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

cos⁻¹ √3/2

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Recognize that $ \cos^{-1} $ is the inverse cosine function, which gives the angle whose cosine is the given value.

Identify the range of the inverse cosine function, which is $[0, \pi]$.

Recall that $ \cos(\pi/6) = \sqrt{3}/2 $.

Since $ \pi/6 $ is within the range of $ \cos^{-1} $, we can conclude that $ \cos^{-1}(\sqrt{3}/2) = \pi/6 $.

Verify that the angle $ \pi/6 $ satisfies the condition by checking that $ \cos(\pi/6) = \sqrt{3}/2 $.

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, cos⁻¹, and tan⁻¹, are used to find angles when given a ratio of sides in a right triangle. For example, cos⁻¹(x) returns 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.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the visualization of the sine and cosine values of angles. Understanding the unit circle helps in determining the angles corresponding to specific cosine and sine values, which is crucial for evaluating inverse trigonometric functions.

Cosine Values

The cosine function relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. For common angles, such as 0°, 30°, 45°, 60°, and 90°, the cosine values are well-known. Recognizing that cos⁻¹(√3/2) corresponds to a specific angle on the unit circle is key to simplifying the expression without a calculator.

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