Find each exact function value.
sin ( ―5π/6)

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Recall the definition of the sine function on the unit circle: \(\sin(\theta)\) represents the y-coordinate of the point on the unit circle at an angle \(\theta\) measured from the positive x-axis.

Recognize that the angle given is \(-\frac{5\pi}{6}\), which is a negative angle. Negative angles are measured clockwise from the positive x-axis.

Convert the negative angle to a positive coterminal angle by adding \(2\pi\): \(-\frac{5\pi}{6} + 2\pi = \frac{7\pi}{6}\).

Identify the reference angle for \(\frac{7\pi}{6}\), which is \(\frac{7\pi}{6} - \pi = \frac{\pi}{6}\), and determine the sign of sine in the third quadrant (where \(\frac{7\pi}{6}\) lies).

Use the known sine value for the reference angle \(\frac{\pi}{6}\), which is \(\frac{1}{2}\), and apply the sign from the third quadrant (sine is negative there) to find \(\sin\left(-\frac{5\pi}{6}\right) = -\frac{1}{2}\).

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Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like -5π/6 on the unit circle is essential for finding exact trigonometric values.

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Introduction to the Unit Circle

Reference Angles

A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps simplify the evaluation of trigonometric functions by relating them to known values in the first quadrant, regardless of the original angle's quadrant or sign.

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Reference Angles on the Unit Circle

Sine Function and Its Sign in Different Quadrants

The sine function corresponds to the y-coordinate on the unit circle. Its value depends on the quadrant of the angle: positive in the first and second quadrants, negative in the third and fourth. Knowing the sign of sine for the angle -5π/6 is crucial for determining the exact value.

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