Find each exact function value. See Example 2.
csc 11π/6

검증된 단계별 안내

Recall that the cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).

Identify the angle given: \(\theta = \frac{11\pi}{6}\). This angle is in radians and is located in the fourth quadrant of the unit circle.

Find the reference angle for \(\frac{11\pi}{6}\) by subtracting it from \(2\pi\): \(2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}\).

Determine the sine of the reference angle \(\frac{\pi}{6}\), which is \(\sin \frac{\pi}{6} = \frac{1}{2}\). Since \(\frac{11\pi}{6}\) is in the fourth quadrant where sine is negative, \(\sin \frac{11\pi}{6} = -\frac{1}{2}\).

Calculate \(\csc \frac{11\pi}{6}\) by taking the reciprocal of \(\sin \frac{11\pi}{6}\): \(\csc \frac{11\pi}{6} = \frac{1}{-\frac{1}{2}}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Understanding the Unit Circle

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It helps determine the sine, cosine, and other trigonometric values for angles measured in radians. Knowing the coordinates of points on the unit circle corresponding to specific angles allows you to find exact trigonometric function values.

Reciprocal Trigonometric Functions

Cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). To find csc for an angle, first find the sine value and then take its reciprocal. This relationship is essential for converting between sine and cosecant values.

Reference Angles and Quadrants

Reference angles help find trigonometric values for angles not on the first quadrant by relating them to acute angles. The sign of the function depends on the quadrant where the angle lies. For 11π/6, understanding its quadrant and reference angle is key to determining the correct sign and exact value.

Find each exact function value. See Example 2.csc 11π/6