Find each exact function value. See Example 3.
sin π/2

Verified step by step guidance

1. Recall that the sine function, \( \sin \theta \), gives the y-coordinate of the point on the unit circle corresponding to the angle \( \theta \).
2. Identify the angle given: \( \frac{\pi}{2} \) radians, which corresponds to 90 degrees.
3. Visualize or recall the unit circle: at \( \frac{\pi}{2} \), the point on the unit circle is at the top, with coordinates \( (0, 1) \).
4. Since sine corresponds to the y-coordinate, \( \sin \frac{\pi}{2} = 1 \).
5. Therefore, the exact value of \( \sin \frac{\pi}{2} \) is 1.

Key Concepts

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 points on this circle, where π radians equal 180 degrees. Understanding the unit circle helps in finding exact trigonometric values for common angles like π/2.

Sine Function Definition

The sine of an angle in the unit circle is the y-coordinate of the corresponding point on the circle. For an angle θ, sin(θ) gives the vertical position on the unit circle, which allows exact evaluation of sine values at standard 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 and cosine values. For example, sin(π/2) equals 1. Memorizing these exact values avoids approximation and is essential for solving trigonometric problems precisely.