In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 1/2 csc x/2

Verified step by step guidance

1. Identify the given function: \(y = \frac{1}{2} \csc \left( \frac{x}{2} \right)\).
2. Recall that the cosecant function \(\csc \theta\) is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).
3. Determine the period of the function inside the cosecant. The standard period of \(\csc x\) is \(2\pi\). Since the argument is \(\frac{x}{2}\), the period \(P\) is given by \(P = \frac{2\pi}{\frac{1}{2}} = 4\pi\).
4. Since the problem asks for two periods, calculate the interval for \(x\) to graph: from \(0\) to \(2 \times 4\pi = 8\pi\).
5. Plot the graph by first sketching the sine function \(y = \sin \left( \frac{x}{2} \right)\) over \([0, 8\pi]\), then draw the cosecant as the reciprocal of sine, scaled by \(\frac{1}{2}\), noting vertical asymptotes where \(\sin \left( \frac{x}{2} \right) = 0\).

Key Concepts

Understanding the Cosecant Function

The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is undefined where sin(x) = 0, leading to vertical asymptotes. Recognizing its periodicity and behavior near these asymptotes is essential for accurate graphing.

Effect of Transformations on Trigonometric Graphs

Transformations such as vertical scaling and horizontal stretching/compression alter the graph's shape and period. In y = (1/2) csc(x/2), the factor 1/2 scales the graph vertically, while the argument x/2 stretches the period horizontally, doubling it compared to the basic csc(x) function.

Period of the Cosecant Function

The period of the basic cosecant function is 2π, matching the sine function's period. When the input is modified to x/2, the period becomes 4π, calculated by dividing 2π by the coefficient of x inside the function. Knowing the period helps in plotting the correct length for two full cycles.