Sine and Cosine Curves by nkc8ft

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									     8.2 Sine and Cosine Curves
Objective To find equations of different sine
and cosine curves and to apply these equations.
          Sine and Cosine Curves
In section 4.4, we have learned that the graph of y = cf(x) can
be obtained by vertically stretching or shrinking and even
reflecting the graph of y = f(x) .
          Sine and Cosine Curves
Also in section 4.4, we learned that the graph of y = f(cx) can
be obtained by horizontally stretching or shrinking the graph
of y = f(x) .




In general, we can determine useful information about the
graphs of y = AsinBx and y = AcosBx by analyzing the
factors of A and B. Before we further discuss the graphs of
sine and cosine, let’s review the properties of sine and cosine.
Properties of Sine and Cosine Functions

The graphs of y = sin x and y = cos x have similar properties:

1. The domain is the set of all real numbers.
2. The range is the set of y values such that –1 y  1.
3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
   over an x-interval of 2.
6. The cycle repeats itself indefinitely in both directions of
   the x-axis.
 Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.                                 3
                  x       0                       2
                                 2                        2
                sin x         0       1         0       -1     0
 Then, connect the points on the graph with a smooth curve
 that extends in both directions beyond the five points. A
 single cycle is called a period.
                                  y           y = sin x
           3               1                           3        5
                    
            2          2                2               2   2    2
                                                                         x

                             1

Remember that y = sin x is an odd function whose graph is
symmetry to origin.
Here is the graph y = f(x) = sin x showing from -2
to 6. Notice it repeats with a period of 2.

                2       2       2       2




It has a maximum of 1 and a minimum of -1 (remember that is
the range of the sine function)
From Algebra recall that an odd function (which
the sine is) is symmetric with respect to the origin
as can be seen here
What are the x intercepts?              Where does sin x = 0?
            …-3, -2, -, 0, , 2, 3, 4, . . .
                  7          3            5
                         
                   2           2    2         2




                   3    2     0      2    3   4




Where is the function maximum?              Where does sin x = 1?
                7    3  5
                 ,   , ,   
                 2     2 2 2
Where is the function minimum?                             Where does sin x = -1?

                5    3 7
                 , ,  ,   
                 2   2 2 2

                   7                3                        5
                                
                    2                 2           2              2




                    3  2                    0           2    3   4




                            5                           3         7
                                         
                             2                2            2          2
  Graph of the Cosine Function
  To sketch the graph of y = cos x first locate the key points.
  These are the maximum points, the minimum points, and the
  intercepts.                                 3
                   x       0                        2
                                         2                 2
                    cos x        1       0       -1        0        1

  Then, connect the points on the graph with a smooth curve
  that extends in both directions beyond the five points. A
  single cycle is called a period.
                          y        y = cos x
          3               1                        3                5
                   
           2          2            2                2       2        2
                                                                             x

                            1
Remember that y = cos x is an even function whose graph is
symmetry to y-axis.
Here is the graph y = f(x) = cosx showing from -2
to 6. Notice it repeats with a period of 2.

          2          2        2          2




 It has a maximum of 1 and a minimum of -1 (remember that is
 the range of the cosine function)
Recall that an even function (which the cosine is) is
symmetric with respect to the y axis as can be seen
here
What are the x intercepts?                Where does cos x = 0?

             3     3 5
              , , ,  ,   
              2   2 2 2 2

                        2       0             2



                       3               3        5
                           2
                        2             2     2         2




Where is the function maximum?              Where does cos x = 1?
          …-4, -2, 0, 2, 4, . . .
Where is the function minimum?             Where does cos x = -1?

                  …-3, -, , 3, . . .


                        2      0            2



                        3            3        5
                            2                          4
                         2           2    2         2



                3                             3
 What would happen if we multiply the function by a
 constant?    All function values would be twice as high

             y = 2 sin x
                               amplitude is here
amplitude
  of this                                  y = 2 sin x
graph is 2


                                                   y = sin x




 The highest the graph goes (without a vertical shift) is called
 the amplitude.
Example: Sketch the graph of y = 3cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
                                                              3
                                0                                         2
                                          2                     2
                 x              3        0          -3         0            3
           y = cosx             1        0          -1         0            1
          y = 3cosx             3        0          -3         0            3
                           max          x-int      min        x-int        max

        (0, 3)
                 y                                        (2, 3)
             2
           1                                     2                3         4 x

           1                                 ( 3 , 0)
           2        (  , 0)                    2
                       2
           3                   ( , –3)
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
                       amplitude = |a|
     If |a| > 1, the amplitude stretches the graph vertically.
       If 0 < |a| < 1, the amplitude shrinks the graph vertically.
       If a < 0, the graph is reflected in the x-axis.
                          y
                             4


                y = 2sin x                   3
                                  2           2    2      x
                 1
            y=   2
                   sin x
             y = – 4 sin x                               y = sin x
reflection of y = 4 sin x                           y = 4 sin x
                         4
For y = A cos x and y = A sin x, A  is the amplitude.

