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Math Help - Simplifing large trigonometric expression with unit circle?!?

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    Simplifing large trigonometric expression with unit circle?!?

    I got stuck again on this problem.

    -(cos(-9pi/4)-sin(-9pi/4))2-(cos 4pi/3 - sin 4pi/3)2

    I used the unit circle to simplify and got this far:

    Bracket 1: -(sqrt2/2-(-sqrt 2/2))2=(-sqrt 2+sqrt 2/2)2=(2 sqrt 2/2)2= 2


    Bracket 2:-(-1/2-sqrt 2/2)2= -(-1-sqrt 2/2)2

    After this im not getting any further and the answer is supposed to be c+ sqrt a/b, I know it's complicated but i really need help!!
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    Re: Simplifing large trigonometric expression with unit circle?!?

    You have made LOTS of mistakes. Be VERY careful with your negative signs!

    $\displaystyle \begin{align*} - \left[ \frac{\sqrt{2}}{2} - \left( - \frac{\sqrt{2}}{2} \right) \right] ^2 &= - \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \right) ^2 \\ &= - \left( \frac{2\sqrt{2}}{2} \right) ^2 \\ &= -\left( \sqrt{2} \right) ^2 \\ &= -2 \end{align*}$

    See if you can fix your second bracket - you have similar mistakes there too...
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    Re: Simplifing large trigonometric expression with unit circle?!?

    I understand the first bracket but the second one is IMPOSSIBLE, i keep getting that i am supposed to either add or subtract a rootnumber that is not possible to simplify, such as sqrt 2 with a whole number. I thought that was not possible to do??
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    Re: Simplifing large trigonometric expression with unit circle?!?

    The second one is far from impossible. Surely you know that

    $\displaystyle \begin{align*} \cos{ \left( \frac{4\pi}{3} \right) } &= \cos{ \left( \pi + \frac{\pi}{3} \right) } \\ &= -\cos{ \left( \frac{\pi}{3} \right) } \\ &= - \frac{1}{2} \end{align*}$

    and

    $\displaystyle \begin{align*} \sin{ \left( \frac{4\pi}{3} \right) } &= \sin{ \left( \pi + \frac{\pi}{3} \right) } \\ &= -\sin{ \left( \frac{\pi}{3} \right) } \\ &= -\frac{\sqrt{3}}{2} \end{align*}$

    So what is $\displaystyle \begin{align*} \cos{ \left( \frac{4\pi}{3} \right) } - \sin{ \left( \frac{4\pi}{3} \right) } \end{align*}$? What do you get when you square it? What do you get when you negate that?
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