Question

State the period for each periodic function, in degrees and in radians. Sketch the graph of each function. a) \(y=\sin 4 \theta\) b) \(y=\cos \frac{1}{3} \theta\) c) \(y=\sin \frac{2}{3} x\) d) \(y=\cos 6 x\)

Short answer

a) 90° or \(\frac{\pi}{2}\) radians, b) 1080° or 6\(\pi\) radians, c) 540° or 3\(\pi\) radians, d) 60° or \(\frac{\pi}{3}\) radians

Step-by-step solution

Determine the Period Formula for Sine and Cosine Functions

The period for the basic sine and cosine functions, \(y = \sin \theta\) and \(y = \cos \theta\), is \(360^{\circ}\) or \(2\pi\) radians. When the function is modified by a coefficient, the period changes. The general formula for the period of \(y = \sin (k\theta)\) or \(y = \cos (k\theta)\) is \(\frac{360^{\circ}}{k}\) in degrees or \(\frac{2\pi}{k}\) in radians.

Calculate the Period for Function a

Given \(y = \sin 4\theta\), the coefficient \(k = 4\). Using the formula: \[\text{Period} = \frac{360^{\circ}}{4} = 90^{\circ}\] \[\text{Period} = \frac{2\pi}{4} = \frac{\pi}{2} \text{radians}\]

Calculate the Period for Function b

Given \(y = \cos \frac{1}{3}\theta\), the coefficient \(k = \frac{1}{3}\). Using the formula: \[\text{Period} = \frac{360^{\circ}}{\frac{1}{3}} = 1080^{\circ}\] \[\text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi \text{radians}\]

Calculate the Period for Function c

Given \(y = \sin \frac{2}{3}x\), the coefficient \(k = \frac{2}{3}\). Using the formula: \[\text{Period} = \frac{360^{\circ}}{\frac{2}{3}} = 540^{\circ}\] \[\text{Period} = \frac{2\pi}{\frac{2}{3}} = 3\pi \text{radians}\]

Calculate the Period for Function d

Given \(y = \cos 6x\), the coefficient \(k = 6\). Using the formula: \[\text{Period} = \frac{360^{\circ}}{6} = 60^{\circ}\] \[\text{Period} = \frac{2\pi}{6} = \frac{\pi}{3} \text{radians}\]

Sketch Graphs of Each Function

For each function: (a) Sketch one complete cycle from 0 to the calculated period for \(y = \sin 4\theta\). (b) Sketch one complete cycle from 0 to the calculated period for \(y = \cos \frac{1}{3}\theta\). (c) Sketch one complete cycle from 0 to the calculated period for \(y = \sin \frac{2}{3}x\). (d) Sketch one complete cycle from 0 to the calculated period for \(y = \cos 6x\).

Key concepts

sine function period

In trigonometry, the sine function, represented as \( \sin \theta \), is a periodic function. A periodic function repeats its values in regular intervals or periods. The period of the basic sine function \( y = \sin \theta \) is \( 360^{\circ} \) or \( 2\pi \) radians. When the sine function is modified by a coefficient like \( k \) in \( y = \sin (k\theta) \), the period is recalculated using the formula: \(\text{Period} = \frac{360^{\circ}}{k}\) in degrees or \(\text{Period} = \frac{2\pi}{k}\) in radians. For example, in the function \( y = \sin 4\theta \), the coefficient \( k = 4 \). Applying our formula we get: \(\text{Period} = \frac{360^{\circ}}{4} = 90^{\circ}\) in degrees and \(\frac{2\pi}{4} = \frac{\pi}{2} \) in radians. This means the function repeats every \( 90^{\circ} \) or \( \frac{\pi}{2} \) radians.

cosine function period

Like the sine function, the cosine function \( \cos \theta \) is also periodic. The basic cosine function \( y = \cos \theta \) has a period of \( 360^{\circ} \) or \( 2\pi \) radians. When the cosine function includes a coefficient \( k \), such as in \( y = \cos (k\theta) \), its period changes based on the same formulas used for the sine function: \(\text{Period} = \frac{360^{\circ}}{k}\) in degrees or \(\text{Period} = \frac{2\pi}{k}\) in radians. In the example \( y = \cos \frac{1}{3}\theta \), the coefficient \( k = \frac{1}{3} \), which changes the period to: \(\text{Period} = \frac{360^{\circ}}{\frac{1}{3}} = 1080^{\circ}\) in degrees and \(\frac{2\pi}{\frac{1}{3}} = 6\pi \) in radians. This indicates that the function's cycle repeats every \( 1080^{\circ} \) or \( 6\pi \) radians.

graphing periodic functions

Graphing periodic functions involves plotting their values over one or more periods. For trigonometric functions like sine and cosine, graphs typically show oscillatory behavior. Let's consider the function \( y = \sin 4\theta \), which has a period of \( 90^{\circ} \) or \( \frac{\pi}{2} \) radians. To graph this function:

• Start from \( 0 \) and plot points for \( \theta \) values within one period, up to \( 90^{\circ} \) or \( \frac{\pi}{2} \) radians.


• The sine function starts at \( 0 \), reaches a maximum at \( \frac{\pi}{4} \), returns to \( 0 \) at \( \frac{\pi}{2} \), and continues to oscillate.

Similarly, for the function \( y = \cos \frac{1}{3}\theta \), having a period of \( 1080^{\circ} \) or \( 6\pi \) radians, plot points from \( 0 \) to beyond a complete cycle to see the cosine curve oscillate. Remember, in each cycle both functions repeat their distinctive, wave-like patterns.

period calculation

Calculating the period of a sine or cosine function involves identifying and applying the coefficient \( k \). If you have a function \( y = \sin (k\theta) \) or \( y = \cos (k\theta) \), here’s what you do:

• Extract the coefficient \( k \).


• Apply the formula for the period in degrees: \(\text{Period} = \frac{360^{\circ}}{k}\).


• For radians, use: \(\text{Period} = \frac{2\pi}{k}\).

For instance, consider \( y = \sin \frac{2}{3}x \). Here \( k = \frac{2}{3} \). Using the formula, we get: \(\text{Period} = \frac{360^{\circ}}{\frac{2}{3}} = 540^{\circ}\) in degrees and \(\frac{2\pi}{\frac{2}{3}} = 3\pi\) in radians. The function repeats every \( 540^{\circ} \) or \( 3\pi \) radians, respectively. This demonstrates the importance of correctly identifying \( k \) and applying the period formulas.

radians and degrees

Radians and degrees are two units to measure angles. Degrees are more commonly used in everyday settings, while radians are often used in advanced mathematics and physics.

• One complete revolution around a circle is \( 360^{\circ} \).


• In radians, the same revolution is \( 2\pi \).

Converting between them is simple: \(\text{Degrees} = \frac{180}{\pi} \times \text{Radians}\) and \(\text{Radians} = \frac{\pi}{180} \times \text{Degrees}\). For example, \( 180^{\circ} \) is \( \pi \) radians because \(\text{Radians} = \frac{\pi}{180} \times 180 = \pi\). When dealing with periodic functions, understanding both units allows you to plot graphs and find periods accurately. For instance, \( y = \cos 6x \) has \( k = 6 \). The period in degrees: \(\text{Period} = \frac{360^{\circ}}{6} = 60^{\circ}\), and in radians: \(\frac{2\pi}{6} = \frac{\pi}{3}\). Thus, the function repeats every \( 60^{\circ} \) or \( \frac{\pi}{3} \) radians.