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Chapter 4: 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

Understand the function

The given function is the inverse cosine function, written as \(y = \arccos(x)\).
02

Define the range of the inverse cosine function

The inverse cosine function, \(y = \arccos(x)\), gives the angle \(y\) whose cosine is \(x\). By definition, this angle \(y\) is in the interval \([0, \pi]\).
03

Fill in the blank

The blank in the sentence must be the term that represents the set of all possible output values (or angles) of the function \(y = \arccos(x)\). This term is 'range'.

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Key Concepts

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

arccos function
When you see the notation \(y = \arccos(x)\), it represents the angle \(y\) such that \(\cos(y) = x\).
In simpler words, if you know a value of the cosine and want to find the corresponding angle, you would use the arccos function.
For example, if \(\cos(\pi \ 2) = 0\), then by definition \(\arccos(0) = \pi \slash 2\).
range of functions
The range of a function is the complete set of all possible output values.
Simply put, it tells you what values you can expect from the result of a function.
In the case of the arccos function, \(y = \arccos(x)\), the range is limited to the set of values \[0, \pi\].
This means that any angle you get from the arccos function will be between 0 and \pi radians.
trigonometric functions
Trigonometric functions relate the angles of a triangle to the lengths of its sides.
These functions include sine (sin), cosine (cos), and tangent (tan), among others.
The cosine function, for instance, relates the adjacent side of a right triangle to its hypotenuse.
The inverse trigonometric functions (like arccos) are used to find the angles when these ratios are known.

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Most popular questions from this chapter

Find the exact value of each expression without using a calculator or table. a. \(\arcsin (1 / 2)\) b. \(\cos ^{-1}(-1 / 2)\) c. \(\tan ^{-1}(-1)\) d. \(\sin (\pi / 3)\) e. \(\cos (-\pi / 2)\) f. \(\sin ^{-1}(-1)\)

Simplify \(\sin (-x) \cos (-x) \tan (-x)\)

The shortest side of a right triangle is \(7 \mathrm{~cm},\) and one of the acute angles is \(64^{\circ} .\) Find the length of the hypotenuse and the length of the longer leg. Round to the nearest tenth of a centimeter.

Find the exact value of each composition without using a calculator or table. $$ \cot (\operatorname{arccot}(0)) $$

Simplify \(\cos (2 y) \cos (y)-\sin (2 y) \sin (y)\)
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