Questions: Determine the period for the function y=-8 csc (13 pi/8 x+4 pi/13) Period =

Transcript text: Determine the period for the function \[ y=-8 \csc \left(\frac{13 \pi}{8} x+\frac{4 \pi}{13}\right) \] Type 'pi' for $\pi$ in your answer(s), if needed. Period $=$ $\square$

Solution

Solution Steps

To find the period of the function $ y = -8 \csc \left(\frac{13 \pi}{8} x + \frac{4 \pi}{13}\right) $, we need to determine the period of the function inside the cosecant, which is a sine function. The period of $\csc(kx)$ is the same as the period of $\sin(kx)$, which is $\frac{2\pi}{|k|}$.

Identify the coefficient $k$ of $x$ in the argument of the cosecant function.
Calculate the period using the formula $\frac{2\pi}{|k|}$.

Step 1: Identify the Coefficient $ k $

The function given is $ y = -8 \csc \left(\frac{13 \pi}{8} x + \frac{4 \pi}{13}\right) $. The coefficient $ k $ of $ x $ in the argument of the cosecant function is $\frac{13\pi}{8}$.

Step 2: Calculate the Period

The period of the cosecant function $\csc(kx)$ is the same as the period of the sine function $\sin(kx)$, which is given by the formula:

\[ \text{Period} = \frac{2\pi}{|k|} \]

Substituting $ k = \frac{13\pi}{8} $ into the formula:

\[ \text{Period} = \frac{2\pi}{\left|\frac{13\pi}{8}\right|} = \frac{2\pi \times 8}{13\pi} = \frac{16}{13} \]