Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.

cos⁻¹ (- 1/2 )

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Understand that $ \cos^{-1}(x) $ represents the angle whose cosine is $ x $.

Recall that the range of $ \cos^{-1}(x) $ is $[0, \pi]$.

Identify the angle $ \theta $ in the range $[0, \pi]$ such that $ \cos(\theta) = -\frac{1}{2} $.

Recognize that $ \cos(\pi - \theta) = -\cos(\theta) $, and since $ \cos(\frac{\pi}{3}) = \frac{1}{2} $, then $ \cos(\pi - \frac{\pi}{3}) = -\frac{1}{2} $.

Conclude that $ \theta = \pi - \frac{\pi}{3} $ is the angle in the range $[0, \pi]$ such that $ \cos(\theta) = -\frac{1}{2} $.

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Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine and arccosine, are used to find angles when given a ratio of sides in a right triangle. For example, cos⁻¹(x) gives the angle whose cosine is x. These functions are essential for solving problems where the angle is unknown, and they have specific ranges to ensure each output is unique.

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Range of Inverse Cosine

The range of the inverse cosine function, cos⁻¹(x), is restricted to the interval [0, π]. This means that when evaluating cos⁻¹(-1/2), the resulting angle must fall within this range. Understanding this range is crucial for correctly interpreting the output of the inverse cosine function.

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Unit Circle

The unit circle is a fundamental concept in trigonometry that helps visualize the values of trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. By using the unit circle, one can determine the angles corresponding to specific cosine values, such as -1/2, which corresponds to angles in the second and third quadrants.

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