## Graph of Sine, Cosine and Tangent Functions

 6.3 Graph of Sine, Cosine and Tangent Functions

• The graphs of $$y = \text{sin } x$$ and $$y = \text{cos } x$$ are sinusoidal and have the following properties:

 (a) The maximum value is $$1$$ while the minimum value is $$-1$$, so the amplitude of the graph is $$1$$ unit. (b) The graph repeats itself every $$360^{\circ}\text{ or }2\pi \text{ rad}$$, so $$360^{\circ}\text{ or }2\pi \text{ rad}$$ is the period for both graphs.

• The graph $$y = \text{tan } x$$ is not sinusoidal and the properties are as follows:

 (a) This graph has no maximum or minimum value. (b) The graph repeats itself every $$180^{\circ}\text{ or }\pi \text{ rad}$$ interval, so the period of a tangent graph is $$180^{\circ}\text{ or }\pi \text{ rad}$$. (c) The function $$y = \text{tan } x$$ is not defined at $$x = 90^{\circ} \text{ and } x = 270^{\circ}$$. The curve approaches the line but does nottouch the line. This line is called an asymptote.

• The graphs for these three functions are as follows:

Graph $$y = \text{cos } x$$ for $$-2\pi \leqslant x \leqslant 2\pi$$

 (a) Amplitude $$=1$$ The maximum value of $$y=1$$ The minimum value of  $$y=-1$$ (b) Period $$=360^{\circ}\text{ or }2\pi$$ (c) $$x$$-intercepts: $$-\dfrac{3}{2}\pi, \ -\dfrac{1}{2}\pi, \ \dfrac{1}{2}\pi, \ \dfrac{3}{2}\pi$$ (d) $$y$$-intercepts: $$1$$

Graph $$y = \text{sin } x$$ for $$-2\pi \leqslant x \leqslant 2\pi$$

 (a) Amplitude $$=1$$ The maximum value of $$y=1$$ The minimum value of $$y=-1$$ (b) Period $$=360^{\circ}\text{ atau }2\pi$$ (c) $$x$$-intercepts:  $$-2\pi, \ -\pi, \ 0, \ \pi, \ 2\pi$$ (d) $$y$$-intercepts: $$0$$

Graph $$y = \text{tan } x$$ for $$-2\pi \leqslant x \leqslant 2\pi$$

 (a) No amplitude There are no maximum and minimum values of $$y$$ (b) Period $$=180^{\circ}\text{ or }\pi$$ (c) $$x$$-asymptotes: $$-\dfrac{3}{2}\pi, \ -\dfrac{1}{2}\pi, \ \dfrac{1}{2}\pi, \ \dfrac{3}{2}\pi$$ (d) $$x$$-intercepts: $$-2\pi, \ -\pi, \ 0, \ \pi, \ 2\pi$$ (e) $$y$$-intercepts: $$0$$

• The values of $$a$$, $$b$$ and $$c$$ in the function $$y = a \text{ sin } bx + c$$ affect the amplitude, the period and the position of the graph

• The effects of changing the values of $$a$$$$b$$ and $$c$$ on the graph can be summarised as follows:

Change in Effects
 $$a$$

 The maximum and minimum values of the graphs (except for the graph of $$y = \text{tan } x$$ where there is no maximum or minimum value)
 $$b$$

 Number of cycles in the range $$0^{\circ} \leqslant x \leqslant 360^{\circ} \text{ or }0^{\circ} \leqslant x \leqslant 2\pi$$: Graphs $$y = \text{sin } x$$ and $$y = \text{cos } x$$ $$\begin{pmatrix} \text{period } = \dfrac{360^{\circ}}{b} \text{ or } \dfrac{2}{b}\pi\end{pmatrix}$$ Graph $$y = \text{tan } x \ \begin{pmatrix} \text{period } = \dfrac{180^{\circ}}{b} \text{ or } \dfrac{1}{b}\pi\end{pmatrix}$$
 $$c$$
 The position of the graph with reference to the $$x$$-axis as compared to the position of the basic graph

Example:

 Example State the cosine function represented by the graph above. Solution: Note that the amplitude is $$4$$. So, $$a=4.$$ Two cycles in the range of $$0^{\circ} \leqslant x \leqslant 2\pi$$. The period is $$\pi$$, that is, $$\dfrac{2\pi}{b} = \pi, \text{ so }b=2.$$ Hence, the graph represents $$y= 4\text{ cos }2x$$.