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 \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{\sin\left(-\frac{2\pi}{3}\right)} = \frac{1}{-\sin\left(\frac{2\pi}{3}\right)} = \frac{1}{-\frac{\sqrt{3}}{2}}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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.