Evaluate each expression without using a calculator.
cos (arccos (-1))

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Recognize that the expression involves the composition of the cosine function and its inverse, arccosine. Specifically, you have \(\cos(\arccos(x))\), where \(x = -1\) in this case.

Recall the definition of the arccosine function: \(\arccos(x)\) gives the angle \(\theta\) in the range \([0, \pi]\) such that \(\cos(\theta) = x\).

Apply this definition to \(\arccos(-1)\), which means finding the angle \(\theta\) where \(\cos(\theta) = -1\) and \(\theta\) is between \(0\) and \(\pi\).

Identify the angle \(\theta\) that satisfies \(\cos(\theta) = -1\) within the principal range of arccosine. This angle is a well-known special angle on the unit circle.

Finally, substitute back into the original expression: \(\cos(\arccos(-1)) = \cos(\theta)\). Since \(\theta\) was chosen so that \(\cos(\theta) = -1\), the expression simplifies accordingly.

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arccos, reverse the effect of their corresponding trigonometric functions. For example, arccos(x) returns the angle whose cosine is x, typically within the range 0 to π for arccos.

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The cosine function relates an angle to the ratio of the adjacent side over the hypotenuse in a right triangle. It is periodic and defined for all real numbers, with values ranging between -1 and 1.

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When composing a function with its inverse, such as cos(arccos(x)), the result simplifies to x within the domain of the inverse function. This property helps evaluate expressions without a calculator.

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