Answer Sheet for Activity #3
Question #1:
This question is simply instructions to familiarize with the applet. Thus there is no solution to this problem.
Question #2:
The purpose of this question is to have students see how changing the angle affects the sine and cosine values. For this question, students simply need to report what they are observing. In other words by using the applet to drag theta, students can physically see how changing the angle affects the sine and cosine values. Thus for this question, the solution would be that as you are increasing the angle the sine values are increasing from 0-90 and 270-360 and decreasing from 90-180 and 180-270, while the cosine values are increasing from 180-270 and 270-360 and decreasing from 0-90 and 90-180.
Question #3:
The purpose of this question is to again have student investigating how changing the angle affects the sine and cosine values. The only difference is that for this question, students are focusing only on the cosine values. Just as the previous question, students can use the the applet to drag theta in order to physically see how changing the angle affects the cosine values. Thus a solution for this question would have to be along the lines of as the angle increases the cosine values are increasing from 180-270 and 270-360 and decreasing from 0-90 and 90-180.
Question #4:
The purpose of this question is for students to begin relating the cosine values to the coordinate plane. In other words, this question helps students realize that the cosine values represent the x-coordinate on the unit circle. With this question, students can use the applet to drag theta and watch as the cosine values change. These values can be seen in the upper right hand corner of the applet. Another way a student could find this solution is to watch the red segment located on the circle. With this students can see that indeed the cosine values represent the x-coordinates through the rise over run rule. Thus he solution to this question is that the cosine value represents the x-coordinate.
Question #5:
The purpose of this question is that of questions 3 and 4. The only difference is that now instead of focusing on cosine, the students are focusing on sine. With this question, students can use the applet to drag theta and watch as the sine values change or how changing the angle affects the values. Thus the solution for this question will be that the sine values represent the y-coordinates through the rise over run rule and as the angle increases the sine values are increasing from 0-90 and 270-360 and decreasing from 90-180 and 180-270.
Question #6:
The purpose of this question is to reiterate what the sine and cosine values represent in the coordinate plane. Through the investigations of the previous questions students can simply restate the fact. Thus the solution for this question is that the sine values represent the y-coordinates and the cosine values represent the x-coordinates of the unit circle.
Question #7:
The purpose of this question is to have students practicing how to write the coordinate values of the unit circle. With this question, students will use the technology to move the slider to the appropriate value and record the values showcased in the upper right hand corner of the applet. Thus for this question the solution will be that the coordinates of a 30 degree angle are (0.5, 0.87) or (1/2, sqrt(3)/2).
Question #8:
The purpose of this question again is to have students practicing how to write the coordinate values of the unit circle. With this question, students will use the technology to move the slider to the appropriate value and record the values showcased in the upper right hand corner of the applet. Thus for this question the solution will be that the coordinates of a 45, 60, 90, 120, 135, 150, and 180 degree are: 45 = (0.71, 0.71); 60 = (0.87, 0.5); 90 = (0, 1); 120 = (-0.5, 0.87); 135 = (-0.71, 0.71); 150 = (-0.87, 0.5); and 180 = (-1, 0)
This question is simply instructions to familiarize with the applet. Thus there is no solution to this problem.
Question #2:
The purpose of this question is to have students see how changing the angle affects the sine and cosine values. For this question, students simply need to report what they are observing. In other words by using the applet to drag theta, students can physically see how changing the angle affects the sine and cosine values. Thus for this question, the solution would be that as you are increasing the angle the sine values are increasing from 0-90 and 270-360 and decreasing from 90-180 and 180-270, while the cosine values are increasing from 180-270 and 270-360 and decreasing from 0-90 and 90-180.
Question #3:
The purpose of this question is to again have student investigating how changing the angle affects the sine and cosine values. The only difference is that for this question, students are focusing only on the cosine values. Just as the previous question, students can use the the applet to drag theta in order to physically see how changing the angle affects the cosine values. Thus a solution for this question would have to be along the lines of as the angle increases the cosine values are increasing from 180-270 and 270-360 and decreasing from 0-90 and 90-180.
Question #4:
The purpose of this question is for students to begin relating the cosine values to the coordinate plane. In other words, this question helps students realize that the cosine values represent the x-coordinate on the unit circle. With this question, students can use the applet to drag theta and watch as the cosine values change. These values can be seen in the upper right hand corner of the applet. Another way a student could find this solution is to watch the red segment located on the circle. With this students can see that indeed the cosine values represent the x-coordinates through the rise over run rule. Thus he solution to this question is that the cosine value represents the x-coordinate.
Question #5:
The purpose of this question is that of questions 3 and 4. The only difference is that now instead of focusing on cosine, the students are focusing on sine. With this question, students can use the applet to drag theta and watch as the sine values change or how changing the angle affects the values. Thus the solution for this question will be that the sine values represent the y-coordinates through the rise over run rule and as the angle increases the sine values are increasing from 0-90 and 270-360 and decreasing from 90-180 and 180-270.
Question #6:
The purpose of this question is to reiterate what the sine and cosine values represent in the coordinate plane. Through the investigations of the previous questions students can simply restate the fact. Thus the solution for this question is that the sine values represent the y-coordinates and the cosine values represent the x-coordinates of the unit circle.
Question #7:
The purpose of this question is to have students practicing how to write the coordinate values of the unit circle. With this question, students will use the technology to move the slider to the appropriate value and record the values showcased in the upper right hand corner of the applet. Thus for this question the solution will be that the coordinates of a 30 degree angle are (0.5, 0.87) or (1/2, sqrt(3)/2).
Question #8:
The purpose of this question again is to have students practicing how to write the coordinate values of the unit circle. With this question, students will use the technology to move the slider to the appropriate value and record the values showcased in the upper right hand corner of the applet. Thus for this question the solution will be that the coordinates of a 45, 60, 90, 120, 135, 150, and 180 degree are: 45 = (0.71, 0.71); 60 = (0.87, 0.5); 90 = (0, 1); 120 = (-0.5, 0.87); 135 = (-0.71, 0.71); 150 = (-0.87, 0.5); and 180 = (-1, 0)