/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 90 Solve for \(\boldsymbol{x}\) : ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 90

Solve for \(\boldsymbol{x}\) : $$ \cos ^{-1} x>\cos ^{-1} x^{2} $$

Short Answer

Expert verified
The inequality is valid when \(0 \leq x < 0.5\)

Step by step solution

01

Identifying the Problem

Here, we have an inequality: \(\cos ^{-1}(x) > \cos ^{-1}(x^{2})\). The first step is to focus on the term \(\cos ^{-1}(x)\) and \(\cos ^{-1}(x^{2})\). Considering the range of \(\cos ^{-1}(z)\) is \([0 ,\pi]\), we know that \(0 \leq x \) and \(x \leq 1 \). Hence, we have that \(x^{2}\) lies within the interval of \([ 0 , 1]\) too.
02

Solving the inequality

After identifying the range, we can use this in our inequality. If \(x\) and \(x^{2}\) must be in the interval \([0, 1]\), \(x\) must be less than or equal to \(x^{2}\) in the mentioned interval if \(x\) >= 0.5. This means that \(\cos ^{-1}(x) > \cos ^{-1}(x^{2})\) for \(0 \leq x < 0.5\) as \(\cos^{-1}(x)\) is decreasing in the interval \([0,1]\).
03

Conclusion

Putting everything together, we have that the original inequality is valid when \(0 \leq x < 0.5\)

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical expressions involving the symbols ">", "<", ">=", or "<=".
They compare two values or expressions to determine their relative size. In the context of the exercise, we're given an inequality:
  • \(\cos^{-1}(x) > \cos^{-1}(x^2)\)
Here, we compare the inverse cosine of \(x\) with the inverse cosine of \(x^2\). This requires us to think about how these values change within their domain.

When solving inequalities:
  • Always consider the domain and range of the involved functions.
  • Be careful when manipulating the inequality as it can affect the direction of the inequality sign.
Handling inequalities with trigonometric functions requires understanding their key properties and behaviors.
Exploring the Arccosine Function
The arccosine function, represented as \(\cos^{-1}(x)\), is the inverse of the cosine function. It allows you to find the angle whose cosine is \(x\).

Key Characteristics of Arccosine:
  • The output or range of \(\cos^{-1}(x)\) is \([0, \pi]\), which represents angles in radians.
  • The arccosine function is decreasing in the interval \([0, 1]\), meaning larger values of \(x\) result in smaller angle sizes.
Understanding these properties helps us see why \(\cos^{-1}(x) > \cos^{-1}(x^2)\) when \(0 \leq x < 0.5\). Since the function is decreasing, a smaller value of \(x\) results in a larger angle, hence the inequality.
Clarifying Domain and Range
The domain and range are fundamental concepts in understanding inverse trigonometric functions.

Domain of Arccosine Function:
  • For \(\cos^{-1}(x)\), the domain is \([-1, 1]\).
  • In our exercise, the focus is within the domain \([0, 1]\) as we consider real numbers for \(x\).

Range of Arccosine Function:
  • The range is \([0, \pi]\). Values from this range can help us solve the inequality correctly.
These concepts are crucial when analyzing and solving the inequality involving \(\cos^{-1}(x)\). By ensuring that \(x\) and \(x^2\) both stay within \([0, 1]\), we can correctly interpret their arccosine values and apply them to the inequality analysis.

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Most popular questions from this chapter

Prove that: $$ \tan ^{-1}\left(\frac{1-x}{1+x}\right)-\tan ^{-1}\left(\frac{1-y}{1+y}\right)=\sin ^{-1}\left(\frac{y-x}{\sqrt{\left(1+x^{2}\right)\left(1+y^{2}\right)}}\right) $$

Prove that: If \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\pi\) then prove that \(x \sqrt{1-x^{2}}+y \sqrt{1-y^{2}}+z \sqrt{1-z^{2}}=2 x y z\)

Find the values of: $$ \cos ^{-1}(\sin (-5))+\sin ^{-1}(\cos (-5)) $$

Prove that \(\sin \left(3 \sin ^{-1}\left(\frac{1}{3}\right)\right)=\frac{23}{27}\)

Find the values of: \(\tan ^{-1}(\tan 1)+\tan ^{-1}(\tan 2)\) \(\quad+\tan ^{-1}(\tan 3)+\tan ^{-1}(\tan 4)\)
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