3. why is there no amplitude for the secant and cosecant graph?\n4. how are the graphs for secant and…

Answer:

Question 3:

The secant function is $y = \sec(x)=\frac{1}{\cos(x)}$ and the cosecant function is $y=\csc(x)=\frac{1}{\sin(x)}$. Amplitude is defined as half the distance between the maximum and minimum values of a periodic - function. For $\cos(x)$ and $\sin(x)$, the range is $[- 1,1]$. But for $\sec(x)$ and $\csc(x)$, their ranges are $(-\infty,-1]\cup[1,\infty)$. Since there is no finite maximum and minimum value (they approach infinity and negative - infinity), there is no well - defined amplitude.

Question 4:

1. Both secant ($y = \sec(x)=\frac{1}{\cos(x)}$) and cosecant ($y=\csc(x)=\frac{1}{\sin(x)}$) are periodic functions. The period of $y = \sec(x)$ and $y=\csc(x)$ is $2\pi$.
2. They both have vertical asymptotes. The secant function $y = \sec(x)$ has vertical asymptotes at $x=(2n + 1)\frac{\pi}{2},n\in\mathbb{Z}$ (where $\cos(x)=0$), and the cosecant function $y=\csc(x)$ has vertical asymptotes at $x = n\pi,n\in\mathbb{Z}$ (where $\sin(x)=0$).
3. Both functions have values that are either greater than or equal to $1$ or less than or equal to $-1$ (their ranges are $(-\infty,-1]\cup[1,\infty)$).

Brief Explanations:

For question 3, amplitude requires finite max and min, which secant and cosecant lack due to their definitions in terms of cosine and sine. For question 4, similarities are based on periodicity, presence of vertical asymptotes, and range characteristics.