Amplitude and Period of Sine and Cosine Functions

Chapter 42

Amplitude and Period of Sine and Cosine Functions Concept Explanation

The Amplitude of a sine or cosine function represents the maximum displacement from the equilibrium (midline) of the wave. It is the vertical stretch or compression of the graph. The amplitude is given by ( |A| ) in the equation $y = A \sin(Bx)$ or $y = A \cos(Bx)$ .

The Period refers to the distance (along the x-axis) over which the function completes one full cycle. For sine and cosine functions, the period is calculated using the formula:

$\text{Period} = \frac{2\pi}{|B|}$

where ( B ) is the coefficient of ( x ) in the function.

*** QuickLaTeX cannot compile formula:
\begin{tikzpicture}\begin{axis}[axis x line=middle,axis y line=middle,samples=100,xmin=-2<em>pi, xmax=2</em>pi,ymin=-2, ymax=2,xlabel={$x$}, ylabel={$y$},domain=-2<em>pi:2</em>pi,width=10cm,height=5cm]\addplot [blue, thick] {sin(deg(x))};\end{axis}\end{tikzpicture}

*** Error message:
Package PGF Math Error: Unknown function `em' (in '-2<em>pi').
leading text: ...n=-2<em>pi:2</em>pi,width=10cm,height=5cm]
Package PGF Math Error: Could not parse input '' as a floating point number, sorry. The unreadable part was near ''. (in '2</em>pi').
leading text: ...n=-2<em>pi:2</em>pi,width=10cm,height=5cm]
Package PGF Math Error: Unknown function `em' (in '2</em>pi').
leading text: ...n=-2<em>pi:2</em>pi,width=10cm,height=5cm]
Package PGF Math Error: Unknown function `em' (in '-2<em>pi').
leading text: ...=5cm]\addplot [blue, thick] {sin(deg(x))};
Package PGF Math Error: Could not parse input '' as a floating point number, sorry. The unreadable part was near ''. (in '2</em>pi').
leading text: ...=5cm]\addplot [blue, thick] {sin(deg(x))};
Package PGF Math Error: Unknown function `em' (in '2</em>pi').

In the graph above, the amplitude is 1, and the period is $2\pi$ .

Common Mistakes:

Confusing Amplitude with Period: Some students mistakenly associate the period with the vertical stretch.
Forgetting Absolute Value: When calculating the amplitude, always take the absolute value of ( A ).

Helpful Tips:

Use the Graph to Visualize: Plot the sine or cosine graph to clearly see the amplitude and period.
Remember the Formula: Use $\text{Period} = \frac{2\pi}{|B|}$ to calculate the period.

Hard Questions:

Q1: For the function $y = 3 \sin(2x)$ , find the amplitude and period.

Step-by-Step Solution:

The amplitude is $|A| = 3$ .
The period is:
$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi$

Answer:

Amplitude: 3
Period: $\pi$

Q2: For the function $y = -4 \cos\left(\frac{x}{3}\right)$ , determine the amplitude and period.

Step-by-Step Solution:

The amplitude is $|A| = 4$ (the negative sign affects the direction, but not the amplitude).
The period is:
$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{\frac{1}{3}} = 6\pi$