Graph of Sine, Cosine and Tangent Functions

6.3

Graph of Sine, Cosine and Tangent Functions

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Graphs of Trigonometric Functions

$y=\sin{x}$ and $y=\cos{x}$

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

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

$y=\tan{x}$

The graph $y=\tan{x}$ is not sinusoidal. The properties of $y=\tan{x}$ are as follows:

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

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

Figure

1. A graph depicting the sine function, y=sin x, showcasing a wave pattern from -2π to 2π with numerical values indicated.

Properties

Amplitude $=1$.
The maximum value of $y=1$.
The minimum value of $y=-1$.
Period $=360^\circ$ or $2\pi$.
$x$-intercepts: $-2\pi$, $-\pi$, $0$, $\pi$, $2\pi$.
$y$-intercepts: $0$.

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

Figure

A graph displaying the waveform of y=cos x, featuring multiple points from -2π to 2π on the x-axis.

Properties

Amplitude $=1$.
The maximum value of $y=1$.
The minimum value of $y=-1$.
Period $=360^\circ$ or $2\pi$.
$x$-intercepts: $-\dfrac{3}{2}\pi$, $-\dfrac{1}{2}\pi$, $\dfrac{1}{2}\pi$, $\dfrac{3}{2}\pi$.
$y$-intercepts: $1$.

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

Figure

A graph displaying the line and curve of y=tan x, illustrating its behavior from -2π to 2π.

Properties

No amplitude.
There is no maximum value of $y$.
There is no minimum value of $y$.
Period $=180^\circ$ or $\pi$.
$x$-asymptotes: $-\dfrac{3}{2}\pi$, $-\dfrac{1}{2}\pi$, $\dfrac{1}{2}\pi$, $\dfrac{3}{2}\pi$.
$x$-intercepts: $-2\pi$, $-\pi$, $0$, $\pi$, $2\pi$.
$y$-intercepts: $0$.

Effect of Values $a$, $b$ and $c$ in the Function $y=a\sin{bx}+c$

The values of $a$, $b$ and $c$ in the function $y=a\sin{bx}+c$ 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 $y=\tan{x}$ where there is no maximum or minimum value).
$b$	Number of cycles in the range $0^\circ \leq x \leq 360^\circ$ or $0 \leq x \leq 2\pi$: Graphs $y=\sin{x}$ and $y=\cos{x}$ (period $=\dfrac{360^\circ}{b}$ or $\dfrac{2}{b}\pi$). Graph $y=\tan{x}$ (period $=\dfrac{180^\circ}{b}$ or $\dfrac{1}{b}\pi$).
$c$	The position of the graph with reference to the $x$-axis as compared to the position of the basic graph.

Example

Question

Graph depicting the wave function y=4cos(2x) with a red line and a blue line, spanning from -π to 2π.

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, $\dfrac{2\pi}{b} = \pi$, so $b=2$.

Hence, the graph represents $y= 4\text{ cos }2x$.