Question

√2/2 csc x − 1 = 0

Solve the equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)

Answer 1

This correct?
\[\large \frac{ \sqrt{2} }{ 2 }*\csc(x)-1=0\]

Answer 2

Yes!

Answer 3

\[\csc(x)=\frac{2}{\sqrt{2}}\]

Right?
And then
\[sin(x)=\frac{\sqrt{2}}{{2}}\]

Right?? Follow?

Answer 4

And I think you should've memorized the unit circle for this problem. Right?

Answer 5

Yes!!!

Answer 6

What is the next step though?

Answer 7

you can take the inverse sine of both sides..

\[x=\sin^{-1}( \frac{ \sqrt{2} }{ 2 })=\]

Or recall the unit circle. Are you familiar with the properties on the circle?

Answer 8

Are you asking what sine goes with one the unit circle?

Answer 9

\[\frac{ \pi }{ 4 }\]

Answer 10

7pi/4

Answer 11

remember, each point on the unit circle is (x,y)=(cos x, sin x)

Answer 12

The sine value is the y value of any point on the circle..

Answer 13

Oh! then it will go with 3pi/4. Right?

Answer 14

yes, this prob says sin(x)=root(2)/2, that occurs at 2 angles where the y coordinate value is root(2)/2

pi/4 and 3pi/4

Answer 15

Yes! I follow

Answer 16

That is the solutions if you limit the angle value to be 0 to 2pi. You can still keep going around the circle more times and get different angle values.. like pi/4 + 2pi

Answer 17

if you add a full revolution to both the angles, you are back at the same point on the circle

Answer 18

Yes, I understand!

Answer 19

So really the answer is those two angles, plus any multiple of 2pi

pi/4 + 2pi*n
3pi/4 + 2pi*n

for n=0,1,2,3...

Answer 20

So those are the final answers?

Answer 21

Also how do I know when to write + 2pi n or some problems the answer is just + 2 pi?

Answer 22

yes, and it depends on the solution. For this one, you got pi/4 and 3pi/4. a full revolution around the circle is 2pi. So if you are at pi/4 and add 2pi, you are back to the same spot with angle pi/4 + 2pi = 9pi/4

Answer 23

You can add as many full revolutions to both the angles and still be in the same point..

pi/4 + 2pi*n
3pi/4 + 2pi*n

n=0,1,2...