8-2 Sine and Cosine Curves
 
 
 
         From previous sections, we worked with horizontal and vertical stretching and shrinking.  Recall that vertical stretching/shrinking happens by changing the amplitude of the graph.  Horizontal stretching/shrinking happens by changing the period length of the graph.   We will apply these concepts to the Sine and Cosine functions.
 
 
 
For functions y = A sin Bx and y = A cos Bx (A not = 0 and B > 0)
amplitude = | A |    and  period = 2p/B
 
Demonstration of Graph of y = sin Bx (Manipula Math)
 
Graphing Examples
1)  Graph y = 3 sin 2x
 
        Solution:  Remember that the basic sine curve has zeros at the beginning, middle and end of a cycle.  Reaches its maximum at the 1/4 mark and has a minimum value at the 3/4 mark.
            This graph has an amplitude = 3 and a period length = 2p/2 =  p
Thus, we have zeros at (0, 0), (p/2, 0) and (p, 0)
                         maximum point at ( p/4, 3)
                         minimum point at (3p/4, -3)
Plot these points and draw a smooth curve to get:
 
2)  Graph y = (1/2)Cos 3x
 
        Solution:  Remember that the basic cosine graph begins and ends at its maximum point.  In the middle, it is at its minimum value, and has zeros at the 1/4 and 3/4 mark.
                        For this graph, the amplitude = 1/2
                                                              period = 2p/ 3
        The maximums  are at the beginning point  (0, 1/2) and
                                                    end point (2p/ 3, 1/2)
                          minimum point at ( p/3, -1/2)
                          Zeros at ( p/ 6, 0)  and ( p/ 2, 0)
Plot these points and smoothly connect to get:
 
3)  Graph y = -2 Sin (p/2)x
 
            Solution:  The amplitude = | -2 | = 2
                                     Period = 2p/ (p/ 2) = 4
                    Note that this graph is a reflection about the x-axis.  This interchanges the maximum and minimum values.
                        zeros at (0, 0), ( 2, 0), ( 4, 0)
                        minimum ( 1, -2)
                        maximum ( 3, 2)
Plot these points and:
 
4) Graph y = -3 cos (p/ 5)x
 
                Solution:  This again is a reflection about the x-axis.
                                  amplitude = | -3 | = 3
                                  period = 2p/ (p/ 5) = 10
                Minimum values at (0, -3), (10, -3)
                Maximum value at ( 5, 3)
                Zeros at ( 2.5, 0) and ( 7.5, 0)
Connect the points and:
 

 
Determining the equation from the graph
 
1) Find the amplitude and period length of each function.  Then write an equation.
 
            Solution:  The graph is a sine curve because it begins at (0, 0).  The amplitude = 3 and the period length is (1/2)p.
                        Since the amplitude = 3, A = 3.
                        Because period = (1/2)p, B = 2p/(1/2)p = 4
                        Thus, the equation is:  y = 3 sin 4x
 
2)  Find the amplitude and period for the function.  Write an equation for the graph.
 
                    Solution:  amplitude = 2 and period length = 1
                        The graph is a cosine graph reflected about the x-axis.  The graph starts at a minimum.
                        A = -2  because it is reflected about the x-axis.
                        B = 2p/1 = 2p
                        Thus the equation is:  y = -2 cos 2px

Solving an equation algebraically or graphically
 
1)  Solve the equation for 0 < x < 2p.   3 sin 2x = 1
                    a)  Graphically using the TI-82 calculator
                                Set calculator to radian mode.  Go to  y=  screen.
                                Type 3 sin 2x for y1  and 1 for y2  Press graph.  Zoom in.
 
Notice that the graphs cross four times between 0 and 2p.  Using your calculator, press the calc button, then the intersect button.  Guess first curve, second curve, then best guess.  Answer to the point furthest to the left is :
(.17, 1)  Repeat the process for the other 3 points.
Other answers are:  (1.40, 1), (3.31, 1) and (4.54, 1).
 
                    b)  Algebraically.
 
3 sin 2x = 1
sin 2x = 1/3
2x = sin-1 (1/3)
The reference angle is .34 (using your calculator)
We need to check through 2 periods to find all four answers.
sine is also positive in quad II (3.14 - .34) = 2.80
Now add 6.28 to each of these values to get all four answers.
2x = .34, 2,80, 6.62, 9.08
x = .17, 1.40, 3.31, 4.54

In the next section, we will add translations to the graph of sine and cosine!!