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

Verified step by step guidance

1. Step 1: Recognize that the cosine function is periodic with a period of \$2\pi\(. This means \)\cos(\theta) = \cos(\theta + 2\pi k)\( for any integer \)k$.
2. Step 2: Simplify the angle \(-\frac{35\pi}{6}\) by adding \$2\pi\( repeatedly until the angle is within the range \)[0, 2\pi)$.
3. Step 3: Calculate the equivalent positive angle by finding the remainder of \(-\frac{35\pi}{6}\) divided by \$2\pi$.
4. Step 4: Once the angle is simplified to a positive angle within \([0, 2\pi)\), use the unit circle to find the cosine of the resulting angle.
5. Step 5: Determine the exact value of the cosine for the simplified angle using known values from the unit circle.

Key Concepts

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 interpretation of the sine, cosine, and tangent functions. Angles measured in radians correspond to points on the unit circle, allowing for the determination of exact values for trigonometric functions.

Cosine Function

The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to an angle θ. It is periodic with a period of 2π, meaning that cos(θ) = cos(θ + 2kπ) for any integer k. Understanding the properties of the cosine function, including its even nature (cos(-θ) = cos(θ)), is essential for evaluating expressions involving negative angles.

Angle Reduction

Angle reduction involves simplifying angles to find their equivalent values within a standard range, typically between 0 and 2π. For example, to evaluate cos(-35π/6), one can add or subtract multiples of 2π to bring the angle within this range. This technique is crucial for finding exact trigonometric values without a calculator, as it allows for easier reference to known angles on the unit circle.