To find the value of \(\cos \theta\) for the given point \((12\sqrt{2}, -12\sqrt{2})\) on the terminal side of \(\theta\), we first need to ensure that this point lies on a unit circle.
In a unit circle, the radius \(r\) is equal to 1. For any point \((x, y)\) on or associated with a terminal side in the coordinate system, the cosine and sine values can be derived using the radius:
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Calculate the radius \(r\) from the point \((x, y)\) using the formula: \[ r = \sqrt{x^2 + y^2} \]
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From the given point, we have \(x = 12\sqrt{2}\) and \(y = -12\sqrt{2}\).
First, calculate \(r\): \[ r = \sqrt{(12\sqrt{2})^2 + (-12\sqrt{2})^2} = \sqrt{(144 \cdot 2) + (144 \cdot 2)} = \sqrt{288 + 288} = \sqrt{576} = 24. \]
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The cosine of the angle \(\theta\) will be: \[ \cos \theta = \frac{x}{r} = \frac{12\sqrt{2}}{24} = \frac{12\sqrt{2}}{24} = \frac{\sqrt{2}}{2}. \]
Thus, the value of \(\cos \theta\) is:
\[ \cos \theta = \frac{\sqrt{2}}{2}. \]
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