Graph of Sine, Cosine and Tangent Functions

6.3 Graph of Sine, Cosine and Tangent Functions
 
The image shows a diagram with the title 'Trigonometric Functions' in a dark blue box. Below the title, there are three arrows pointing to three separate light blue boxes. Each box contains a different trigonometric function: 1. \( y = a \sin bx + c \) 2. \( y = a \cos bx + c \) 3. \( y = a \tan bx + c \) At the top of the image, there is a logo with the text 'Pandai'.
 
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 \(y= 4\text{ cos }2x\).

 

Graph of Sine, Cosine and Tangent Functions

6.3 Graph of Sine, Cosine and Tangent Functions
 
The image shows a diagram with the title 'Trigonometric Functions' in a dark blue box. Below the title, there are three arrows pointing to three separate light blue boxes. Each box contains a different trigonometric function: 1. \( y = a \sin bx + c \) 2. \( y = a \cos bx + c \) 3. \( y = a \tan bx + c \) At the top of the image, there is a logo with the text 'Pandai'.
 
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 \(y= 4\text{ cos }2x\).