Find the exact value of each expression.
sin (π/12)

Verified step by step guidance

1. Recognize that \( \frac{\pi}{12} \) is an angle that can be expressed as a difference of two special angles: \( \frac{\pi}{4} - \frac{\pi}{6} \).
2. Use the sine difference identity: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
3. Substitute \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{6} \) into the identity: \( \sin(\frac{\pi}{4} - \frac{\pi}{6}) = \sin(\frac{\pi}{4}) \cos(\frac{\pi}{6}) - \cos(\frac{\pi}{4}) \sin(\frac{\pi}{6}) \).
4. Recall the exact values: \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), and \( \sin(\frac{\pi}{6}) = \frac{1}{2} \).
5. Substitute these values into the expression: \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, making it essential for evaluating trigonometric functions like sin(π/12).

Angle Sum and Difference Identities

Angle sum and difference identities are formulas that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine and cosine of those angles. For example, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). These identities are particularly useful for finding the exact values of trigonometric functions for angles that are not standard, such as π/12, by expressing them as the sum or difference of known angles.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often expressed in terms of square roots. For instance, sin(π/12) can be calculated using known angles like π/6 and π/4. Understanding how to derive these exact values using identities and the unit circle is crucial for solving trigonometric problems accurately.