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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Unit Circle and Angle Measurement in Radians

The unit circle is a circle with radius 1 centered at the origin, used to define sine and cosine for all angles. Angles measured in radians correspond to arc lengths on this circle, where 2π radians equal 360 degrees. Understanding how angles wrap around the circle helps determine their position in different quadrants.

Sine and Cosine Values and Their Signs

Sine and cosine functions give the y- and x-coordinates of a point on the unit circle for a given angle. The sign (positive or negative) of sin and cos values indicates the quadrant: for example, sin is positive in quadrants I and II, while cos is positive in quadrants I and IV.

Quadrant Determination Using Trigonometric Signs

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 (-, +).

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