please help me solve the following exercise:

Graphs of Trigonometric Functions Using Derivatives Grade 11 13 Math A&A June 2022 Before you start the activity read the instructions carefully. In this activity you will work in pairs (or groups of three if necessary). The main goal is to nd a way to draw the graphs of the trigonometric functions sine and cosine using your knowledge of derivatives. For this activity you shouldn't use calculators or computers, all the calculations needed can (and should) be carried out by hand. All information you need is presented in the beginning of the activity, but feel free to ask if you need any help. This investigation is divided in four parts, you should go through the activity in the right order. 1. The rst part consists of simple questions about your prior knowledge, answer the questions as a group. 2. The second part present steps to draw the graph of the sine function, if you follow these steps you should get a good sketch of the graph. 3. The third part is very similar to the second one, but you won't have the steps description this time, a good strategy is to adapt the steps from the second part. 4. The last part is a quick reection about what you have learned and your general thoughts on this activity. @: o The values of the functions f (as) = sin(.r) and 9(3) = cos(a:) can be calculated for any x E R. o The functions f(w) = serum) and 9(3) = cos(a:) are both continuous and differentiable in its domains. 0 360 = 2n rad. 1 Prior Knowledge (a) Does anyone in your group know how to draw the graph of trigonometric functions? The graph of which functions can you draw? (b) What is a root of a function? Do you remember how to nd the roots of any function? (e) Describe briey how you can nd the local maximum or local minimum points of a function. Do you know how to do that for trigonometric functions? 2 The Sine Function Now you are expected to draw the graph of the trigonometric function f (as) = sin($). As a group reect on how can you do this. Try to come up with a strategy to draw the graph. If you don't have many ideas answering the following questions may help. What is the best way to nd points in the graph such that no important feature of this graph is left out? How many roots does this function have? How many local maximum points does this function have? How many local minimum points does this function have? After reecting for a while or coming up with your strategy follow the steps listed below. Steps: 1. Using values of 33 in radians, nd the roots(zeros) of the function f (9:) = sin(a:). Remember that you can take any a: E R. Find an expression to describe all the roots. 2. Plot the points corresponding to the mintercepts in a plane. 3. Find the critical points of the function and after that determine which of those points are local maximum or local minimum points. 4. Plot the maximum and minimum points in the plane you have. 5. Connect the points to form a graph, keep in mind that the graph of such a function has no corners or discontinuities. 3 The Cosine Function Now you are expected to draw the graph of the trigonometric function g($) = cos($). As a group reect on how can you do this. Try to come up with a strategy to draw the graph. If you don't have many ideas answering the following questions may help. What is the best way to nd points in the graph such that no important feature of this graph is left out? How many roots does this function have? How many local maximum points does this function have? How many local minimum points does this function have? After you reected for a while and came up with your strategy try to use this strategy to draw the graph of 9(93) = want)- 4 Final Thoughts (a) What features of the graphs you've drawn are the most interesting to you? (b) Can you apply a method like the one you used do draw the graph of sine and cosine to other kinds of functions? If yes, for which functions you believe this method would work? (c) What are your nal thoughts on this activity