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Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$ $$\cos^2x-\sin^2x=\sin$$

Edit $$1-\sin^2x-\sin^2x=\sin x$$ $$2\sin^2 x+\sin x-1=0$$ $\sin x=a$ $$2a^2+a-1=0$$ $$(a+1)(2a-1)=0$$ $$x=-1,\dfrac{1}{2}$$ $$x=\sin^{-1}(.5)=30^{\circ}=\dfrac{\pi}{6}$$ $$x=sin^{-1}(-1)=-90^{\circ}=-\dfrac{\pi}{2}$$

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What are you doing from the 2nd to the 3rd line? It looks like you're trying to add $sin x$ even though it is within a product in parantheses. I'm pretty sure you know that you can't do that. –  us2012 Jul 16 '12 at 23:53
    
I was just trying to get something accomplished on here so I can get useful help. What do I do after the second line? –  Austin Broussard Jul 16 '12 at 23:54
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2 Answers 2

up vote 2 down vote accepted

What you have is:

$$2\sin^2 x+\sin x-1=0$$

And let $\sin x = a$ so you'll have to solve a quadratic equation for $a$.

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@AustinBroussard: Sorry, I don't understand. you just substitute $\sin x$ with something more familiar and less annoying! If that's what you meant. When you find the values of $a$ you'll have values of $\sin x$ accordingly. –  Gigili Jul 17 '12 at 0:09
    
@AustinBroussard: Suppose $\sin x =a$, and solve the equation $a^2+a-1=0$. What values of $a$ you can find? –  Gigili Jul 17 '12 at 0:15
    
@AustinBroussard: Yes, exactly. But your answer is not correct, how you got $a=\frac{\sqrt 2}2$? –  Gigili Jul 17 '12 at 0:19
    
@AustinBroussard: You got: $(a+1)(2a-1)=0$, right? What are the values of $a$? Please first make sure you understand what I am saying and you have the correct answer, then you can accept my answer! –  Gigili Jul 17 '12 at 0:22
    
$a=-1$ and $\dfrac{1}{2}$. Can you delete your older comments because the site hates an overload of comments. –  Austin Broussard Jul 17 '12 at 0:23
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The calculation is almost completely correct. You reached the two possibilities $\sin x=\frac{1}{2}$ and $\sin x=-1$.

We are interested in solutions in the interval $-\pi \lt x\le \pi$.

Certainly $x=\frac{\pi}{6}$ is a solution, since $\sin(\pi/6)=\frac{1}{2}$. But there is another $x$ in our interval whose sine is $\frac{1}{2}$, namely $x=\pi-\frac{\pi}{6}=\frac{5\pi}{6}$. A look at the graph of $y=\sin x$ shows this. You can do a partial verification by calculator, by asking it to compute $\sin(5\pi/6)$, the sine of $150^\circ$.

There is only one place $x$ in our interval where $\sin x=-1$, so that part is fully correct.

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So the two answers I have are incorrect? $\dfrac{\pi}{6}$ and $-\dfrac{\pi}{2}$ –  Austin Broussard Jul 17 '12 at 1:19
    
The two answers are correct, I wrote that they are correct. You missed one answer, $\frac{5\pi}{6}$. So there are three answers, of which you got two. –  André Nicolas Jul 17 '12 at 1:23
    
$30^\circ -180^\circ =150^\circ$? Is that why there is a third answer? –  Austin Broussard Jul 17 '12 at 1:26
    
You have it sort of backwards, $180^\circ-30^\circ=150^\circ$, which is why $150^\circ$ is a solution. –  André Nicolas Jul 17 '12 at 1:28
    
But why is it so important to find an answer in the second quadrant? I'm just confused as to why you have to have the $150^\circ$ –  Austin Broussard Jul 17 '12 at 1:29
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