For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.
s = 65

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Understand that the angle given, \(s = 65\) radians, is quite large since one full rotation (circle) is \(2\pi\) radians, approximately \(6.283\) radians. To find the equivalent angle within one full rotation, reduce \(s\) by subtracting multiples of \(2\pi\) until the result lies between \(0\) and \(2\pi\) radians.

Calculate the equivalent angle \(s_{\text{equiv}}\) using the formula: \(s_{\text{equiv}} = s - 2\pi \times n\), where \(n\) is the largest integer such that \(s_{\text{equiv}}\) is between \(0\) and \(2\pi\). This step helps to find the coterminal angle within the first rotation.

Use a calculator to find \(\sin(s_{\text{equiv}})\) and \(\cos(s_{\text{equiv}})\). These values will help determine the quadrant of the angle because the signs of sine and cosine vary by quadrant.

Recall the sign rules for sine and cosine in each quadrant: - Quadrant I: \(\sin > 0\), \(\cos > 0\) - Quadrant II: \(\sin > 0\), \(\cos < 0\) - Quadrant III: \(\sin < 0\), \(\cos < 0\) - Quadrant IV: \(\sin < 0\), \(\cos > 0\)

Based on the signs of \(\sin(s_{\text{equiv}})\) and \(\cos(s_{\text{equiv}})\), identify the quadrant in which the angle \(s\) lies.

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Each quadrant of the coordinate plane has a unique combination of signs for sine and cosine. By calculating sin s and cos s, and noting their signs, one can identify the quadrant where the angle s lies: Quadrant I (+, +), II (+, -), III (-, -), or IV (-, +).

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