Graphs of trigonometric functions

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

$- 1 \leq sin θ \leq + 1 and - 1 \leq cos θ \leq + 1$ 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 ( $θ \to - \infty$ ) and right ( $θ \to + \infty$ ).

Task!

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

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

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

(a)

$tan 0 = \frac{sin 0}{cos 0} = \frac{0}{1} = 0$

$tan π = \frac{sin π}{cos π} = \frac{0}{- 1} = 0$

$tan 2 π = \frac{sin 2 π}{cos 2 π} = \frac{0}{1} = 0$

(b)

$tan θ$ is not be defined when $cos θ = 0$ i.e. when $θ = \pm \frac{π}{2}, \pm \frac{3 π}{2}, \pm \frac{5 π}{2}, \dots$

(c)

$1st quadrant: tan θ = \frac{sin θ}{cos θ} = \frac{+ ve}{+ ve} = + ve$

$2nd quadrant: tan θ = \frac{sin θ}{cos θ} = \frac{+ ve}{- ve} = - ve$

$3rd quadrant: tan θ = \frac{sin θ}{cos θ} = \frac{- ve}{- ve} = + ve$

$4th quadrant: tan θ = \frac{sin θ}{cos θ} = \frac{- ve}{+ ve} = - ve$

3.2 The graph of $tan θ$

The graph of $tan θ$ against $θ$ , for $- 2 π \leq θ \leq 2 π$ 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.

3 Graphs of trigonometric functions

3.1 Graphs of sin θ and cos θ

Task!

Answer

3.2 The graph of tan θ

Task!

Answer

3.1 Graphs of sin $θ$ and cos $θ$

3.2 The graph of $tan θ$