To find \(\cos \theta\) given that the y-coordinate of the point where the terminal side of \(\theta\) intersects the unit circle is \(2\sqrt{2}\), we need to clarify the context. In a unit circle, the coordinates of the point of intersection are represented as \((\cos \theta, \sin \theta)\) and the radius is 1.
The fact that the y-coordinate (which is \(\sin \theta\)) is given as \(2\sqrt{2}\) suggests an inconsistency because the maximum value for \(\sin \theta\) (or any y-coordinate on the unit circle) should be 1.
However, assuming you mean to determine the \(\cos\) value based on the scenario depicted with an acute angle and actual unit circle properties, I suspect you may have intended a different value that is possible within \([-1, 1]\).
To work through this theoretically:
- If we imagine a larger circle or stretch the values, we would use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).
- Since \(\sin \theta = 2\sqrt{2}\), this value cannot correspond to any point on the unit circle but rather suggests a misunderstanding.
Please double-check the context or rephrase the values related to angles or coordinates in the unit or standard circles you’re working within. Remember: for \(\sin \theta\), its value must absolutely remain confined to \([-1, 1]\).
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