In Exercises 29–51, find the exact value of each expression. Do not use a calculator. cos⁻¹ (− 1/2)

Verified step by step guidance

1. Recognize that \( \cos^{-1}(x) \) represents the angle whose cosine is \( x \).
2. Identify the range of \( \cos^{-1}(x) \), which is \([0, \pi]\) radians or \([0, 180^\circ]\).
3. Recall that \( \cos(\theta) = -\frac{1}{2} \) corresponds to angles in the second quadrant within the specified range.
4. Determine the reference angle where \( \cos(\theta) = \frac{1}{2} \), which is \( \frac{\pi}{3} \) radians or \( 60^\circ \).
5. Find the angle in the second quadrant: \( \pi - \frac{\pi}{3} \) radians or \( 180^\circ - 60^\circ \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find the angle whose cosine is a given value. For example, cos⁻¹(−1/2) asks for the angle θ where cos(θ) = −1/2. The range of the arccosine function is [0, π], which is crucial for determining the correct angle.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and their corresponding sine and cosine values. It is a circle with a radius of 1 centered at the origin of a coordinate plane. Understanding the unit circle helps in identifying the angles that yield specific cosine values, such as −1/2.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are essential for determining the exact values of trigonometric functions in different quadrants. For cos⁻¹(−1/2), the reference angle is π/3, and since cosine is negative in the second quadrant, the angle corresponding to cos(θ) = −1/2 is 2π/3.