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 (-11𝜋/6)

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Identify the reference angle for \(-\frac{11\pi}{6}\) on the unit circle. This angle is equivalent to \(-\frac{11\pi}{6} + 2\pi = \frac{\pi}{6}\).

Locate the point on the unit circle corresponding to \(\frac{\pi}{6}\), which is \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).

Recall that the tangent function is defined as \(\tan(\theta) = \frac{y}{x}\), where \(x\) and \(y\) are the coordinates of the point on the unit circle.

Calculate \(\tan\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\).

Use the odd property of the tangent function, \(\tan(-\theta) = -\tan(\theta)\), to find \(\tan\left(-\frac{11\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}}\).

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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 values of these functions for various angles, allowing for easy calculation of trigonometric values.

Trigonometric Function Properties

Trigonometric functions exhibit specific properties, including even and odd functions. The cosine function is even, meaning cos(-θ) = cos(θ), while the sine and tangent functions are odd, so sin(-θ) = -sin(θ) and tan(-θ) = -tan(θ). These properties are useful for simplifying calculations and finding values of trigonometric functions at negative angles.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. One full rotation (360 degrees) is equivalent to 2π radians. Understanding how to convert between these two units is essential for accurately interpreting angles on the unit circle and performing calculations involving trigonometric functions.

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