Question

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

Answer 1

x= pi/4

x= 5pi/6

Explanation:

Answer 2

`(\pi)/(4)`

`(5\pi)/(6)`

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=(-7\pi)/(6)`.

Let's check the second choice.

`\csc((\pi)/(4))=(2)/(√(2)) \text{ since } \sin((\pi)/(4))=(√(2))/(2)`.

`\csc((\pi)/(4))>1 \text{ since } (2)/(√(2))>1`

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

So we can eliminate second choice.

Let's look at the third.

`\csc((5\pi)/(6))=2 \text{ since } \sin((5\pi)/(6))=(1)/(2)` which means
`\csc((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((7\pi)/(6))=-2 \text{ since } \sin((7\pi)/(6))=(-1)/(2)` which means
`\csc((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=-(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
`(\pi)/(4)` worked but not
`(\pi)/(2)`

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

We said
`(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.