Analysis of Trigonometric Functions - Assignment Example

School Trigonometric Functions and Their Derivatives Number Examination Session Draw a table and calculate 10 values foreach of the and functions between the range of. (5 marks) -1.5707 0.0000 -1.1780 0.3826 -2.4142 -0.7853 0.7071 -1.0000 -0.3926 0.9238 -0.4142 0.0000 1.0000 0.0000 0.3141 0.9510 0.3249 0.3926 0.9238 0.4142 0.7853 0.7071 1.0000 1.1780 0.3826 2.4142 1.5707 0.0000 2) Using the values obtained in part 1, draw a graph of the and functions for. Make sure you use one whole page for this. You can do all the functions in the same page. (5 marks) 3) Prove using the unit circle graph that, and. (10 marks) Solution: In order to define cosine and sine function, we draw a unit circle and an angle with radian measure measured counterclockwise from the x-axis or we have the negative angle measured clockwise. Thus, cosine and sine function for the angle are given by the adjacent side, opposite side and hypotenuse of right triangle, see figure: The point is described by the - point, the point is describe by the -point and the point is at , because: and On the other hand, we observed that the point where the angle crosses the unit circle lies vertically above the point where the negative angle crosses, so the components of the point for the negative angle are when the components for the positive angle are, thus: The point is described by the - point, the point is describe by the -point and the point is at , because: and Finally, we have the follow conclusion (Johnston2002), Cosine function is an even function: Sine function is an odd function: Q.E.D. 4) In your own words, explain the relationship between the function and its derivative. (You can use graphs to illustrate your answer and first principles, but you must have at least one paragraph with a minimum of 30 words (your own) of explanations). (10 marks) In order to describe the sine function and its derivative, we first need to find the critical points of sine function. The sketched sine function is shown; we noted that the slope of the graph is zero at and at. So we know that the graph of the derivative of touches the -axis at those two -values. Now we need to find the inflection points. We noted that we have three inflection points in this closed interval: at , , and at . We also noted that the value of the derivative of is 1 at , at and again at . This tell us that the graph of the derivative has a local maxima at the point , a local minimum at the point and a local maxima again at . If we plot these points, and sketch the graph of the derivative of using all of this information. The curve that we get looks very familiar: it is the graph of . The derivative of is (Johnston2002): The functions sine and cosine are related each other. 5) Differentiate (10 marks) Solution: Using the trigonometric identity (Lehmann, 1994; Moyer, 1998): Thus, We can calculate: then, which simplifies to: Then by the chain rule (Lehmann, 1994; Moyer, 1998): Which yields: Using the identity (Lehmann, 1994; Moyer, 1998), Thus, Finally, Q.E.D 6) Differentiate (10 marks) Solution: Given: and Form the derivatives of sin and cosine and using ratio’s rule (Lehmann, 1994; Moyer, 1998), we have: Thus, After some algebra, Using the identity: (Lehmann, 1994; Moyer, 1998) Q.E.D. References: Moyer R.E. and Ayres F., 1998 “Schaum’s Outline of Trigonometry”, McGraw-Hill Johnston E.H. and Mathews J., 2002 “Calculus” Pearson Lehmann C. H., 1994 “College Algebra”, John Wiley and Sons, Inc. Read More