Find the solution set for the trigonometric equation below. Please show your steps and the identity that is applicable to this problem.

sin2 theta = sin theta in the domain of 0

Question

Find the solution set for the trigonometric equation below. Please show your steps and the identity that is applicable to this problem.

sin2 theta = sin theta in the domain of 0 < theta < 2 pi   (that's less/greater than or equal to, not just less/greater than.)

Thank you!!

tkhunny · Answer

$\sin^{2}(\theta) = \sin(\theta)$

Is that it?  What's your plan?

How would you find the two solutions to this?  $x^{2} = x$?

anonymous · Answer

I guess I would take the square root x?

anonymous · Answer

Also the original equation is sin times 2 theta, not sin ^ 2 theta

tkhunny · Answer

Not a good plan.  How about this?  $x^{2} - x= 0$

tkhunny · Answer

Okay, then $\sin(2\theta) = \sin(\theta)$.

In that case, how would you find the TWO answers to this?  $x^{2} - x = 0$??

anonymous · Answer

Would I plug in a value for x?

tkhunny · Answer

1) "Plug in" doesn't mean anything.
2) How many quadratic equations have you factored in your lifetime?

anonymous · Answer

Oh, would it be (x^2 + 0 ) (x + 0) ?

tkhunny · Answer

??  You should not forget your algebra.  It is a factoring exercise.

$x^{2} - x = 0$

$x(x-1) = 0$

$x = 0\;or\;x = 1$

This should be ringing bells.  Is it?

anonymous · Answer

Yes, I remember that.

tkhunny · Answer

That is what we need for this problem.

$\sin(2\theta) - \sin(\theta) = 0$

Can you see how to factor that?  You may need an identity for $\sin(2\theta)$.

anonymous · Answer

If i factored it, would it be sin^2(sin - 1) ?
And would I need a pythagorean identity to solve?

tkhunny · Answer

No.  That makes no sense at all.  "sin" doesn't mean anything.  Where did you get the exponent?

What did you do with my hint about $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$?

anonymous · Answer

I'm not sure. I really don't know how to solve this problem. Is it possible that you could show me the first few steps? I'm having trouble understanding where to start.

tkhunny · Answer

I already have shown you much of it.  Let's do the rest, but first, I must worn you that you will have to do better than this on your exam.  Trig identities are very important.  All your algebra studies are very important.  You can't learn the new material if you have forgotten the old.

$\sin(2\theta) = \sin(\theta)$

Algebra -- Addition

$\sin(2\theta) - \sin(\theta) = 0$

Trigonometry -- Double Angle Identity

$2\sin(\theta)\cos(\theta) - \sin(\theta) = 0$

Algebra -- Factoring (Distributive Property)

$\sin(\theta)\left[2\cos(\theta) - 1\right] = 0$

I'm guessing that there is nothing there that you could not have done.  Now what?

anonymous · Answer

Now do I add 1 and divide by 2 in order to isolate sine and cosine?

tkhunny · Answer

You are forgetting your algebra.  Remember why we factored things?!  You must gain this skill.  It WILL be required of you an an exam.

$\sin(\theta)\left[2\cos(\theta) - 1\right] = 0$

Algebra -- The Zero Property of Multiplication

$\sin(\theta) = 0\;or\;2\cos(\theta) - 1 = 0$

We're not even using any trigonometry.  When was your last algebra class?

tkhunny · Answer

Grades are one thing, but if you go to Stanford not knowing how to factor or how to use algebra to solve simple equations, you will fall behind until you gain these skills.  Would you like to complete the assignment effectively?  To me, this means understanding what you are doing and learning the skills you will need in the future.