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.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3

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Identify the angle given: \(t = \frac{4\pi}{3}\). Locate this angle on the unit circle diagram.

Find the coordinates corresponding to \(t = \frac{4\pi}{3}\) on the unit circle. From the image, the coordinates are \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).

Recall that for any angle \(t\) on the unit circle, \(\sin t\) is the y-coordinate of the point. So, \(\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}\).

The cosecant function is the reciprocal of sine, so \(\csc t = \frac{1}{\sin t}\). Therefore, \(\csc \frac{4\pi}{3} = \frac{1}{\sin \frac{4\pi}{3}}\).

Substitute the sine value into the reciprocal to express \(\csc \frac{4\pi}{3}\) as \(\csc \frac{4\pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}\). Simplify this expression to get the final form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t (in radians) measured from the positive x-axis. The coordinates (x, y) of each point represent the cosine and sine of the angle t, respectively.

Trigonometric Functions and Their Values

Trigonometric functions such as sine, cosine, and cosecant are defined based on the coordinates of points on the unit circle. For an angle t, sin(t) = y, cos(t) = x, and csc(t) = 1/sin(t). Understanding these relationships allows calculation of function values using the unit circle coordinates.

Evaluating Cosecant Function

The cosecant function, csc(t), is the reciprocal of sine: csc(t) = 1/sin(t). To find csc(4π/3), first identify the sine value at 4π/3 from the unit circle coordinates, then take its reciprocal. If sin(t) = 0, csc(t) is undefined.

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