Question

Solve for theta sin  θ/2  − cos θ = 0

Answer 1

help please

Answer 2

Check me on this, but one possible answer is 2pi/3.

Answer 3

I don't know the best pedagogical approach to helping, except that I recommend you review your unit circle....

Answer 4

\[\large \sin \frac{\theta}{2}-\cos \theta=\sin \frac{\theta}{2}-\cos 2(\frac{\theta}{2})\]
Using your double angle results. Expand/Distribute cos2(theta/2).
\[\large \cos 2t=\cos^2 t-\sin^2 t\]
Apply that rule to your given question.
\[\large \sin\frac{\theta}{2}-(\cos^2 \frac{\theta}{2}-\sin^2 \frac{\theta}{2})\]

Answer 5

Using trigonometric identities: Make cos^2(theta/2) in terms of sine.
\[\huge \sin \frac{\theta}{2}-(1-\sin^2 \frac{\theta}{2}-\sin^2 \frac{\theta}{2})=0\]
\[\huge \sin \frac{\theta}{2}-1+2\sin^2 \frac{\theta}{2}=0\]
Solve it as a quadratic by factorising.
\[\huge 2\sin^2 \frac{\theta}{2}+\sin \frac{\theta}{2}-1=0\]

Answer 6

How can I factor if it's still theta/2?

Answer 7

You look at it as if it were x. \frac{\theta}{2} is simply x. And you get to the end before finding theta.

Answer 8

\[\huge (2\sin \frac{\theta}{2}-1)(\sin \frac{\theta}{2}+1)=0\]

Answer 9

That's the factored form.

Answer 10

I think you're afraid of theta/2 because of you think when you multiply, something is going to happen inside the sine. Well fortunately, it doesn't. So theta/2 is not different to theta. You just find theta when you solve for theta/2.
\[\sin\frac{\theta}{2}\times \sin \frac{\theta}{2}=\sin^2\frac{\theta}{2}\]

Answer 11

because you think*

Answer 12

Yeah lol! Thank you so much though! Very in depth and helpful explanation ! Can't thank you enough! I get it!! :)

Answer 13

Okay. Now let's continue on: There are two solutions to the above quadratic as you can see.
\[\huge \sin \frac{\theta}{2}=\frac{1}{2}\]
or
\[\huge \sin \frac{\theta}{2}=-1\]

Answer 14

But I chose my angles as 60 180 and 300 and it's still wrong

Answer 15

We haven't finished yet. Using your calculator we can find what theta/2 is equal to.
\[\frac{\theta}{2}=30, 150, 270\]
Now we find theta. And may I ask, do you have a range to what theta can become?

Answer 16

because theta can just get bigger and bigger without any range. is theta in between the range 0<theta<360?

Answer 17

Oh yeah 0<theta<360

Answer 18

Nice! I got it right! THANKS SO MUCH!

Answer 19

No worries mate.