Problem 40 - Algebra and Trigonometry

Question

Find the period and graph the function. $$ y=\csc 2\left(x+\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The period of the function is \( \pi \).

Step 1: Understanding the Function

We begin by identifying that the function given is a cosecant function, which is the reciprocal of the sine function. The function is written as \( y = \csc(2(x+\frac{\pi}{2})) \). The general form of the cosecant function is \( y = \csc(bx+c) \). Here, \( b = 2 \) and \( c = 2 \cdot \frac{\pi}{2} \).

Step 2: Determine the Period

The period of the cosecant function is determined by the formula \( \frac{2\pi}{|b|} \). For the given function, \( b = 2 \) so the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate Phase Shift

The phase shift of the function can be found using \(-\frac{c}{b}\). Here, \( c = 2 \cdot \frac{\pi}{2} = \pi \), so the phase shift is \(-\frac{\pi}{2} \times \frac{1}{2} = -\frac{\pi}{2}\). This means the graph shifts \( \frac{\pi}{2} \) units to the left.

Step 4: Graph the Function

To graph \( y = \csc(2(x+\frac{\pi}{2})) \), start by making the sine base graph \( y = \sin(2(x+\frac{\pi}{2})) \), considering the same period and phase shift. Plot vertical asymptotes at every \(\pi\) interval starting from previous results and sketch the cosecant curve, which will dip down or rise based on sine’s maxima and minima within its interval between asymptotes.

Key Concepts

Period of Trigonometric Functions

When working with trigonometric functions, understanding their period is essential. The period of a function is the distance over which the function completes one full cycle. For basic trigonometric functions like sine, cosine, and cosecant, this tells us how often the function repeats its values. The general formula to find the period of a trigonometric function like \( y = \ ext{csc}(bx + c) \) is \( \frac{2\pi}{|b|} \). In our case, with the function \( y = \ ext{csc}(2(x+\frac{\pi}{2})) \), the constant \( b \) is 2. Thus, the period is \( \frac{2\pi}{2} = \pi \). This means that the cosecant function completes one full cycle over an interval of \( \pi \) instead of the usual \( 2\pi \) you might expect with regular sine or cosine functions, due to the factor of 2 affecting the frequency.

Phase Shift

The phase shift of a trigonometric function refers to the horizontal movement of the function along the x-axis. It's like sliding the graph left or right on the graph paper. For the function \( y = \ ext{csc}(2(x+\frac{\pi}{2})) \), the phase shift can be calculated using the formula \( -\frac{c}{b} \). In this situation, substitute \( c = \pi \) (since \( c = 2 \cdot \frac{\pi}{2} \)) and \( b = 2 \). Plug these into the formula: \( -\frac{\pi}{2} \times \frac{1}{2} = -\frac{\pi}{2} \). The negative sign indicates that the shift is to the left, shifting the whole graph by \( \frac{\pi}{2} \) units. Adjusting for phase shift helps in accurately positioning the trigonometric function before graphing it.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their curves on a coordinate system, showing their periodic nature. In the case of the cosecant function, which is the reciprocal of the sine function, the process includes several steps:1. **Start with the Base Function:** Begin by sketching the graph of its corresponding sine function. For \( y = \text{csc}(2(x+\frac{\pi}{2})) \), that would be \( y = \sin(2(x+\frac{\pi}{2})) \). 2. **Determine Key Points:** Identify important locations for sin(x), like zeros, peaks, and troughs, within one period. Note where these intervals shift due to phase shift. 3. **Add Vertical Asymptotes:** The cosecant function will have vertical asymptotes at the x-values where the sine function equals zero (within its modified intervals due to phase shift and period). For sine, these would occur at every interval of \( \pi \); thus, draw vertical asymptotes at every \( \pi \) starting from the adjusted phase-shifted point. 4. **Sketch the Cosecant Curve:** Between asymptotes, the cosecant function will follow a pattern dipping below or arching above, shaped by the peaks and valleys created from the sine function.Graphing with these steps allows you to visualize trigonometric functions accurately, getting a clear picture of their behavior across their period and as they shift horizontally.