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

Verified step by step guidance

1. 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.
2. Recall the definition of the arccosine function: \(\arccos(x)\) gives the angle \(\theta\) in the range \([0, \pi]\) such that \(\cos(\theta) = x\).
3. Apply this definition to \(\arccos(-1)\), which means finding the angle \(\theta\) where \(\cos(\theta) = -1\) and \(\theta\) is between \(0\) and \(\pi\).
4. 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.
5. 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.

Key Concepts

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.

Cosine Function

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.

Function Composition and Simplification

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.