Question

sin4theta=cos2theta
How do I solve this equation? I can only find 15 degrees and 75 degrees, but I know there are more answers.

Answer 1

\[\large\rm \sin4\theta=\cos2\theta\]Recall your sine double angle formula:\[\large\rm \color{orangered}{\sin2x=2\sin x \cos x}\]We can apply this in the following way,\[\large\rm 2\sin2\theta\cos2\theta=\cos2\theta\]Ya?

Answer 2

yes

Answer 3

The trick here is,
to not cancel things out.
You lose information when you cancel things out.

So yes, it might seem like cancel cos2theta is the most natural thing to do,
but we're going to avoid that.

Answer 4

Instead let's subtract cos2theta to the left side,\[\large\rm 2\sin2\theta \cos2\theta-\cos2\theta=0\]and do some factoring.

Answer 5

If we factor cos2theta out of each term,\[\large\rm \cos2\theta(2\sin2\theta-1)=0\]And then apply your Zero-Factor Property,\[\large\rm \cos2\theta=0\qquad\qquad\qquad\qquad\qquad 2\sin2\theta-1=0\]setting each factor equal to zero,
and solving for theta in each case.

Answer 6

Confused by any of those steps? :o

Answer 7

Do I substitute the cos2theta and sin2theta now to solve them?

Answer 8

No, you don't need to apply any more Double Angle Formulas,
if that's what you're asking.

Answer 9

I'm not sure how to find the angles for cos2theta=0.

Answer 10

Well if you had cos(x)=0,
would you remember which `special angles` give you a cosine of 0?

Answer 11

Gotta think back to your unit circle.

Answer 12

90 degrees and 270 degrees

Answer 13

Ok same idea with our problem,
but instead of the angle x = 90 and 270,
we have that our angle is 2theta = 90 and 270.

\(\large\rm 2\theta=90,~270\)
So we have another step to apply in order to fully solve for this value theta.
We'll divide by 2.

Answer 14

So 45 degrees and 135 degrees

Answer 15

Good good good.
And it looks like you already found the 15 and 75 earlier! :)
Those values come from the sine factor.

Answer 16

This is the weird thing in math sometimes,
cancelling out causes problems.

Gotta get used to working around that.

Answer 17

so I simplified it to sin2theta=1/2, but angles in the third and fourth quadrant make sine
-1/2

Answer 18

Yes, good.
Only the Q1 and Q2 angles seem to work there.

Answer 19

I have the answer key and it says the answers are 15, 45, 75, 135, 195, 225, 255, and 315
How do I find the remaining answers?

Answer 20

So you needed all of the values within 360 degrees.
Ok we should take a step back then.

Answer 21

\[\large\rm \cos2\theta=0\]Cosine gives us 0 at angles 90 and 270, yes.
But if we kept spinning around the circle, it would give us more angles, right?
So if we're at 90 and do a full spin and land at the same spot,
we're now at an angle of 90+360 = 450.
And from 270 if we spin around the circle, we end up at 270+360=630.

So we could actually (and we'll need to for this problem) write our result like this,\[\large\rm 2\theta=90,~270,~450,~630\]

Answer 22

Keep in mind that we could keep on spinning.
What is it that told me to stop right there?

Well, we're dividing by 2.
So I know that we will find `every angle within two rotations of the circle`.

Answer 23

If we had 3theta, you'd have to spin further,
little bit more work.

Answer 24

Oh I see. Thank you!!

Answer 25

So try that with your sine, ya? :)
Adding 360 to the values before you cut them in half.
You should end up with all of them.

Answer 26

Yes. I have them all now. Thank you so much!

Answer 27

cool :)