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Chapter 4: Problem 82

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cos \frac{35 \pi}{6}$$

Short Answer

Expert verified
The exact value of the cosine of \(35 \pi /6\) is \(-\sqrt{3}/2\).

Step by step solution

01

Convert to Unit Circle

The first step is to convert the angle to a value that falls within the first rotation of the unit circle (from \(0\) to \(2 \pi\)). This can be done by finding the remainder when dividing \(35 \pi /6\) by \(2 \pi\). This is equivalent to finding the remainder when dividing \(35/6\) by \(2. Using modular arithmetic, we find that: \[35/6 \mod 2 = 35/6 - 2*5 = 5/6\]So, the angle \(35 \pi /6\) is equivalent to \(5 \pi /6\) within the first rotation of the unit circle.
02

Find the Reference Angle

The reference angle is the acute angle the terminal side of the angle makes with the x-axis. Because our angle is now \(5 \pi /6\), the terminal side lies in the second quadrant making an acute angle of \(\pi - 5\pi /6 = \pi/6\) with the negative x-axis.
03

Evaluate the Cosine

The cosine of an angle in the second quadrant is negative, and the absolute value will be the same as that computed in the first quadrant. In the first quadrant, \(\cos (\pi /6) = \sqrt{3}/2\). Therefore \[\cos (35 \pi /6) = \cos (5 \pi /6) = -\sqrt{3}/2\]This is the exact value for the cosine.

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