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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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