Graphs of the Sine and Cosine FunctionsReview of Prerequisites:1. The unit circle is the circle that has as its center the origin of the coordinate plane and aradius of length one. Using a rectangular coordinate system, the unit circle is the set ofpoints described by {(x,y) | x2 + y2 = 1}. Using a polar coordinate system, the unit circle isdescribed by {(r,) | r = 1}.2. The wrapping function is the function P that assigns to each number t on the number line,the ordered pair representing the coordinates of the point on the unit circle to which it isattached when the number line is wrapped around the unit circle as described: 0 is attachedto the point (1,0); the positive half of the number line is then wrapped around the unitcircle in the counterclockwise direction and the negative half wrapped in the clockwisedirection.The abscissa of the point on the unit circle to which the point t on the number line hasbeen attached is defined to be the cosine of t, written cos t. The ordinate of this point isdefined to be the sine of t, written sin t.3. Equivalently, for any real number t, the cosine of t (cos t) and the sine of t (sin t) may bedefined as the first and second coordinates respectively of the image of the point (1,0) undera rotation of magnitude t about the origin.Objectives:1. To use the wrapping function, the Pythagorean Theorem, and the symmetries of the unitcircle to calculate exact values of the sine and cosine functions of a selected set of numbers.2. To graph the sine and cosine functions.Materials:1. TI82 CalculatorSine and Cosine GraphsActivity:1.. Use the following diagrams, the Pythagorean Theorem, and the geometry of the circle to findthe exact values of the sine and cosine functions indicated in the tables that follow thediagrams. Use your calculator to approximate t, sin t, and cos t each to two decimalplaces.1 a.P41(1,0)(0,1)(-1,0)(0,-1)In the diagram above, label the lengths of the legs of the right triangle that is shown. Discussthe geometric concepts that are used in finding these lengths.List two additional values of t that the wrapping function P assigns to the point P(/4).2b.P61(1,0)(0,1)(-1,0)(0,-1)In the diagram above, label the lengths of the legs of the right triangle that is shown. Discussthe geometric concepts that are used in finding these lengths.List two additional values of t that the wrapping function P assigns to the point P(/6).c.tExact 0/6 /4 /3 /2 2/3 3/4 5/6 Approx.cos tExactApprox.sin tExactApprox.What symmetry of the unit circle may be used to obtain the values of sin t and cos t forvalues of t in the table that are larger than /2 from those corresponding to values of tsmaller than /2 ? Explain.d.tExact 0-/6 -/4 -/3 -/2 -2/3 -3/4 -5/6 -Approx.cos tExactApprox.sin tExactApprox.What symmetry of the unit circle may be applied to obtain the values in this table fromthose in (c)? Explain.e.tExact 7/6 5/4 4/3 3/2 5/3 7/4 11/6 2Approx.cos tExactApprox.sin tExactApprox.What symmetry of the unit circle may be applied to obtain the values in this table from thosein (c)? Explain.4. On the coordinate axes provided, approximate the graphs of y = cos t between t = -2 andt = 2 by plotting the points in the preceding tables, and drawing a smooth curve through thepoints. y = cos t-10.51-0.535. On the coordinate axes provided, approximate the graphs of y = sin t between t = -2 andt = 2 by plotting the points in the preceding tables, and drawing a smooth curve through thepoints.y = sin t-10.51-0.546. What do you observe about the graph of each from t = -2 and t = 0 as compared with thegraph from t = 0 to t = 2? Explain why this is the case.7. How would you describe the graph of each function from t = 2 to t = 4? How would youextend the graphs to include all values of t on the number line? Explain how, and why youranswers are mathematically valid.8. What, if any, symmetries do each of these graphs possess? Explain.9. If you know that the sine of a number is 0.7, can you find its cosine? Explain how and why.10. A toddler's swing set is constructed with the seat supported by 6' metal rods that arerestricted from rotating more than 45 from the vertical position. The seat is 2' above theground. How high above the ground is it possible for an energetic child to swing? (Use theideas in this activity, circles and the concept of similarity to solve this