Problem 64 - Algebra 2

Question

Sketch one cycle of the graph of each sine function. $$ y=\sin 4 \theta $$

Step-by-Step Solution

Answer

The graph of the function \(y = \sin 4\theta\) shows a periodic wave between -1 and 1, starting from zero, reaching a maximum, crossing zero again, reaching a minimum, and returning back to zero. The period is \(\frac{\pi}{2}\).

Step 1

Define the standard characteristics of the sine function: period, amplitude, frequency, maximum, minimum. For the given function \(y = \sin 4\theta\), the period \(T = \frac{2\pi}{4} = \frac{\pi}{2}\), amplitude \(A = 1\), frequency \(f = \frac{1}{T} = 2\), max value \(V_{max} = 1\), and min value \(V_{min} = -1\).

Step 2

Find the four critical points that appear in one period of any sine function: when it starts, reaches its maximum, crosses zero again, reaches its minimum, and returns to zero. Based on the period calculated in the previous step, divide \(\frac{\pi}{2}\) by 4 to calculate the position of these points. Here are the results: \(\theta = 0\), \(\theta = \frac{\pi}{8}\), \(\theta = \frac{\pi}{4}\), \(\theta = \frac{3\pi}{8}\), \(\theta = \frac{\pi}{2}\), which correspond to \(y = 0\), \(y = 1\), \(y = 0\), \(y = -1\), \(y = 0\), respectively.

Step 3

Plotting the values. Since the function is periodic, only one cycle needs to be plotted. Plot the values of \(\theta\) on the x-axis and \(y\) on the y-axis. Start at the point (0,0), which represents the first critical point. Proceed to the next critical point, where \(\theta = \frac{\pi}{8}\) and \(y = 1\). Proceed similarly for the next critical points: \((\frac{\pi}{4}, 0)\), \((\frac{3\pi}{8}, -1)\), and \((\frac{\pi}{2}, 0)\). Connect the points with a smooth curve, ensuring that it looks like a wave with a peak and a trough. This completes one cycle of the function. Repeat this pattern to complete the graph.

Key Concepts

Period of Sine Function

The period of a sine function is an essential characteristic that tells us how long it takes the function to complete one full cycle. In a regular sine function, like \(y = \sin \theta\), the period is \(2\pi\).
This means that the sine wave repeats itself after \(2\pi\) radians.For the function \(y = \sin 4\theta\), the coefficient \(4\) affects the period. The formula to find the period is \(T = \frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(\theta\). So, we calculate:

• \(T = \frac{2\pi}{4}\)
• \(T = \frac{\pi}{2}\)

This means the sine wave completes a cycle every \(\frac{\pi}{2}\) radians, making it four times faster than the standard sine wave.

Amplitude of Sine Function

The amplitude of a sine function indicates its height from the centerline to a peak. In simpler terms, the amplitude is the wave's maximum displacement from its equilibrium position.For the function \(y = \sin 4\theta\), the amplitude remains \(1\) because there is no coefficient directly multiplying the \(\sin\) function. This tells us the wave reaches a maximum height of \(1\) and a minimum of \(-1\).
In general:

• If \(y = A \sin b\theta\), the amplitude is \(|A|\).
• The larger the amplitude, the taller the wave.

Understanding amplitude helps in sketching the graph, ensuring the peaks and troughs are correct.

Frequency of Sine Function

Frequency is the number of cycles a sine wave completes in a unit interval, usually interpreted as cycles per unit of \(\theta\). The frequency \(f\) is connected to the period \(T\) using the formula \(f = \frac{1}{T}\).For \(y = \sin 4\theta\), we've already calculated the period to be \(\frac{\pi}{2}\). Now, let's find the frequency:

• \(f = \frac{1}{\frac{\pi}{2}}\)
• \(f = 2\) (in radians per unit interval)

This means the function completes 2 full cycles between 0 and \(2\pi\), making it quicker than the standard \(y = \sin \theta\) function.

Graphing Trigonometric Functions

Graphing trigonometric functions may seem challenging, but it's straightforward once we know their key features like period, amplitude, and critical points.To graph \(y = \sin 4\theta\):

• Begin at the origin \((0,0)\).
• Mark critical points: maximum \((\frac{\pi}{8}, 1)\), zero \((\frac{\pi}{4}, 0)\), minimum \((\frac{3\pi}{8}, -1)\), and end \((\frac{\pi}{2}, 0)\).
• These points help you outline the basic "wave" shape.

Connect these using a smooth curve. Ensure the curve captures the wave-like nature. Visualizing it like a series of hills (peaks) and valleys (troughs) is helpful.Repeat the process if a longer graph is needed. Recognizing these patterns will ease plotting other trigonometric functions.

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, are fundamental in mathematics and are part of the foundations of trigonometry. They model periodic phenomena such as oscillations and waves. Characteristics of Sine Function:

• It is periodic, consistently repeating its pattern.
• The amplitude affects the wave's height.
• The period determines the wave's length.
• Frequency indicates how fast the wave oscillates.

Applications of sine functions span different fields, from physics to engineering, showcasing their wide utility in describing real-world periodic events.