In Exercises 49–59, find the exact value of each expression. Do not use a calculator. csc(-2𝜋/3)

검증된 단계별 안내

Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\).

Use the property of sine for negative angles: \(\sin(-\theta) = -\sin \theta\). So, \(\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right)\).

Find \(\sin\left(\frac{2\pi}{3}\right)\) by recognizing that \(\frac{2\pi}{3}\) is in the second quadrant where sine is positive, and it corresponds to \(\pi - \frac{\pi}{3} = \frac{2\pi}{3}\). So, \(\sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right)\).

Recall the exact value \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), so \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Combine these results to find \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{\sin\left(-\frac{2\pi}{3}\right)} = \frac{1}{-\sin\left(\frac{2\pi}{3}\right)} = \frac{1}{-\frac{\sqrt{3}}{2}}\).

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주요 개념

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

Understanding the Cosecant Function

Cosecant (csc) is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). To find csc(θ), you first determine sin(θ) and then take its reciprocal. This relationship is fundamental when evaluating trigonometric expressions without a calculator.

Evaluating Trigonometric Functions at Negative Angles

Trigonometric functions have specific properties for negative angles. For sine, sin(-θ) = -sin(θ), meaning the sine function is odd. This helps simplify expressions like csc(-2π/3) by relating them to positive angle values.

Reference Angles and Unit Circle Values

Using the unit circle, angles are measured in radians, and their sine values correspond to y-coordinates. The reference angle for 2π/3 is π/3, whose sine value is √3/2. Understanding reference angles allows exact evaluation of trigonometric functions at various angles.