In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(2𝜋/3)

Verified step by step guidance

1. Identify the given angle: \(\frac{2\pi}{3}\). This angle is in radians and is between \(\pi/2\) and \(\pi\), which means it lies in the second quadrant.
2. Find the reference angle for \(\frac{2\pi}{3}\). The reference angle \(\theta_r\) in the second quadrant is calculated by subtracting the angle from \(\pi\): \(\theta_r = \pi - \frac{2\pi}{3}\).
3. Simplify the reference angle: \(\theta_r = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3}\).
4. Recall the sine value of the reference angle \(\frac{\pi}{3}\). From the unit circle, \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
5. Determine the sign of sine in the second quadrant. Since sine is positive in the second quadrant, \(\sin \frac{2\pi}{3} = +\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).

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 angles in different quadrants to their acute counterparts, allowing the use of known values for sine, cosine, and tangent.

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin, where angles are measured in radians. Understanding the position of angles like 2π/3 on the unit circle helps determine the sine and cosine values based on coordinates of points on the circle.

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent depends on the quadrant in which the angle lies. For example, sine is positive in the first and second quadrants and negative in the third and fourth, which is essential when using reference angles to find exact values.