circle was drawn in quadrant III? Would we make any changes if the circle was drawn in quadrant IV? Transfer this information to Circle B, dealing with the angle measurement that corresponds to the base angle (this will be the lower point on Circle B, not the upper point). Labeling the points that the triangle makes.

6 If we put the two circles together (A and B), we form the Unit Circle. The Unit Circle deals with three special right triangles and the coordinates of the quadrants. The points that go around the circle are always: (cos,sin ). The Unit Circle that we just created has a radius of one. In this scenario, the x-length is the cosine function and the y- length is the sine function. So to find any trigonometric function from a unit circle with a radius of one, the following equations are used: If the Unit circle does not have a radius of one, then the x-length is NOT the value of the cosine function. The x-length is just the adjacent of the triangle and must be placed over the hypotenuse to generate the answer for cosine. Likewise, the sine in no longer the y-length. The y-length is the opposite of the triangle and must be placed over the hypotenuse to generate the answer for sine. So to find any trigonometric function, from a unit circle of ANY radius, the following equations are used:

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