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

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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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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.

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Introduction to the Unit Circle

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.

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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.

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