Given a point on the unit circle at an angle of $frac{5pi}{4}$ radians, find the coordinates of this point. Then, if the unit circle is flipped about the y-axis, determine the new coordinates of the original point after the flip.

First, find the coordinates of the point on the unit circle at $\frac{5\pi}{4}$ radians. This point can be represented as:

$ (\cos(\frac{5\pi}{4}), \sin(\frac{5\pi}{4})) $

We know that:

$ \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} $

$ \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} $

Thus, the coordinates at $\frac{5\pi}{4}$ radians are:

$ (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $

Next, when the unit circle is flipped about the y-axis, the x-coordinate of the point changes sign, but the y-coordinate remains the same. Therefore, the new coordinates after the flip are:

$ (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $

First, find the coordinates of the point at $frac{5pi}{4}$ radians on the unit circle:

$ (cos(frac{5pi}{4}), sin(frac{5pi}{4})) $

We calculate:

$ cos(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

$ sin(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

So, the coordinates are:

$ (-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $

After flipping the unit circle about the y-axis, the coordinates become:

$ (frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $

At $frac{5pi}{4}$ radians, the coordinates are:

$ (-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $

After flipping about the y-axis:

$ (frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}) $

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