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

Key Concepts

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 example, the cosecant function is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). Understanding these functions is crucial for evaluating expressions like csc(7π/6) by determining the sine value at the specified angle.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They help in determining the values of trigonometric functions for angles greater than 90 degrees or less than 0 degrees. For instance, to find csc(7π/6), one would first identify its reference angle, which is π/6, and then use the sine value of this angle to compute the cosecant.