In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.
6 3 2 3 6 6 3 2 3 6

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
csc 7𝜋/6

검증된 단계별 안내

Identify the angle given: \(t = \frac{7\pi}{6}\). This angle is in radians and corresponds to a point on the unit circle.

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

Locate the coordinates of the point on the unit circle corresponding to \(t = \frac{7\pi}{6}\). The coordinates are \((x, y) = (\cos t, \sin t)\).

Find the \(y\)-coordinate (which is \(\sin \frac{7\pi}{6}\)) from the unit circle. This value will be used to calculate \(\csc \frac{7\pi}{6}\).

Calculate \(\csc \frac{7\pi}{6}\) by taking the reciprocal of \(\sin \frac{7\pi}{6}\), i.e., \(\csc \frac{7\pi}{6} = \frac{1}{\sin \frac{7\pi}{6}}\). If \(\sin \frac{7\pi}{6} = 0\), then \(\csc \frac{7\pi}{6}\) is undefined.

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주요 개념

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Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where one full rotation equals 2π radians. Understanding how to locate an angle t on the unit circle is essential for determining the corresponding coordinates (x, y) that represent cosine and sine values.

Trigonometric Functions and Their Coordinates

On the unit circle, the x-coordinate corresponds to cos(t) and the y-coordinate corresponds to sin(t). Other trigonometric functions like cosecant (csc t) are defined in terms of sine, with csc t = 1/sin t. Knowing how to use the coordinates to find these values is crucial, especially to identify when functions are undefined (e.g., when sin t = 0).

Evaluating Trigonometric Functions at Specific Angles

To evaluate functions like csc(7π/6), first locate the angle 7π/6 on the unit circle, find the sine value from the y-coordinate, and then compute the reciprocal for cosecant. Recognizing common reference angles and their sine values helps simplify this process and determine if the function is defined or undefined.