The unit circle has been divided into twelve equal arcs, corresponding to t-values of

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

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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cos 5𝜋/6

Verified step by step guidance

Identify the position of \(5\pi/6\) on the unit circle. It is in the second quadrant.

Recall that the cosine of an angle in the unit circle is the x-coordinate of the corresponding point.

In the second quadrant, cosine values are negative.

The reference angle for \(5\pi/6\) is \(\pi/6\).

Use the known cosine value for \(\pi/6\), which is \(\sqrt{3}/2\), and apply the sign for the second quadrant to find \(\cos(5\pi/6)\).

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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 and cosine functions. Each point on the unit circle corresponds to an angle measured in radians, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value of that angle.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. On the unit circle, the cosine of an angle is the x-coordinate of the corresponding point, while the sine is the y-coordinate. Understanding these functions is essential for evaluating trigonometric expressions at specific angles, such as 5π/6.

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 in different quadrants. For example, the reference angle for 5π/6 is π/6, which allows us to find the cosine value by considering the sign based on the quadrant in which the angle lies.

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