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

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Recall that the sine, cosine, and tangent functions are periodic with period \(2\pi\). This means for any real number \(s\), \(\sin(s + 2\pi) = \sin s\), \(\cos(s + 2\pi) = \cos s\), and \(\tan(s + 2\pi) = \tan s\).

Since \(s = 2\pi\), recognize that this corresponds to one full rotation around the unit circle, bringing the angle back to the starting point.

Evaluate \(\sin(2\pi)\) by considering the \(y\)-coordinate of the point on the unit circle at angle \(2\pi\). The point is \((\cos 2\pi, \sin 2\pi)\).

Evaluate \(\cos(2\pi)\) by considering the \(x\)-coordinate of the point on the unit circle at angle \(2\pi\).

Find \(\tan(2\pi)\) using the definition \(\tan s = \frac{\sin s}{\cos s}\), substituting the values found for \(\sin(2\pi)\) and \(\cos(2\pi)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on the unit circle, where the x-coordinate gives the cosine value and the y-coordinate gives the sine value of the angle.

Periodic Properties of Trigonometric Functions

Sine, cosine, and tangent functions are periodic, meaning their values repeat at regular intervals. For sine and cosine, the period is 2π, so sin(s + 2π) = sin s and cos(s + 2π) = cos s, which helps find exact values for angles like s = 2π.

Definition of Tangent in Terms of Sine and Cosine

Tangent of an angle s is defined as tan s = sin s / cos s, provided cos s ≠ 0. Understanding this relationship allows calculation of tangent values once sine and cosine are known, especially for standard angles on the unit circle.

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