Problem 43
Question
Find the period and sketch the graph of the equation. Show the asymptotes. $$ y=\csc \frac{1}{3} x $$
Step-by-Step Solution
Verified Answer
Period is \(6\pi\); asymptotes at \(x = 3n\pi\).
1Step 1: Identify the Basic Function
The given function is derived from the cosecant function, which is the reciprocal of the sine function. The general form of the cosecant function is \(y = \csc(kx)\), where \(k\) affects the function's period.
2Step 2: Determine the Period of the Cosecant Function
The period of the basic sine and cosecant function \(y = \csc(x)\) is \(2\pi\). When the function is transformed to \(y = \csc\left(\frac{1}{3}x\right)\), the period becomes \(\frac{2\pi}{k}\). Here, \(k = \frac{1}{3}\), so the period is \(\frac{2\pi}{\frac{1}{3}} = 6\pi\).
3Step 3: Identify the Asymptotes
The asymptotes of the cosecant function occur where the sine function (its reciprocal) is equal to zero. For \(y = \csc\left(\frac{1}{3}x\right)\), find points where \(\sin\left(\frac{1}{3}x\right) = 0\), which is when \(\frac{1}{3}x = n\pi\) where \(n\) is an integer. Solving for \(x\) gives \(x = 3n\pi\), indicating vertical asymptotes at \(x = 3n\pi\).
4Step 4: Sketch the Graph
To sketch the graph, plot the asymptotes at \(x = 3n\pi\) where \(n\) is an integer. Between each pair of asymptotes, the graph of \(y = \csc\left(\frac{1}{3}x\right)\) will follow the shape of downward and upward opening branches diverging towards the asymptotes. The graph touches or approaches infinity at the asymptotes and reaches its peaks at halfway between them.
Key Concepts
Cosecant FunctionPeriod of Trigonometric FunctionsVertical AsymptotesReciprocal Trigonometric Functions
Cosecant Function
The cosecant function, denoted as \( \csc(x) \), is a trigonometric function that is defined as the reciprocal of the sine function. This means that \( \csc(x) = \frac{1}{\sin(x)} \).
It is important to note that \( \csc(x) \) is undefined wherever \( \sin(x) = 0 \), because division by zero is undefined in mathematics. These undefined points result in vertical asymptotes on the graph of the cosecant function.
The basic shape of the cosecant graph features arcs that open either upwards or downwards, between these asymptotes. Each arc of the cosecant function stradles a sine curve, except it expands towards infinity near the vertical asymptotes.
It is important to note that \( \csc(x) \) is undefined wherever \( \sin(x) = 0 \), because division by zero is undefined in mathematics. These undefined points result in vertical asymptotes on the graph of the cosecant function.
The basic shape of the cosecant graph features arcs that open either upwards or downwards, between these asymptotes. Each arc of the cosecant function stradles a sine curve, except it expands towards infinity near the vertical asymptotes.
Period of Trigonometric Functions
The period of a trigonometric function is the smallest interval over which the function repeats its values. For the basic sine and cosecant function \( y = \csc(x) \), the period is \( 2\pi \). This signifies that every \( 2\pi \) units along the x-axis, the graph will repeat the same pattern.
When a trigonometric function is transformed, such as \( y = \csc\left(\frac{1}{3}x\right) \), the period changes. The new period can be found by dividing the original period by the coefficient of \( x \). So, in this case, the period becomes \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).
Understanding the period is crucial because it helps in sketching the graph accurately. By knowing the period, one knows the spacing between the features, like peaks, troughs, and asymptotes within one cycle.
When a trigonometric function is transformed, such as \( y = \csc\left(\frac{1}{3}x\right) \), the period changes. The new period can be found by dividing the original period by the coefficient of \( x \). So, in this case, the period becomes \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).
Understanding the period is crucial because it helps in sketching the graph accurately. By knowing the period, one knows the spacing between the features, like peaks, troughs, and asymptotes within one cycle.
Vertical Asymptotes
Vertical asymptotes are lines where the function approaches infinity; they occur where the function is undefined. For the cosecant function, vertical asymptotes occur where the sine function equals zero.
In the case of \( y = \csc\left(\frac{1}{3}x\right) \), the asymptotes are found by setting \( \sin\left(\frac{1}{3}x\right) = 0 \). Solving \( \frac{1}{3}x = n\pi \) for \( x \), where \( n \) is an integer, gives us \( x = 3n\pi \).
These correspond to each integral multiple of \( 3\pi \), such as \( 0, 3\pi, -3\pi \), and so forth. The presence of vertical asymptotes is crucial to understanding trigonometric function behavior, as they indicate the limits at which the function becomes infinitely large or small.
In the case of \( y = \csc\left(\frac{1}{3}x\right) \), the asymptotes are found by setting \( \sin\left(\frac{1}{3}x\right) = 0 \). Solving \( \frac{1}{3}x = n\pi \) for \( x \), where \( n \) is an integer, gives us \( x = 3n\pi \).
These correspond to each integral multiple of \( 3\pi \), such as \( 0, 3\pi, -3\pi \), and so forth. The presence of vertical asymptotes is crucial to understanding trigonometric function behavior, as they indicate the limits at which the function becomes infinitely large or small.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions are derived from the basic sine, cosine, and tangent functions. The most common ones include cosecant \( \csc(x) \), secant \( \sec(x) \), and cotangent \( \cot(x) \).
These functions are defined as follows:
- \( \csc(x) = \frac{1}{\sin(x)} \)
- \( \sec(x) = \frac{1}{\cos(x)} \)
- \( \cot(x) = \frac{1}{\tan(x)} \)
Being reciprocal means they inherit the behavior of their base functions but exhibit inverse characteristics, particularly where the original functions are zero, creating points of discontinuity and resulting in vertical asymptotes.
Understanding their relationship with primary functions helps in sketching their graphs and predicting where they will be undefined or infinite.
These functions are defined as follows:
- \( \csc(x) = \frac{1}{\sin(x)} \)
- \( \sec(x) = \frac{1}{\cos(x)} \)
- \( \cot(x) = \frac{1}{\tan(x)} \)
Being reciprocal means they inherit the behavior of their base functions but exhibit inverse characteristics, particularly where the original functions are zero, creating points of discontinuity and resulting in vertical asymptotes.
Understanding their relationship with primary functions helps in sketching their graphs and predicting where they will be undefined or infinite.
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