Problem

Find the terminal point on the unit circle determined by $\frac{3 \pi}{4}$ radians.
Use exact values, not decimal approximations.
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(x, y)=\text { (‥) }
\]

Answer

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Answer

Final Answer: The terminal point on the unit circle determined by \(\frac{3 \pi}{4}\) radians is \(\boxed{\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)}\).

Steps

Step 1 :The terminal point on the unit circle determined by an angle in standard position is given by the coordinates \((\cos(\theta), \sin(\theta))\). In this case, the angle is \(\frac{3 \pi}{4}\) radians. We need to find the cosine and sine of this angle.

Step 2 :The cosine and sine of \(\frac{3 \pi}{4}\) radians are both approximately 0.707, but with opposite signs. However, the question asks for exact values, not decimal approximations.

Step 3 :We know that \(\cos(\frac{3 \pi}{4}) = -\frac{\sqrt{2}}{2}\) and \(\sin(\frac{3 \pi}{4}) = \frac{\sqrt{2}}{2}\).

Step 4 :Final Answer: The terminal point on the unit circle determined by \(\frac{3 \pi}{4}\) radians is \(\boxed{\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)}\).

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