Use reference angles to find the exact value of each expression. Do not use a calculator. sin (-17𝜋/3)

Verified step by step guidance

1. First, recognize that the angle given is in radians and is negative: \(-\frac{17\pi}{3}\). To work with this angle, we want to find a coterminal angle between \(0\) and \(2\pi\) by adding multiples of \(2\pi\) until the angle is positive and within one full rotation.
2. Since one full rotation is \(2\pi = \frac{6\pi}{3}\), add \(2\pi\) repeatedly to \(-\frac{17\pi}{3}\) until the angle is between \(0\) and \(2\pi\). Calculate \(-\frac{17\pi}{3} + n \times \frac{6\pi}{3}\) for some integer \(n\).
3. Once you find the positive coterminal angle \(\theta\), determine its reference angle. The reference angle is the acute angle between \(\theta\) and the nearest x-axis (either \(0\), \(\pi\), or \(2\pi\)).
4. Identify the quadrant in which the coterminal angle lies. This is important because the sign of \(\sin(\theta)\) depends on the quadrant: positive in Quadrants I and II, negative in Quadrants III and IV.
5. Use the reference angle to find the exact value of \(\sin(\theta)\) using known sine values of special angles (like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), etc.), and apply the appropriate sign based on the quadrant.

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to a corresponding angle in the first quadrant, where trigonometric values are well-known.

Angle Coterminality and Reduction

Angles that differ by full rotations (multiples of 2π radians) share the same terminal side and thus have the same trigonometric values. Reducing an angle by adding or subtracting 2π simplifies the angle to an equivalent one within a standard interval, making calculations easier.

Sine Function Properties and Sign Determination

The sine function is periodic with period 2π and odd, meaning sin(-θ) = -sin(θ). The sign of sine depends on the quadrant of the angle: positive in the first and second quadrants, negative in the third and fourth. This helps determine the exact value after finding the reference angle.