In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)

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Recognize that the angle given is in radians: \(\frac{11\pi}{6}\). Since \(2\pi\) radians correspond to a full circle, we can use the periodicity of cosine to simplify the angle.

Use the periodic property of cosine: \(\cos(\theta) = \cos(\theta + 2k\pi)\) for any integer \(k\). Subtract \(2\pi\) from \(\frac{11\pi}{6}\) to find a coterminal angle within \([0, 2\pi)\).

Calculate the coterminal angle: \(\frac{11\pi}{6} - 2\pi = \frac{11\pi}{6} - \frac{12\pi}{6} = -\frac{\pi}{6}\). Since cosine is an even function, \(\cos(-\theta) = \cos(\theta)\), so \(\cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right)\).

Recall the exact value of \(\cos\left(\frac{\pi}{6}\right)\) from the unit circle or special triangles: \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\).

Therefore, the exact value of \(\cos\left(\frac{11\pi}{6}\right)\) is the same as \(\cos\left(\frac{\pi}{6}\right)\), which is \(\frac{\sqrt{3}}{2}\).

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주요 개념

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Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Radian measure relates the angle to the length of the arc on the unit circle, where 2π radians equal 360°. Understanding how to locate angles like 11π/6 on the unit circle is essential for evaluating trigonometric functions exactly.

Reference Angles

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It helps simplify the evaluation of trigonometric functions by relating them to known values in the first quadrant. For angles greater than 2π or in other quadrants, finding the reference angle allows use of standard trigonometric values.

Exact Values of Trigonometric Functions

Certain angles on the unit circle correspond to exact trigonometric values expressed in terms of square roots and fractions, such as π/6, π/4, and π/3. Knowing these exact values allows one to find cosine or sine without a calculator. For example, cos(π/6) = √3/2, which can be used with reference angles to find cos(11π/6).