In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use 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 \( \frac{4\pi}{3} \) on the unit circle.

Locate the corresponding point on the unit circle for \( \frac{4\pi}{3} \), which is \( \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \).

Recall that the cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function, \( \sin(\theta) \).

Determine \( \sin\left(\frac{4\pi}{3}\right) \) using the y-coordinate of the point, which is \( -\frac{\sqrt{3}}{2} \).

Calculate \( \csc\left(\frac{4\pi}{3}\right) \) as the reciprocal of \( \sin\left(\frac{4\pi}{3}\right) \), which is \( -\frac{2}{\sqrt{3}} \).

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Key Concepts

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

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of trigonometric functions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and cosecant, relate the angles of a triangle to the lengths of its sides. For any angle θ in standard position, the sine function gives the y-coordinate and the cosine function gives the x-coordinate of the corresponding point on the unit circle. The cosecant function is the reciprocal of sine, defined as csc(θ) = 1/sin(θ), and is undefined when sin(θ) = 0.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are crucial for determining the values of trigonometric functions in different quadrants. For example, the angle 4π/3 is in the third quadrant, and its reference angle can be found by subtracting π from it, which helps in calculating the sine and cosine values based on the signs of these functions in that quadrant.

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