Problem 30 - Algebra and Trigonometry

Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. $$ y=2 \csc x $$

Step-by-Step Solution

Answer

The graph of the function \(y = 2csc(x)\) is the graph of the function \(y = 2/sin(x)\). Therefore, where sin(x) = 0 (at \(-2\pi\), \(-\pi\), 0, \(\pi\), \(2\pi\)), are vertical asymptotes. The maximum value of sin(x) is 1 and its minimum value is -1, so the maximum value of \(2csc(x)\) is 2 and its minimum value is -2. The graph will approach the asymptotes and change direction at its maximum and minimum values.

Step 1: Understanding the Cosecant Function

Cosecant function, csc(x), is the reciprocal of the sine function. In other words, \(csc(x) = 1/ sin(x)\). When \(sin(x) = 0\), the cosecant function becomes undefined and these points become the vertical asymptotes for the csc function.

Step 2: Graphing the Sine Function

Plot the sine function for two periods. The period for sine is \(2\pi\). Therefore, you should plot for x values ranging from \(-2\pi\) to \(2\pi\).

Step 3: Identifying Asymptotes

The asymptotes occur at points where the sine function equals zero during the 2-period range. This would be at \(-2\pi\), \(-\pi\), 0, \(\pi\) and \(2\pi\). Draw vertical dashed lines at these x-values to represent the asymptotes.

Step 4: Graphing the Cosecant Function

At points where sine \(=1\), \(y = 2 csc(x)\) becomes 2 because \(csc(x)\) would equal \(1/sin(x)\). And at points where sine \(=-1\), \(y = 2 csc(x)\) becomes -2. Finally, between every two asymptotes, the cosecant function approaches each asymptote.

Step 5: Plotting the Cosecant Function

Based on step 4, plot the points and draw the curve of y = 2csc(x) for each period

Key Concepts

Reciprocal Functions

When dealing with trigonometric functions, understanding reciprocal functions is crucial. The concept of a reciprocal function involves flipping a given function upside down. In the case of the cosecant function, it is the reciprocal of the sine function. This means that for any given angle \(x\), the value of \(\csc(x)\) is \(1/\sin(x)\).
This relationship is key because it introduces points where the cosecant function is undefined. These occur precisely when the sine function equals zero.
The reciprocal nature of cosecant means that wherever the sine function hits its peaks and troughs, the cosecant function approaches infinity or negative infinity.

• Cosecant: \( \csc(x) = \frac{1}{\sin(x)} \)
• Undefined points: occur at \(x\) values where \(\sin(x) = 0\)

Trigonometric Graphing

Graphing trigonometric functions is a skill that enhances your understanding of periodic behavior and function properties. To visualize the cosecant function, it's initially easier to graph the sine function over a complete cycle, which is \(2\pi\).
Since trigonometric functions like sine are periodic, we repeat this pattern to cover more periods for a comprehensive view. This is a simple approach to prepare the foundation for graphing the cosecant function, as it directly relates to sine.

• Sine function graph: Wave pattern starting at zero, reaching 1 at \(\pi/2\), back to zero at \(\pi\), to -1 at \(3\pi/2\), and cycles back at \(2\pi\).
• Periodicity: Sine and consequently cosecant, repeat every \(2\pi\).

Vertical Asymptotes

In graphing the cosecant function, recognizing where vertical asymptotes occur is pivotal. Vertical asymptotes are lines the graph approaches but never touches or crosses. They appear at the \(x\)-values where the function is undefined.
For the cosecant function, these values coincide with the zeros of the sine function. This means the asymptotes for \(2 \csc(x)\) will occur at points like \(-2\pi\), \(-\pi\), \(0\), \(\pi\), and \(2\pi\).

• Vertical asymptotes: Where \(\sin(x) = 0\)
• Key points: \(-2\pi, -\pi, 0, \pi, 2\pi\)

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent. These functions are fundamental in analyzing periodic phenomena and are pivotal in various mathematical applications.
The cosecant function, as mentioned, is tied to sine through its reciprocal relationship. What makes trigonometric functions intriguing is their cyclical nature, which allows them to model waves, circular motion, and oscillations accurately.
Understanding how to graph and interpret these functions expands one's ability to apply trigonometry in real-world contexts.

• Key functions: sine, cosine, tangent; cosecant (\(\csc\)), secant (\(\sec\)), cotangent (\(\cot\))
• Cyclical behavior: Their repetitive patterns across intervals.