2. Use a Protractor to Measure and Mark Every Around the Circle

Name ______Period ______1. At one end of the paper, construct a circle on a set of axes, which has a radius of one spaghetti length. 2. Use a protractor to measure and mark every around the circle. 3. Adjacent to the circle draw another set of axes with an x-axis that is about 6.5 spaghetti lengths. 4. Place the string along the circle with the end at zero. Mark the marks onto the string. Then stretch the string along the x-axis of the other set of axes and transfer the marks. You have used arc lengths, which are to determine the intervals on the x-axis. Label them accordingly. 5. Place a piece of spaghetti from the origin to the mark on the circle. Take another piece of spaghetti and measure the vertical distance from the “Hypotenuse” to the x-axis. Take this measurement and transfer it to the other set of axes. Place the vertical piece perpendicular to the x-axis above the or interval and place a dot at the end of the spaghetti to record the height of the spaghetti. 6. Continue this process until you have gone completely around the circle. 7. Draw a smooth curve to connect your dots. 8. You have graphed the sine curve. Take out your calculators and graph . Does it look like your drawn graph? 9. Now try to graph the cosine curve. What values do you need to transfer to the axes this time?

Spaghetti Sine Curve Questions:

1. What is the radius of the circle (in spaghetti units)? 2. What is the circumference in spaghetti units? 3. Where would a triangle corresponding to be constructed? 4. What is the period of the sine curve? That is, what is the wavelength – after how many radians does the graph start to repeat? 5. Compared with the radius, what is the height of the triangle at ? This number is the sine of . 6. Compared with the radius, what is the height of the triangle at the , , and ? 7. If you build triangles only at the , , and so forth, marks, what is the smallest number of different triangles that you need to form to obtain the lengths needed to construct the graph of one period of the sine curve? 8. Write a one-paragraph explanation to a classmate about why .

Name ______Period ______Step 1: Hit MODE: Set to radian, parametric (par), and “simultaneous” (simul) graphing modes.

Step 2: Hit WINDOW: Set Tmin = 0, Tmax = 6.3, Tstep = . Set Xmin = -1.2, Xmax = 6.3, Xscl = 1, Ymin = -2.5, Ymax = 2.5, Yscl = 1

Step 3: Hit Y= : Set X1T = and Y1T =. This will graph the unit circle.

Set X2T = T and Y2T = . This will graph the sine curve.

Step 4: Now start the graph and watch the point go counterclockwise around the unit circle as the curve goes from 0 to 2 in the positive direction. You will simultaneously see the y-coordinate of the point on the circle being graphed as a function along the horizontal axis. To watch the drawing again, go into Y=, retype an equation and hit graph. You can do this as many times as you need to in order to answer the following questions.

Step 5: Double the value of Tmax, (Tmax = 12.6). Change the (x, y) window to by . Go into Y = and change the graphing style (far left) to be a moving point. Run the graph and watch how the sine curve tracks the y-coordinate of the point as it moves around the unit circle. You can do this as many times as you need to in order to answer the following questions.