How to find all x in which sin(x) = k (k is constant)

[tapp]

Hello everyone
I have a question, and ill use an example to ask it.


I want to find all 0<x<2π which sin(x) will equal -0.5.
So I punch in a calculator:
arcsin(-0.5)
I get -π/6.
I add 2π to it and I get 11/6π which is cool.
But of course there's one more x that can be used and my question is how to find it.
(Its 7/6π and my book didn't explained how to find it)

Any suggestions?

Thank you very much.

 

[stapel]

I want to find all 0<x<2π which sin(x) will equal -0.5.
So I punch in a calculator: arcsin(-0.5)
I get -π/6.
I add 2π to it and I get 11/6π which is cool.
But of course there's one more x that can be used and my question is how to find it.

Use what you learned back in pre-calculus (or in your stand-alone trig course, if you took one) about the shape of the sine wave.

 

[tapp]

Use what you learned back in pre-calculus (or in your stand-alone trig course, if you took one) about the shape of the sine wave.

I know there has to be some simple logic behind it but I can't recall it...

 

[stapel]

I know there has to be some simple logic behind it...

Yes; it's called "looking at the sine wave".

 

[tapp]

But I looked and still can't figured it out...

 

[fcabanski]

But I looked and still can't figured it out...

It might be easier if you look at a unit circle.

Also, recall that sin is negative in the 3rd and 4th quadrants. -pi/6 gives you a reference angle of pi/6. Check that reference angle in the 3rd and 4th quadrants.

 

[stapel]

But I looked and still can't figured it out...

Did you not learn about the periodicity of the sine wave? that the sine wave repeats values as it cycles through its period? That places where the angle-value is a "mirror" of the first-quadrant value (such as pi/4 and 7pi/4 = 2pi - pi/4) the sine has the same value?

 

[tapp]

It might be easier if you look at a unit circle.

Also, recall that sin is negative in the 3rd and 4th quadrants. -pi/6 gives you a reference angle of pi/6. Check that reference angle in the 3rd and 4th quadrants.

Alright so I can take pi/6 add a pi to it and then I get the result that's in the 3rd quadrant. Great.

Did you not learn about the periodicity of the sine wave? that the sine wave repeats values as it cycles through its period? That places where the angle-value is a "mirror" of the first-quadrant value (such as pi/4 and 7pi/4 = 2pi - pi/4) the sine has the same value?

I did of course and I tried thinking about it, I always go back to the unit circle when dealing with sine and cosine but I couldn't remember the trick there.

Thank you both.