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

Verified step by step guidance

Recognize that the problem asks to express \( \sin \frac{2\pi}{3} \) using a trigonometric identity involving \( -\frac{\pi}{6} \).

Recall the sine difference identity: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

Set \( a = \pi \) and \( b = \frac{\pi}{6} \) so that \( a - b = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \), which is equivalent to \( \frac{2\pi}{3} \) after simplification or by considering angle equivalences.

Use the identity to write \( \sin \left( \pi - \frac{\pi}{6} \right) = \sin \pi \cos \frac{\pi}{6} - \cos \pi \sin \frac{\pi}{6} \).

Identify the trigonometric functions in the expression and fill in the blanks accordingly, noting the signs and values of sine and cosine at these standard angles.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Angle Identities

These identities relate the values of trigonometric functions at different angles, often involving sums, differences, or multiples of angles. They allow simplification or transformation of expressions, such as expressing sin(2π/3) in terms of angles like π/6.

Reference Angles and Quadrants

Understanding the reference angle and the quadrant in which an angle lies helps determine the sign and value of trigonometric functions. For example, 2π/3 is in the second quadrant, where sine is positive and cosine is negative.

Co-function and Negative Angle Identities

Co-function identities relate sine and cosine of complementary angles, while negative angle identities express functions of negative angles in terms of positive angles, e.g., sin(-θ) = -sin(θ). These are useful for rewriting expressions involving negative angles.

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