Use identities to fill in each blank with the appropriate trigonometric function name.
sin 2π/3 = _____ (- π/6)

Verified step by step guidance

1. Recognize that \( \sin \frac{2\pi}{3} \) can be expressed using the identity \( \sin(\pi - \theta) = \sin \theta \).
2. Identify that \( \frac{2\pi}{3} = \pi - \frac{\pi}{3} \), so \( \sin \frac{2\pi}{3} = \sin(\pi - \frac{\pi}{3}) \).
3. Use the identity \( \sin(\pi - \theta) = \sin \theta \) to find that \( \sin(\pi - \frac{\pi}{3}) = \sin \frac{\pi}{3} \).
4. Recall that \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).
5. Recognize that \( \sin(-\theta) = -\sin \theta \), so \( \sin(-\frac{\pi}{6}) = -\sin \frac{\pi}{6} \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. The sine function, for example, is defined as the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is essential for evaluating angles and solving trigonometric equations.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of trigonometric functions, where the x-coordinate represents the cosine and the y-coordinate represents the sine of an angle. This concept is crucial for determining the values of trigonometric functions at various angles, including those in different quadrants.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. These identities are useful for simplifying expressions and solving trigonometric equations, making them vital for filling in blanks in trigonometric problems.