Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.
s = π/2

검증된 단계별 안내

Identify the angle given: here, the angle \( s = \frac{\pi}{2} \) radians, which corresponds to 90 degrees on the unit circle.

Recall the definitions of sine, cosine, and tangent on the unit circle: for an angle \( s \), \( \sin s \) is the y-coordinate, \( \cos s \) is the x-coordinate, and \( \tan s = \frac{\sin s}{\cos s} \) provided \( \cos s \neq 0 \).

Locate the point on the unit circle corresponding to \( s = \frac{\pi}{2} \). This point is at the top of the circle, where the coordinates are \( (0, 1) \).

From the coordinates, determine \( \sin s = 1 \) and \( \cos s = 0 \).

Calculate \( \tan s = \frac{\sin s}{\cos s} = \frac{1}{0} \), which is undefined because division by zero is not possible.

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주요 개념

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

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to arc lengths on this circle, where π radians equal 180 degrees. Understanding the position of an angle like π/2 on the unit circle helps determine the sine, cosine, and tangent values.

Definition of Sine, Cosine, and Tangent

Sine of an angle is the y-coordinate, cosine is the x-coordinate of the corresponding point on the unit circle. Tangent is the ratio of sine to cosine (tan s = sin s / cos s). These definitions allow exact evaluation of trigonometric functions for special angles such as π/2.

Exact Values of Trigonometric Functions at Special Angles

Certain angles like 0, π/6, π/4, π/3, and π/2 have well-known exact sine, cosine, and tangent values. For s = π/2, sin s = 1, cos s = 0, and tan s is undefined due to division by zero. Memorizing or deriving these values is essential for solving trigonometric problems precisely.

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.s = π/2