In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.tan 5𝜋/3

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Identify the angle 5\(\pi\)/3 on the unit circle. Notice that 5\(\pi\)/3 is equivalent to 2\(\pi\) - \(\pi\)/3, which places it in the fourth quadrant.

Locate the coordinates on the unit circle for the angle \(\pi\)/3, which are (1/2, \(\sqrt{3}\)/2).

Since 5\(\pi\)/3 is in the fourth quadrant, the coordinates are (1/2, -\(\sqrt{3}\)/2) because the y-coordinate is negative in the fourth quadrant.

Recall that the tangent function is defined as \(\tan\)(\(\theta\)) = \(\frac{y}{x}\).

Substitute the coordinates (1/2, -\(\sqrt{3}\)/2) into the tangent function: \(\tan\)(5\(\pi\)/3) = \(\frac{-\sqrt{3}\)/2}{1/2}.

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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 from the positive x-axis, allowing for easy calculation of trigonometric functions.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, these functions can be defined as follows: sine is the y-coordinate, cosine is the x-coordinate, and tangent is the ratio of sine to cosine. Understanding these functions is crucial for solving problems involving angles and their corresponding values.

Even and Odd Properties

Trigonometric functions exhibit specific symmetry properties: sine is an odd function (sin(-x) = -sin(x)), while cosine is an even function (cos(-x) = cos(x)). This means that the sine function reflects across the origin, while the cosine function reflects across the y-axis. These properties can simplify calculations and help find values of trigonometric functions at negative angles or angles greater than 2π.

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