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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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