Find the exact value of each expression.
sin (13π/12)

Verified step by step guidance

1. Recognize that the angle \( \frac{13\pi}{12} \) is not one of the standard angles on the unit circle, so we need to express it as a sum or difference of angles whose sine values we know exactly.
2. Rewrite \( \frac{13\pi}{12} \) as \( \pi + \frac{\pi}{12} \) or as a sum of two angles such as \( \frac{3\pi}{4} + \frac{\pi}{3} \) or \( \pi - \frac{\pi}{12} \). For this problem, use the sum \( \frac{3\pi}{4} + \frac{\pi}{3} \) because both \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) are standard angles.
3. Apply the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Here, \( a = \frac{3\pi}{4} \) and \( b = \frac{\pi}{3} \).
4. Substitute the known exact values: \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), and \( \cos \frac{\pi}{3} = \frac{1}{2} \).
5. Combine these values into the formula: \( \sin \left( \frac{3\pi}{4} + \frac{\pi}{3} \right) = \sin \frac{3\pi}{4} \cos \frac{\pi}{3} + \cos \frac{3\pi}{4} \sin \frac{\pi}{3} \), then simplify the expression step-by-step to find the exact value.

Key Concepts

Unit Circle and Radian Measure

The unit circle represents angles in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like 13π/12 on the unit circle helps in determining the sine value by relating it to known reference angles.

Angle Sum and Difference Identities

These identities allow the calculation of trigonometric functions for angles expressed as sums or differences of standard angles. For example, sin(13π/12) can be rewritten as sin(π + π/12) or sin(3π/4 + π/6) to use known sine and cosine values.

Exact Values of Sine and Cosine for Special Angles

Certain angles like π/6, π/4, and π/3 have well-known exact sine and cosine values. Using these values in combination with angle sum or difference formulas enables finding the exact sine of non-standard angles such as 13π/12.