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

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Recall that the function \(\cos^{-1}(x)\), also known as arccosine, gives the angle \(\theta\) in the range \(0 \leq \theta \leq \pi\) such that \(\cos(\theta) = x\).

Identify the value inside the arccosine function: here, it is \(-\frac{1}{2}\), so we want to find \(\theta\) such that \(\cos(\theta) = -\frac{1}{2}\).

Recall the unit circle values where cosine equals \(-\frac{1}{2}\). Cosine corresponds to the x-coordinate on the unit circle, so find the angles in \([0, \pi]\) where the x-coordinate is \(-\frac{1}{2}\).

From known special angles, \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\) and \(\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}\), but since the range of \(\cos^{-1}\) is \([0, \pi]\), only \(\frac{2\pi}{3}\) is valid.

Therefore, the exact value of \(\cos^{-1}\left(-\frac{1}{2}\right)\) is the angle \(\theta = \frac{2\pi}{3}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function returns the angle whose cosine is a given value. It is defined for inputs between -1 and 1, and its output range is from 0 to π radians (0° to 180°). Understanding this helps find the angle corresponding to a specific cosine value without a calculator.

Unit Circle and Reference Angles

The unit circle represents angles and their cosine and sine values on a circle of radius 1. Knowing key points where cosine values are common fractions, like -1/2, helps identify exact angle measures. Reference angles simplify finding angles in different quadrants.

Exact Values of Cosine for Special Angles

Certain angles, such as 30°, 60°, and 120°, have well-known exact cosine values like ±1/2 or ±√3/2. Recognizing these values allows you to determine the exact angle for cos⁻¹(-1/2) without approximation or calculators.

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