3 Graphs of trigonometric functions

3.1 Graphs of sin  θ   and cos  θ

Since we have defined both sin θ and cos θ in terms of the projections of the radius vector O P of a circle of unit radius it follows immediately that

1 sin θ + 1 and 1 cos θ + 1 for any value of θ .

We have discussed the behaviour of sin θ and cos θ 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

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We have extended the graphs to negative values of θ using the relations sin ( θ ) = sin θ , cos ( θ ) = cos θ . Both graphs could be extended indefinitely to the left ( θ ) and right ( θ + ).

Task!

(a) Using the graphs in Figure 29 and the fact that tan θ sin θ cos θ calculate

  the values of tan 0 , tan π , tan 2 π .

(b) For what values of θ is tan θ undefined?

(c) State whether tan θ is positive or negative in each of the four quadrants.

(a)

tan 0 = sin 0 cos 0 = 0 1 = 0

tan π = sin π cos π = 0 1 = 0

tan 2 π = sin 2 π cos 2 π = 0 1 = 0

(b)

tan θ is not be defined when cos θ = 0 i.e. when θ = ± π 2 , ± 3 π 2 , ± 5 π 2 ,

(c)

1st quadrant: tan θ = sin θ cos θ = + ve + ve = + ve

2nd quadrant: tan θ = sin θ cos θ = + ve ve = ve

3rd quadrant: tan θ = sin θ cos θ = ve ve = + ve

4th quadrant: tan θ = sin θ cos θ = ve + ve = ve

3.2 The graph of tan θ

The graph of tan θ against θ , for 2 π θ 2 π is then as in Figure 30. Note that whereas sin θ and cos θ have period 2 π , tan θ has period π .

Figure 30

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Task!

On the following diagram showing the four quadrants mark which trigonometric quantities cos , sin , tan , are positive in the four quadrants. One entry has been made already.

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