3 Graphs of trigonometric functions
3.1 Graphs of sin and cos
Since we have defined both and in terms of the projections of the radius vector of a circle of unit radius it follows immediately that
for any value of .
We have discussed the behaviour of and in each of the four quadrants in the previous subsection.
Using all the above results we can draw the graphs of these two trigonometric functions. See Figure 29. We have labelled the horizontal axis using radians and have shown two periods in each case.
Figure 29
We have extended the graphs to negative values of using the relations Both graphs could be extended indefinitely to the left ( ) and right ( ).
Task!
(a) Using the graphs in Figure 29 and the fact that calculate
the values of .
(b) For what values of is undefined?
(c) State whether is positive or negative in each of the four quadrants.
(a)
(b)
is not be defined when i.e. when
(c)
3.2 The graph of
The graph of against , for is then as in Figure 30. Note that whereas and have period , has period .
Figure 30
Task!
On the following diagram showing the four quadrants mark which trigonometric quantities are positive in the four quadrants. One entry has been made already.