In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos(-35𝜋 / 6)

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Recognize that cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\). So, rewrite the expression as \(\cos\left(\frac{35\pi}{6}\right)\).

Since the angle \(\frac{35\pi}{6}\) is greater than \(2\pi\), find a coterminal angle by subtracting multiples of \(2\pi\) until the angle lies between \(0\) and \(2\pi\). Use the formula \(\theta_{coterminal} = \theta - 2\pi k\) where \(k\) is an integer.

Calculate the coterminal angle: \(\frac{35\pi}{6} - 2\pi \times k\). Since \(2\pi = \frac{12\pi}{6}\), find \(k\) such that the result is between \(0\) and \(2\pi\).

Once you have the coterminal angle in the first rotation, determine the reference angle by finding the acute angle between the coterminal angle and the nearest x-axis multiple of \(\pi\) (like \(\pi\) or \(2\pi\)).

Use the unit circle and the sign of cosine in the quadrant of the coterminal angle to find the exact value of \(\cos\left(\frac{35\pi}{6}\right)\), which will be the same as the original expression.

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Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles on the unit circle, including negative angles which represent clockwise rotation, is essential for evaluating trigonometric functions like cosine.

Coterminal Angles

Coterminal angles differ by full rotations of 2π radians but share the same terminal side on the unit circle. To simplify trigonometric expressions, you can add or subtract multiples of 2π to find an equivalent angle within a standard interval, making it easier to evaluate functions exactly.

Exact Values of Trigonometric Functions

Certain angles on the unit circle correspond to well-known exact values for sine and cosine, often involving fractions and square roots. Memorizing these values or knowing how to derive them allows you to find exact trigonometric values without a calculator, which is crucial for problems requiring exact answers.