# Thread: cos^2(x) = 1/2 ... [o,2π)

1. ## cos^2(x) = 1/2 ... [o,2π)

Anyone have any idea how to solve it? Sqrt of both sides? But it's not a memory value angle... :\

cos^2(x) = 1/2

2. $\sqrt{cos^2x}=cos(x)=\sqrt{\frac{1}{2}}\rightarrow cos(x)=\frac{1}{\sqrt{2}}\rightarrow cos(x)=\frac{\sqrt{2}}{2}\rightarrow x=\frac{\pi}{4},\frac{7\pi}{4}$

3. Originally Posted by dwsmith
$cos(x)=\frac{\sqrt{2}}{2}=\frac{\pi}{4}$
I think you mean $\cos(x)=\frac{\sqrt{2}}{2}\implies x = \frac{\pi}{4}$

What about other solutions on $[0,2\pi)$ ??

4. Originally Posted by dwsmith
$\sqrt{cos^2x}=cos(x)=\sqrt{\frac{1}{2}}\rightarrow cos(x)=\frac{1}{\sqrt{2}}\rightarrow cos(x)=\frac{\sqrt{2}}{2}\rightarrow x=\frac{\pi}{4},\frac{7\pi}{4}$
The other case to consider is $\cos(x)=-\sqrt{\frac{1}{2}}$ and this is left for the OP to do.

5. Hello : $\sqrt{A^{2}}=\left | A \right |$

6. $\displaystyle \left[cos\left(\frac{3\pi}{4}\right)\right]^2=\left(\frac{-\sqrt{2}}{2}\right)^2=\frac{1}{2}$

Therefore, the angles that incorporate the negative values satisfy the equation.

7. Originally Posted by mathnoob2
Anyone have any idea how to solve it? Sqrt of both sides? But it's not a memory value angle...
As others have pointed out, yes, it is!

:\

cos^2(x) = 1/2