Problem 12
Question
Identify the period and tell where two asymptotes occur for each function. $$ y=\tan \frac{3 \theta}{2} $$
Step-by-Step Solution
Verified Answer
The period of the function \(y=tan(\frac{3θ}{2})\) is \(\frac{2\pi}{3}\). Two asymptotes can be found at \(-\frac{2\pi}{3}\) and \(\frac{2\pi}{3}\), and two additional ones at \(-\frac{4\pi}{3}\) and \(\frac{4\pi}{3}\).
1Step 1: Identify the form of the function
Firstly, understand that the given function is in the form of \(y=tan(cθ)\), where \(c\) is the coefficient of \(\theta\). The standard period of the tangent function, denoted as \(y=tan(\theta)\), is \(\pi\). So, here our function is \(y=tan(\frac{3θ}{2})\), meaning \(c=\frac{3}{2}\).
2Step 2: Calculate the period
To find the period of the function, we divide the standard period \(\pi\) by the absolute value of the coefficient of \(\theta\). In our case, the period of the function is, \(P=\frac{\pi}{\left|\frac{3}{2}\right|}=\frac{2}{3}\pi\).
3Step 3: Identify the asymptotes
The asymptotes of a tangent function are found at the extrema within each period. In a standard tangent function, they are located at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). The coefficient of the function stretches the period, hence asymptotes would occur sparsely or densely. Here, with \(c=\frac{3}{2}\), the initial asymptotes locate at \(-\frac{\pi}{2c}=-\frac{2\pi}{3}\) and \(\frac{\pi}{2c}=\frac{2\pi}{3}\). By moving by one period on either side, two more asymptotes can be obtained at \(-\frac{2\pi}{3}-\frac{2\pi}{3}=-\frac{4\pi}{3}\) and \(\frac{2\pi}{3}+\frac{2\pi}{3}=\frac{4\pi}{3}\).
Key Concepts
Period of a functionAsymptotesTangent function
Period of a function
When we speak about the period of a function, we are referring to how long it takes for the function to repeat its values. Specifically, for trigonometric functions like the tangent, the period is the length over which the function starts repeating its cycle.
In a standard tangent function, which is denoted by \(y = \tan(\theta)\), the period is \(\pi\). This means that every \(\pi\) units along the x-axis, \(\tan(\theta)\) begins anew, producing the same set of y-values. However, when modifications are made to the function, such as multiplying \(\theta\) by a coefficient, the period changes.
For example, in the function \(y = \tan\left(\frac{3\theta}{2}\right)\), the coefficient \(\frac{3}{2}\) is applied to \(\theta\). The effect of this is to compress the x-axis. To find the new period, we divide the original period \(\pi\) by the absolute value of the coefficient: \(P = \frac{\pi}{\left|\frac{3}{2}\right|} = \frac{2\pi}{3}\). Hence, the function \(\tan\left(\frac{3\theta}{2}\right)\) repeats its cycle every \(\frac{2\pi}{3}\).
**Key Points**:
In a standard tangent function, which is denoted by \(y = \tan(\theta)\), the period is \(\pi\). This means that every \(\pi\) units along the x-axis, \(\tan(\theta)\) begins anew, producing the same set of y-values. However, when modifications are made to the function, such as multiplying \(\theta\) by a coefficient, the period changes.
For example, in the function \(y = \tan\left(\frac{3\theta}{2}\right)\), the coefficient \(\frac{3}{2}\) is applied to \(\theta\). The effect of this is to compress the x-axis. To find the new period, we divide the original period \(\pi\) by the absolute value of the coefficient: \(P = \frac{\pi}{\left|\frac{3}{2}\right|} = \frac{2\pi}{3}\). Hence, the function \(\tan\left(\frac{3\theta}{2}\right)\) repeats its cycle every \(\frac{2\pi}{3}\).
**Key Points**:
- The period is the distance over which the function's behavior repeats.
- For \(\tan(\theta)\), the default period is \(\pi\).
- To find a new period in \(y = \tan(c\theta)\), calculate \(\frac{\pi}{|c|}\).
Asymptotes
Asymptotes are important in understanding the behavior of functions as they approach certain x-values. In trigonometric functions, asymptotes denote where the function heads towards infinity, and never quite reaches a definite value.
For the basic tangent function \(y = \tan(\theta)\), vertical asymptotes occur at values of \(\theta\) where the function is undefined. These typically hold true at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), repeating every \(\pi\). The interesting part comes in when a constant is involved, stretching or compressing the period. This modifies the asymptote locations.
In \(y = \tan\left(\frac{3\theta}{2}\right)\), the positions of the asymptotes depend on both the period and the coefficient of \(\theta\). Initially, these are at \(-\frac{2\pi}{3}\) and \(\frac{2\pi}{3}\). Moving by one period in either direction provides additional points: \(-\frac{4\pi}{3}\) and \(\frac{4\pi}{3}\).
**In Summary**:
For the basic tangent function \(y = \tan(\theta)\), vertical asymptotes occur at values of \(\theta\) where the function is undefined. These typically hold true at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), repeating every \(\pi\). The interesting part comes in when a constant is involved, stretching or compressing the period. This modifies the asymptote locations.
In \(y = \tan\left(\frac{3\theta}{2}\right)\), the positions of the asymptotes depend on both the period and the coefficient of \(\theta\). Initially, these are at \(-\frac{2\pi}{3}\) and \(\frac{2\pi}{3}\). Moving by one period in either direction provides additional points: \(-\frac{4\pi}{3}\) and \(\frac{4\pi}{3}\).
**In Summary**:
- Asymptotes for tangent indicate undefined values.
- In standard form, they occur at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
- New asymptotes are identified by updating these based on any changes in the period.
Tangent function
The tangent function is one of the primary functions in trigonometry and is central to understanding angles and lengths in right-angle triangles. It is defined as the ratio of the opposite side to the adjacent side in a triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
This function is unique due to its periodicity and the presence of vertical asymptotes. The tangent function displays regular patterns repeating every \(\pi\), unlike sine and cosine which have periods of \(2\pi\). Its graph is continuous, but at intervals—specifically at \(\theta = (2n+1)\frac{\pi}{2} \) for integer \(n\)—it heads towards infinity, creating vertical asymptotes.
Enhancing its complexity, when the angle \(\theta\) is multiplied by a coefficient such as \(\frac{3}{2}\), the tangent function alters both in period and position of asymptotes. These coefficients allow the tangent curve to fit into different scales or situations efficiently.
**Remember**:
This function is unique due to its periodicity and the presence of vertical asymptotes. The tangent function displays regular patterns repeating every \(\pi\), unlike sine and cosine which have periods of \(2\pi\). Its graph is continuous, but at intervals—specifically at \(\theta = (2n+1)\frac{\pi}{2} \) for integer \(n\)—it heads towards infinity, creating vertical asymptotes.
Enhancing its complexity, when the angle \(\theta\) is multiplied by a coefficient such as \(\frac{3}{2}\), the tangent function alters both in period and position of asymptotes. These coefficients allow the tangent curve to fit into different scales or situations efficiently.
**Remember**:
- Tangent is a ratio relating to triangle sides.
- It repeats every \(\pi\), shown as its period \(\pi\).
- Modifying \(\theta\) with a coefficient alters the graph's period and asymptotes.
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