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Math Help - halp solving trig formula using unit circles

  1. #1
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    Question halp solving trig formula using unit circles

    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
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  2. #2
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    Re: halp solving trig formula using unit circles

    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?
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  3. #3
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    Re: halp solving trig formula using unit circles

    Quote Originally Posted by MattyDman View Post
    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?
    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

    $\theta = \pm \dfrac \pi 2 \pm 2 k \pi~~k\in\mathbb{Z}$

    $\theta = \pm \dfrac {2\pi} 3 \pm 2 k \pi~~k\in\mathbb{Z}$
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    Re: halp solving trig formula using unit circles

    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.
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    Re: halp solving trig formula using unit circles

    Quote Originally Posted by MattyDman View Post
    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?
    Eliminated post because it duplicated romsek's.
    Last edited by JeffM; May 12th 2014 at 08:25 PM.
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  6. #6
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    Re: halp solving trig formula using unit circles

    Quote Originally Posted by MattyDman View Post
    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.
    I provided 4 base solutions. There are no others.
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