Find the exact value of each expression.
sin (- 5π/12)

Verified step by step guidance

1. \( \sin(-x) = -\sin(x) \). Therefore, \( \sin(-\frac{5\pi}{12}) = -\sin(\frac{5\pi}{12}) \).
2. Notice that \( \frac{5\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \).
3. \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \). Here, \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \).
4. \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \), \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
5. \( \sin(\frac{5\pi}{12}) = \sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) - \cos(\frac{\pi}{3})\sin(\frac{\pi}{4}) \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the sine, cosine, and tangent functions based on the coordinates of points on a circle with a radius of one. The angle is measured from the positive x-axis, and the coordinates of any point on the circle correspond to the cosine and sine values of that angle. Understanding the unit circle is essential for evaluating trigonometric functions for any angle, including negative angles.

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. For negative angles, the reference angle can be found by adding 2π until the angle is positive, or by considering the equivalent positive angle. This concept is crucial for determining the sine value of angles in different quadrants, as it allows us to relate the sine of the original angle to that of its reference angle.

Sine Function Properties

The sine function is periodic and has specific properties that are important for evaluating its values. It is an odd function, meaning that sin(-θ) = -sin(θ), which simplifies calculations for negative angles. Additionally, the sine function has a range of [-1, 1] and is positive in the first and second quadrants, while negative in the third and fourth quadrants. Understanding these properties helps in accurately finding the sine of angles like -5π/12.