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Chapter 5: Problem 2

Fill in the blank. The_______of \(y=\arccos (x)\) is \([0, \pi]\)

Short Answer

Expert verified
The range of \(y = \arccos(x)\) is \([0, \pi]\).

Step by step solution

01

Identify the function properties

First, recognize that the function given is the arccosine function, denoted as \(y = \arccos(x)\). This function is the inverse of the cosine function.
02

Determine the range of the arccosine function

The arccosine function, \(y = \arccos(x)\), returns the angle whose cosine is \(x\). By definition, this angle is within the interval \([0, \pi]\), which includes all possible outputs of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inverse function
An inverse function reverses the effect of the original function. If you have a function that takes an input and produces an output, the inverse function will take that output and return the original input.

For example, in the case of the cosine function and the arccosine function:
  • The cosine function, denoted as \(\texttt{cos}(x)\), takes an angle \(x\) and returns the ratio of the adjacent side to the hypotenuse in a right triangle.
  • The arccosine function, denoted as \(y = \texttt{arccos}(x)\), does the opposite. It takes a value \(x\) (which is a ratio) and returns the angle \(y\) whose cosine is \(x\).

To put it simply, if \(x = \texttt{cos}(y)\), then \(y = \texttt{arccos}(x)\). This relationship helps us understand the properties and behavior of the arccosine function.
range
The range of a function is the set of all possible outputs or values that the function can produce. For the arccosine function, \(y = \texttt{arccos}(x)\), its range is the set of all angles that the function can return.

Specifically, for \(y = \texttt{arccos}(x)\), the output is an angle \(y\) between \[0, \pi \] radians.
  • This means that when you input a value \(x\) in the domain \([-1, 1]\), the arccosine function will output an angle in radians that falls within this range.
  • The reason for this particular interval is due to the behavior of the cosine function over one period. Cosine achieves all of its possible values (from -1 to 1) exactly once as the input angle \(x\) varies from 0 to \(\pi\) radians.

Therefore, the range of the arccosine function covers all potential angles that have corresponding cosine values between -1 and 1.
cosine function
The cosine function is a fundamental trigonometric function that relates the angles of a right triangle to the lengths of its sides. Defined as \( \texttt{cos}(x) \), it follows several important properties:

  • For any angle \(x\), \( \texttt{cos}(x) \) returns the cosine value, which is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
  • The cosine function is periodic with a period of \(2\pi \), meaning it repeats its values every \(2\pi \) radians.
  • It is an even function, which implies that \( \texttt{cos}(-x) = \texttt{cos}(x) \).

The cosine function鈥檚 range is from -1 to 1 for any real number input, meaning it produces values only within this interval.

When considering the inverse, the arccosine function, we take the output of cosine (which must be between -1 and 1), and use it to find the corresponding angle, which falls within the range \[0, \pi \].

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