What is the amplitude for the following?
     y = 4 cos x                      y = -3 sin x

   amplitude is 4                    amplitude is 3
The last thing we want to see is what happens if we
put a coefficient on the x.
                     y = sin 2x
                                      y = sin 2x




                                     y = sin x

It makes the graph "cycle" twice as fast. It does one
complete cycle in half the time so the period becomes .
The period of a function is the x interval needed for the
function to complete one cycle.
For b  0, the period of y = a sinbx is 2 .
                                         b
For b  0, the period of y = a cosbx is also 2 .
                                               b
If b > 1, the graph of the function is shrunk horizontally.
                         y
    y  sin 2 x                                    period: 2
  period:                                         y  sin x x

                                                 2

If 0 < b < 1, the graph of the function is stretched horizontally.
                        y                                y  cos x
             1
      y  cos x                                           period: 2
             2                    2    3     4      x
    period: 4


 For b < 0, what is the period of y = a sinbx and y = a cosbx ?
Use basic odd-even properties to the trigonometric function

        sin (–u) = – sinu                 cos(–u) = cosu

 The period of y = 3sin (–4x) is the same period as y = – 3sin4x.
                                      
 So the period of y = 3sin (–4x) is
                                      2
 The period of y = 2cos (–5x) is the same period as y = 2cos5x.
                                    2
 So the period of y = 2cos (–5x) is
                                     5
  So if we look at y = sinx the  affects the period.
                                               This will be
                                      2       true for
                     The period T =
                                              cosine as
                                               well.
                                      2 
What is the period of y = cos4x?   T    
                                       4   2
                                               y = cos x
 This means
 the graph will
 "cycle" every
 /2 or 4 times
 as often
                                                  y = cos 4x
What do you think will happen to the graph if we put a
fraction in front?          1
                     y  sin       x
                               2
                             y = sin 1/2 x




                                        y = sin x

The period for one complete cycle is twice as long or 4
     absolute value of this
       is the amplitude


y  A cost          y  A sin t

      Period is 2 divided by
      absolute value of this
        Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the
x-axis.                            y         y = sin (–x)
            Use the identity
            sin (–x) = – sin x                                         x
                        y = sin x                          2


Example 2: Sketch the graph of y = cos (–x).
   The graph of y = cos (–x) is identical to the graph of y = cos x.
                                  y
 Use the identity
  cos (–x) = cos x                                                x
                                                        2
                      y = cos (–x)
Example 3: Sketch the graph of y = 2sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
Use the identity sin (– x) = – sin x:  y = 2 sin (–3x) = –2 sin 3x
                                                       
  amplitude: |a| = |–2| = 2            period: 2 = 2
                                                 b      3
                                      Partition the [0, 2/3] into 4
Calculate the five key points.        subintervals.
                    x                                                2
                                     0         6          3     2        3

              y = –2 sin 3x          0        –2          0     2       0
                        y
                                             (  , 2)
                        2                     2
                                                      2   5
                6                6       3        2        3    6           x
               (0, 0)                    (  , 0) 2
                    2
                                           3     ( , 0)
                            (  , -2)                 3
                             6
 Practice 1. Give the amplitude and period of the function
 y  4sin 3x. Then sketch at least one cycle of its graph.

                                            2 2
   amplitude = 4  4              period =    
                                             B   3
Key points:
   x   sin3x        x    4sin 3x             x   4sin3x
  0    0            0     0                 0    0
                                          
       1            6     4                      4
  6                                         6
                                          
  3
       0            3     0
                                            3    0
                                          
   2   1           2     4
                                                 4
                                            2
 2                 2
        0                                   2
  3                       0
                     3                       3   0
   Practice 2. Give the amplitude, period, and an equation
   of the curve shown.




Use the cosine curve, y  A cos Bx, with amplitude 3 and period 8.
    2         2 
8     B        
     B          8    4
                                            
             So the equation is y  3cos        x.
                                            4
Practice 3. Solve the equation 6sin 2 x  5 for 0  x  2 .
Give the answer to the nearest hundredth of a radian
                                          5
         6sin 2x  5            sin 2 x 
                                          6
 2 x  .99, 2.16, 7.27, 8.44   If 0  x  2 , then 0  2 x  4
 x  0.49, 1.08, 3.64, 4.22
or graph y1  6sin 2 x and y2  5 and find the intersections over
the interval 0  x  2
Assignment


P. 305 #1 – 10, 11 – 16, 21 – 23

								
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