Module 4  Parametric Equations, Trigonometric and Inverse Trigonometric Functions  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  Self Test  
Lesson 4.2: Trigonometric Functions  
In this lesson you will use parametric equations to illustrate the connection between the graphs of y = sin(x) and the unit circle. You will also see how to transform the graph of y = sin(x) to obtain the graph of y = sin[B(x + C)] + D. Exploring x = cos t, y = sin t Before investigating the relationship between the sine function and the unit circle, examine the parametric equations shown below. For , predict the shape of the curve that is produced by the parametric equations y = sin t 4.2.1 Make sure your calculator is in Radian mode by checking the MODE menu. Graph the parametric equations to verify your prediction. Click here for the answer.
The Sine Function The sine function can be defined in several ways. One way is to let sin t be the ycoordinate of the point on the unit circle whose intercepted arc is t units long, which is another way of saying that the associated central angle is t radians. Recall that the sine wave has the following shape: The relationship between the graph of y = sin x and the unit circle's yvalues can be illustrated by graphing both simultaneously. Sequential and Simultaneous Graphing Modes In the sixth row of the MODE menu, the options are Sequential or Simultaneous (Simul). (See the screen below.) In Sequential mode, graphs are drawn one at a time in the order listed in the Y= editor. In Simultaneous mode, all graphs in the Y= editor are drawn at the same time rather than sequentially. Select the following settings in the MODE menu to have parametric equations graphed simultaneously. Unwrapping the Sine Wave
4.2.2 Describe the relationship you see between the circle and the sine wave. Click here for the answer. A Periodic Function and Its Period A periodic function has yvalues that repeat over specific intervals of xvalues. The period of a periodic function is the length of an xinterval over which the yvalues make one complete cycle. The sine function has a period of 2 , which is the number of radians in one complete revolution. A Transformed Sine Curve The graph of y = Asin[B(x + C)] + D is a transformed version of the graph of y = sin(x). The effects of each of the values of A, B, C, and D are listed below. The transformations were discussed in Module 3. A produces a vertical stretch or shrink by a factor of A; and a reflection about the xaxis if A is negative. A is also called the amplitude of the sine wave, which is half the distance between the highest and lowest yvalues of the graph. B produces a horizontal stretch or shrink, which changes the period of the curve. The period of the transformed sine wave is . If B is negative there is a reflection about the yaxis. C produces a horizontal shift. D produces a vertical shift. The graph of y = 5sin[ (x  1)] + 2 is the transformed graph of y = sin x that has the following characteristics: Amplitude of 5 (stretched vertically by a factor of 5) Period of (compressed horizontally by a factor of ) Shifted horizontally right 1 unit Shifted vertically up 2 units
Press to enter and make sure you enter the parentheses correctly.
4.2.3 Describe the transformations of y = sin(x) that produce the graph of y = 3sin[2(x + 1)] 4 then graph the equation on your calculator. Click here for the answer. 

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