Question

526 chapter 5 an introduction to trigonometric functions 24. the given graphs are of the form y = a sin(bt) and y = a csc(bt). determine the equation of each graph. 25.

Explanation:

Step1: Recall the general form of sine and cosecant functions

The general form of $y = A\sin(Bt)$ and $y=A\csc(Bt)$. The amplitude of $y = A\sin(Bt)$ is $|A|$ and the period of $y = A\sin(Bt)$ is $T=\frac{2\pi}{|B|}$, and $y = A\csc(Bt)=\frac{A}{\sin(Bt)}$.

Step2: Analyze the sine - function graph (top graph)

For the graph of $y = A\sin(Bt)$, the amplitude $A$ is the maximum distance from the $x$ - axis. From the graph, the maximum value is $\frac{1}{2}$ and the minimum value is $-\frac{1}{2}$, so $A=\frac{1}{2}$. The period $T$ is the distance between two consecutive peaks or troughs. The period $T=\pi$. Since $T = \frac{2\pi}{|B|}$ and $T=\pi$, we have $\pi=\frac{2\pi}{|B|}$, which gives $|B| = 2$. Since the graph has no phase - shift and starts at the origin, the equation of the sine function is $y=\frac{1}{2}\sin(2t)$.

Step3: Analyze the cosecant - function graph (bottom graph)

For the graph of $y = A\csc(Bt)=\frac{A}{\sin(Bt)}$, the vertical asymptotes of $y = A\csc(Bt)$ occur where $\sin(Bt)=0$. The period of the related sine function (the reciprocal of the cosecant function) can be determined from the distance between consecutive vertical asymptotes. The distance between consecutive vertical asymptotes is $4$, so the period of the related sine function is $T = 4$. Since $T=\frac{2\pi}{|B|}$ and $T = 4$, we have $4=\frac{2\pi}{|B|}$, so $|B|=\frac{\pi}{2}$. The amplitude of the related sine function (used to find $A$) can be observed from the fact that the minimum and maximum values of the cosecant function are symmetric about the $x$ - axis. The equation of the cosecant function is $y = 1\csc(\frac{\pi}{2}t)$ because when we consider the general form $y = A\csc(Bt)$ and the behavior of the graph, $A = 1$ and $B=\frac{\pi}{2}$.

Answer:

The equation of the first graph (sine - type) is $y=\frac{1}{2}\sin(2t)$. The equation of the second graph (cosecant - type) is $y=\csc(\frac{\pi}{2}t)$.