Question

QUICK! 75 POINTS !!Select all that are part of the solution set of csc(x) > 1 and over 0 ≤ x ≤ 2π.

Answer 1

Answer:

$\frac{\pi}{4}$

$\frac{5\pi}{6}$

Step-by-step explanation:

The answer uses the unit circle and that sine and cosecant are reciprocals.

The first choice doesn't even fit the criteria that $x$ is between $0$ and $2\pi$ (inclusive of both endpoints) because of the $x=\frac{-7\pi}{6}$.

Let's check the second choice.

$\csc(\frac{\pi}{4})=\frac{2}{\sqrt{2}} \text{ since } \sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$.

$\csc(\frac{\pi}{4})>1 \text{ since } \frac{2}{\sqrt{2}}>1$

$\csc(\frac{\pi}{2})=1 \text{ since } \sin(\frac{\pi}{2})=1$ which means $\csc(\frac{\pi}{2})=1$ which is not greater than 1.

So we can eliminate second choice.

Let's look at the third.

$\csc(\frac{5\pi}{6})=2 \text{ since } \sin(\frac{5\pi}{6})=\frac{1}{2}$ which means $\csc(\frac{5\pi}{6})>1$.

$\csc(\pi)$  isn't defined because $\sin(\pi)=0$.

So we are eliminating 3rd choice now.

Let's look at the fourth choice.

$\csc(\frac{7\pi}{6})=-2 \text{ since } \sin(\frac{7\pi}{6})=\frac{-1}{2}$ which means $\csc(\frac{7\pi}{6})<1$ and not greater than 1.

I was looking at the rows as if they were choices.

Let me break up my choices.

So we said $x=-\frac{7\pi}{6}$ doesn't work because it is not included in the inequality $0\le x \le 2\pi$.

How about $x=0$?  This leads to $\csc(0)$ which doesn't exist because $\sin(0)=0$.

So neither of the first two choices on the first row.

Let's look at the second row again.

We said $\frac{\pi}{4}$ worked but not $\frac{\pi}{2}$

Let's look at the choices on the third row.

We said $\frac{5\pi}{6}$ worked but not $x=\pi$

Let's look at at the last choice.

We said it gave something less than 1 so this choice doesn't work.

Answer 2

Answer:

x= pi/4

x= 5pi/6

Step-by-step explanation: