7-4 Evaluating and Graphing Sine and Cosine

  Sines and Cosines of Special Angles
    30o, 45o and 60o angles are used many times in mathematics.  I strongly urge you to memorize, or at least be able to derive the sine and cosine of these special angles.
        In a 30-60-90 triangle, the sides are in ratio of 1  :2
        Look at the triangle below:
 
        Sin 30o = y/r  = 1/2, while the Cos 30o = x/r = 
     Sin 60o = y/r = , while the Cos 60o = x/r = 1/2
 
        In a 45-45-90 triangle, the sides are in ratio of 1 : 1
        Study the triangle below:
 
        Sin 45o = x/r = ,  while Cos 45o

The wise person will memorize the following chart:
 
      Degrees        radians        Sin q        Cos q
0 0 0 1
30 p/6 1/2
45 p/4
60 p/3 1/2
90 p/2 1 0

The graph of Sine and Cosine Functions
y = Sin x
Demonstration of Sine Graph (Manipula Math)
Notice that this graph is a periodic graph.  It repeats the same graph every 2punits.  It is increasing from 0 to half pi, decreasing from half pi to negative 1.5 pi and increasing to 2 pi.  Then the repeat starts.  This matches what happens to the Sine function in the quadrants.  Positive in first and second and negative in the third and fourth.  Maximum value for the graph is 1 and the minimum value is -1.

y = Cos x
Demonstration of Cosine Graph (Manipula Math)
 
This graph is similar to the previous shape.  It is also a periodic graph with the cycle being 2p.  It also matches the signs of the quadrants with quad one being positive, quads two and three, negative and quad 4 back to positive.  The difference in these two graphs is the starting point for the Cosine graph.  It starts at the maximum value.  The Sine curve started at  the origin point.

An easy way to remember these graphs is to know their 5 important points.  The zeros, maximum and minimum points.
The Sine curve has zeros at the beginning, middle and end of a cycle.  The maximum happens at the 1/4 mark and the minimum appears at the 3/4 mark.
The Cosine curve begins and ends with the maximum.  It has a minimum at the middle point.  Zeros appear at the 1/4 and 3/4 mark of the cycle.

 Reference Angles
All angles can be referenced back to an angle in the first quadrant.  This is true because the trig functions are periodic.  Study each of the quadrant formulas below to find the reference angles.
To find the reference angle a, simply use the chart above to locate the angle q.
Example:  If  q =120,then you are in quadrant II.  Thus, use the formula 180 - 120 to get a reference angle of 60.
Example:  If  q = 195, then you are in quadrant III.  Thus, use the formula 195 - 180 to get a reference angle of 15.
Example:  If  q = 300, then you are in quadrant IV.  Thus, use the formula 360 - 300 to get a reference angle of 60.

    Relating this idea of reference angles and Sine and Cosine is easy.  Determine the reference angle as we did above and put the correct sign on each function.  From previous sections the Sine function is positive in quadrants I and II and negative in quadrants III and IV.  The Cosine function is positive in quadrants I and IV, while negative in quadrants II and III.
Examples
Sin 135o = Sin ( 180o - 135o) = Sin 45o
Cos 310o = Cos (360o - 310o) = Cos 50o
Sin 210o = Sin (210o - 180o) = - Sin 30o (Sin is negative in third quad)
Cos 112o = Cos (180o - 112o) = - Cos 68o (Cos is negative in 2nd quad)

Using your TI-82 to evaluate Sine and Cosine
1)  To calculate in degrees:
            Procedure 1
            a)  Press mode button and highlight degrees.  Press enter.
            b)  Type in problem as example:  Sin 45.  Press enter.
            c)  Answer:  .7071067812
    Note:  Most trig answers are round to 4 decimal places.  You can set your calculator to fixed mode by:  press mode.  Press down arrow and highlight 4.  Press enter.  Press clear.   Now enter the problem:  Sin 45
Answer is now .7071
 
            Procedure 2.  (Use the degree button)
            a)  Enter problem:  Sin 45
            b)Press 2nd function.  Press "angle" button.  You get a menu with the degree symbol as choice 1.  Either press 1 or enter.
            c)  Your display should now look like this:  Sin 45o Press enter.
Answer is .7071 if you are still in fixed mode!!

    Note:  Procedure 2 works regardless of the mode you are in.
 

2)  To calculate in radian measure.
 
            Procedure 1
            a)  Press mode button.  Press down arrow twice.  Highlight radian.  Press enter.   You are now in rads.  All problems will now be calculated in rads.
            b)  Enter a problem.  Example Cos 5.  Press enter.
Answer (rounded to 4 decimal places) is .2837
 
            Procedure 2.  (Using the rad button)
            a)  Enter a problem such as  Sin -2     Press 2nd function.  Press angle button.  Press down twice or 3.
            b)  Your display should look like:  sin -2r   Press enter.
Answer is -.9093
 
Note:  The use of the radian button will override the setting in mode.  Just like the degree button does.


We should be ready to take a look at the other trig functions!!
 
Take me back for some brush-up work!!