Problem 57 Determine the period and sketch ... [FREE SOLUTION] (2024)

Short Answer

period of trigonometric functions

The period of a trigonometric function is the distance over which the function's values repeat. For the basic cotangent function, \( y = \cot(x) \), this period is \( \pi \). This implies that every \( \pi \) units along the x-axis, the function will exhibit the same pattern of values. When we alter the argument of the cotangent function, like in \( y = \cot(\frac{x}{2}) \), the period changes accordingly. The general formula for the period of \( y = \cot(bx) \) is: \[ \text{Period} = \frac{\pi}{b} \] Here, \( b = \frac{1}{2} \), so the period is: \[ \frac{\pi}{\frac{1}{2}} = 2\pi \] This tells us that the function \( y = \cot(\frac{x}{2}) \) repeats every \( 2\pi \) units along the x-axis. Identifying the period is crucial for understanding and graphing trigonometric functions, as it helps to establish the points where the function repeats.

graphing trigonometric functions

Graphing trigonometric functions involves plotting their values over one or more periods. For cotangent functions, a few key points and behaviors should be noted. Firstly, cotangent functions have vertical asymptotes where the function is undefined. These occur where the denominator of the cotangent function would be zero. For \( y = \cot(\frac{x}{2}) \), vertical asymptotes are found at: \[ x = 0, \pm 2\pi, \pm 4\pi, \ldots \] Between these vertical asymptotes, the function will cross the x-axis at the zeros, which are spaced evenly according to the period. In our example, zeros occur at: \[ x = 2k+1)\pi \text{ where } k \in \mathbb{Z} \] (i.e., at \( x = \pm \pi, \pm 3\pi, \ldots \)). When graphing, plot these key points and asymptotes first, then sketch the curve of the cotangent function, which decreases from positive infinity to negative infinity as it passes through each zero, starting near the vertical asymptote on one side until it approaches the next vertical asymptote.

properties of cotangent