Problem 23 - Algebra 2

Question

Sketch one cycle of the graph of each sine function. $$ y=\sin 3 \theta $$

Step-by-Step Solution

Answer

The graph of \(y = \sin 3\theta\) completes one cycle between 0 and \(\frac{2\pi}{3}\). It starts from the origin, reaches its peak at \(\frac{\pi}{6}\), crosses the midline at \(\frac{2\pi}{6}\), reaches its valley at \(\frac{3\pi}{6}\), and returns to the midline at \(\frac{4\pi}{6}\) that is equal to \(\frac{2\pi}{3}\).

Step 1: Identify the Period of the Function

The usual period of the sine function is \(2\pi\). As our function has a coefficient of 3 for \(\theta\), this will cause the function to cycle more quickly. The period of our function \(y = sin 3\theta\) becomes \(\frac{2\pi}{3}\).

Step 2: Identify the Amplitude, Midline, Phase Shift

In our function \(y = sin 3\theta\), there is no phase shift or vertical translation, hence the midline is y=0. The function is not preceded by a negative sign, so it's not reflected. Also, there's no coefficient other than 3 affecting the function, so the amplitude (the peak value of sinusoid) is 1.

Step 3: Start Sketching the Graph

Start at the origin, then since the function is positive, the next point should be a crest (peak) of the wave. This occurs at \(\frac{\pi}{6}\) since the period is \(\frac{2\pi}{3}\). The wave then returns to the midline at \(\frac{2\pi}{6}\), goes to a trough (valley) at \(\frac{3\pi}{6}\), and finally returns to the midline at the end of its period, which is \(\frac{4\pi}{6}\) or \(\frac{2\pi}{3}\).

Step 4: Finish Sketching the Graph

Continue sketching the sine curve between the points you have identified. Remember that the sine function is smooth and oscillates between the top and bottom of the wave smoothly.

Key Concepts

Period of a Sine Function

The period of a sine function is the length it takes for the function to complete one full cycle of its wave. In its simplest form, the sine function, represented as \( y = \sin \theta \), has a period of \(2\pi\). This means that it takes \(2\pi\) radians for the sine function to repeat its values. If there is a coefficient, say \( a \), multiplying the variable \( \theta \), the period is modified appearing as \( \frac{2\pi}{a} \). For instance, in the function \( y = \sin 3\theta \), the coefficient 3 means the period becomes \( \frac{2\pi}{3} \). This is because the wave compresses, repeating itself more frequently and within a smaller interval along the \( x \)-axis.
Understanding how to find the period helps in sketching sine functions as it signals when the graph should start repeating itself, marking the end of one complete cycle. This compression or expansion of the wave due to the coefficient is crucial in plotting the graph correctly.

Amplitude of Trigonometric Functions

The amplitude of a sine function refers to the peak height of the wave from its central axis or midline. The standard amplitude of \( y = \sin \theta \) is 1, indicating the wave oscillates 1 unit above and below the midline, which is usually the \( x \)-axis at \( y = 0 \).
For any function of the form \( y = a \sin \theta \), the amplitude is the absolute value of \( a \). This value tells us how high and how low the wave will go from its central position. In our specific function \( y = \sin 3\theta \), the amplitude remains 1 because there is no coefficient other than 3 affecting \( \sin \theta \), and it acts only on the period, not the amplitude. Thus, the sine wave will peak at 1 and dip to -1, staying symmetric around the midline. Grasping the amplitude is crucial for accurately sketching the breadth of oscillation of trigonometric graphs.

Sketching Trigonometric Graphs

Sketching trigonometric graphs involves plotting the wave pattern effectively using insights from its key components: period and amplitude. Start by identifying these characteristics. The function \( y = \sin 3\theta \) should illustrate these well.
1. **Starting Point**: Begin sketching at the origin (0, 0) because there's no horizontal or vertical shift in \( \theta \).
2. **Crest, Troughs, and Midline**: Since the function is positive, the first crest will appear at \( \frac{\pi}{6} \). The wave returns to the midline at \( \frac{\pi}{3} \), hits a trough at \( \frac{\pi}{2} \), and completes its cycle at \( \frac{2\pi}{3} \).
3. **Smooth Oscillation**: The sine wave should always have a smooth and continuous curve, oscillating seamlessly between crests and troughs. Each peak and valley should reflect the calculated amplitude, ensuring the wave hits its peak at 1 and its trough at -1.
By following these steps, you can accurately depict the characteristic sinusoidal shape of trigonometric functions, appreciating how changes in formula components affect the graph.