Question

Find all values of x in the interval [0,2pi] that satisfy the equation.

2sin^2x=1 and

sin2x = cosx

Answer 1

Well, for the top one you need to solve for sin(x) first, which includes getting rid of the 2nd power applied to it. Once we solve for sin(x) we can go from there.

Answer 2

Would you do that by \[\sqrt{\sin^2x}=\sqrt{1}\] square rooting both sides?

Answer 3

You forgor to divide by 2, that disappeared in your equation. But after you divide by 2 you would square root both sides, yes.

Answer 4

so \[\sin x = \sqrt{1/2}\]

Answer 5

Dont forget plus/minus. We always have to have that if we take an even root of things.

Answer 6

Then after that what would be the next step? Or is that the final answer?

Answer 7

No, we actually need x. Solving for sinx doesnt cut it. So it looks best if we simplify what we have. Because once we simplify it, the value will match up with one of the very neat values that can be found on the unit circle:

\[\pm \sqrt{\frac{ 1 }{ 2 }}=\pm \frac{ \sqrt{2} }{ 2 }\]Do you know your unit circle or have one handy to identify where sinx would equal this value for the positive and negative?

Answer 8

3pi /4

Answer 9

That would be one. There are 3 more actually. There are 2 angles for which sin = positive sqrt(2)/2 and 2 angles for which sin = negative sqrt(2)/2

Answer 10

Or would it be \[5\pi/4, \pi/4, and, 7\pi/4\]

Answer 11

Those would be the other 3. So yes, those are all 4 of your answers :3

Answer 12

Ok cool! And then for the other problem sin2x = cosx

Answer 13

Right. We need to use an identity. We cannot have our functions have different angles. One has a 2x and one has an x. We want that 2x to become an x by using a double-angle identity. Would you happen to know thedouble-angle formula for sin2x?

Answer 14

2 sinx cosx = cosx right?

Answer 15

Bingo. Now just divide both sides by cosx and then by 2.

Answer 16

Sinx = 1/2 ?

Answer 17

Yes. So there are two values in which sinx = 1/2 on your unit circle :3

Answer 18

\[\pi/3, and, 5\pi/3?\]

Answer 19

Nope. pi/3and 5pi/3 is where cos is 1/2

Answer 20

Oh cos is in the y right? \[5\pi/6\] and \[pi/6\]

Answer 21

sin is y, cos is x :P
but yes, those are the two correct values :3

Answer 22

Would the steps be the same for inequalities? Like sin x \[\le 1/2\]

Answer 23

Hmm...well, we know that pi/6 and 5pi/6 = 1/2. So which interval keeps it le 1/2?

Is it pi/6 to 5pi/6that is less than 1/2 or is it 5pi/6 to pi/6?

Answer 24

5pi/6 to pi/6