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The cosecant function is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. This means if you know the value of the sine of an angle, you can find the cosecant by taking its reciprocal. Mathematically, if \( y = \sin(x) \), then \( y = \csc(x) = \frac{1}{\sin(x)} \). The cosecant is undefined wherever the sine function is zero, leading to vertical asymptotes, which are crucial features of its graph.

Key characteristics of the cosecant function include:

• The graph has a series of u-shaped curves called branches, finding themselves between vertical asymptotes where the sine function is zero.
• These branches go towards infinity near the asymptotes, demonstrating the undefined nature of the cosecant when sine is zero.
• The function is neither even nor odd; however, it is periodic just like the sine function.

The amplitude of the graph, dictated by the coefficient in front of the cosecant term, affects how tall the branches of the graph appear. In our case, the amplitude is 5, which amplifies the curve vertically.

The period of a function describes how often the function repeats its pattern. For trigonometric functions such as the cosecant, the period is vital in predicting when the function will loop through its values again. In essence, it answers the question, "after how many units along the x-axis does the function start repeating?"

The period of the standard cosecant function \( y = \csc(x) \) is \( 2\pi \). However, in transformations of this function, like in our exercise where \( y = 5 \csc(3x) \), the period is altered by the coefficient of x inside the function. It is calculated using: \[\text{Period} = \frac{2\pi}{B} \]where \( B \) represents the coefficient of \( x \). For \( y = 5 \csc(3x) \), substituting \( B = 3 \), we find the period to be \( \frac{2\pi}{3} \). This means every \( \frac{2\pi}{3} \) units, the pattern of the graph repeats, justifying its cyclical nature.

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. For functions like the cosecant, these occur where the function is undefined. In the case of the cosecant function, vertical asymptotes occur where the sine function is zero, as the cosecant is the reciprocal of sine.

For our specific function \( y = 5 \csc(3x) \), vertical asymptotes are determined by setting the argument of the sine function, \( 3x \), equal to integer multiples of \( \pi \) (zeroes of sine), yielding:\[3x = n\pi \ \Rightarrow \ x = \frac{n\pi}{3} \]where \( n \) is an integer. This calculation tells us the x-values where our graph approaches infinity. These asymptotes guide how we plot the cosecant curve, ensuring that it arches away sharply at these points.

• These vertical asymptotes mark the points of discontinuity in the graph.
• Knowing them helps anticipate the overall shape of the function as it weaves and dances around these invisible barriers.
• They are a crucial part of understanding and sketching the function’s behavior over its period.

Observing how the function behaves near asymptotes can aid greatly in grasping the concept of infinity in mathematics.