Find each exact function value. See Example 2. cos (―4π/3)

Verified step by step guidance

1. Recall that the cosine function is periodic with period \(2\pi\), so \(\cos(\theta) = \cos(\theta + 2k\pi)\) for any integer \(k\). This can help simplify the angle if needed.
2. Identify the angle \(-\frac{4\pi}{3}\) on the unit circle. Since the angle is negative, it means we rotate clockwise from the positive x-axis.
3. Convert the negative angle to a positive coterminal angle by adding \(2\pi\): \(-\frac{4\pi}{3} + 2\pi = \frac{2\pi}{3}\).
4. Evaluate \(\cos\left(\frac{2\pi}{3}\right)\) by recognizing that \(\frac{2\pi}{3}\) is in the second quadrant where cosine values are negative, and it corresponds to a reference angle of \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\).
5. Use the known cosine value for the reference angle \(\frac{\pi}{3}\), which is \(\frac{1}{2}\), and apply the sign based on the quadrant to find \(\cos\left(-\frac{4\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like -4π/3 on the unit circle helps determine the corresponding coordinates and trigonometric values.

Reference Angles and Quadrants

Reference angles are the acute angles formed between the terminal side of a given angle and the x-axis. Knowing the quadrant where the angle lies is essential because the signs of sine and cosine depend on the quadrant. For negative angles, rotation is clockwise, affecting the quadrant placement.

Cosine Function on the Unit Circle

The cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. To find cos(-4π/3), identify the point on the unit circle at -4π/3 radians and read its x-coordinate. This value gives the exact cosine function value.