I'm sorry but the book I'm using doesn't have the U used at any point, do you mind saying what u stands for?
Hi,
I'm working on some practice questions using unit circles in advance of a chapter quiz tomorrow and am stuck on one question that I must have a mental block about. The question is as follows:
Solve the equation on the interval theta is greater than or equal to zero and less than pi:
sin^2 theta - cos^2 theta = 1 + cos theta
Any guidance that you can provide in the way of general rules of thumb that I would use to solve this type of problem would be very useful. Thanks!
MattyD
it's just a substitution, make the polynomial structure clear. I guess there's an easier way to do it.
$\sin^2(\theta)-\cos^2(\theta)=1+\cos(\theta)$
$1-\cos^2(\theta)-\cos^2(\theta)=1+\cos(\theta)$
$-2\cos^2(\theta)=\cos(\theta)$
One immediately obvious solution is $\cos(\theta)=0 \Rightarrow \theta=\pm \dfrac \pi 2$
otherwise we can divide by $\cos(\theta)$ and obtain
$-2\cos(\theta)=1$
$\cos(\theta) =-\dfrac 1 2$
$\theta = \pm \dfrac {2\pi} 3$
now all these solutions are periodic with period $2\pi$
so the final answer is
$\theta = \pm \dfrac \pi 2 \pm 2 k \pi~~k\in\mathbb{Z}$
$\theta = \pm \dfrac {2\pi} 3 \pm 2 k \pi~~k\in\mathbb{Z}$
Thank you for laying it all out for me. I see that you have two of the four answers that the book provides. I will study what you've done along with what the book is showing and see if I can come to the other two solutions on my own. I'll post how I'm doing a little later. Thanks again.