/*! 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 31 Solve \((\mathbf{a}) f(x)=0,(\ma... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 31

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=\sin ^{2} x \cos x-\cos x$$

Short Answer

Expert verified
(a) \(x=\frac{\pi}{2}, \frac{3\pi}{2}\); (b) \(x\in(\frac{\pi}{2}, \frac{3\pi}{2})\); (c) \(x\in[0,\frac{\pi}{2})\cup(\frac{3\pi}{2},2\pi)\)."

Step by step solution

01

Factor the Expression

We start by factoring the expression given for \(f(x)\). The expression is \(f(x) = \sin^2 x \cos x - \cos x\). We notice that \(\cos x\) is a common factor, thus:\[f(x) = \cos x (\sin^2 x - 1)\]
02

Recognize Trigonometric Identity

Recognize that \(\sin^2 x - 1\) can be rewritten using the Pythagorean identity \(\sin^2 x = 1 - \cos^2 x\). Hence,\[\sin^2 x - 1 = -(\cos^2 x - 1) = -(1 - \cos^2 x)\] simplifies to \(-\cos^2 x\). Thus, we rewrite our function:\[f(x) = \cos x (-\cos^2 x) = -\cos^3 x\]
03

Solve for (a) \(f(x)=0\)

Set \(-\cos^3 x = 0\). This gives us \(\cos x = 0\). We solve for \(x\) over the interval \([0, 2\pi)\).\[\cos x = 0 \Rightarrow x = \frac{\pi}{2}, \frac{3\pi}{2}\]
04

Solve for (b) \(f(x)>0\)

Consider where \(-\cos^3 x > 0\). This inequality holds when \(\cos^3 x < 0\), which occurs when \(\cos x < 0\).Find \(x\) where \(\cos x < 0\) within \([0, 2\pi)\):- The cosine function is negative in the second and third quadrants: - Solutions in the interval are: - From \(\pi/2\) to \(3\pi/2\)So, \(x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)\).
05

Solve for (c) \(f(x)

Now consider where \(-\cos^3 x < 0\). This inequality holds true when \(\cos^3 x > 0\), which occurs when \(\cos x > 0\).Find \(x\) where \(\cos x > 0\) within \([0, 2\pi)\):- The cosine function is positive in the first and fourth quadrants: - Solutions in the interval are: - From \(0\) to \(\pi/2\) and from \(3\pi/2\) to \(2\pi\)So, \(x \in [0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi)\).

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Key Concepts

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

Factoring Expressions
When dealing with algebraic or trigonometric expressions, factoring is often one of the first approaches to simplify the problem. It involves breaking down a complex expression into simpler components or factors.
In the given function, \( f(x) = \sin^2 x \cos x - \cos x \), the common factor is \( \cos x \). Factoring it out, we get:
  • \( f(x) = \cos x (\sin^2 x - 1) \)
Factoring simplifies the problem by reducing the number of trigonometric terms we need to consider. It helps us easily identify and apply trigonometric identities, which is our next step.
Trigonometric Identities
Trigonometric identities are equations that relate various trigonometric functions to one another. They are crucial when simplifying expressions or proving other trigonometric equations. The Pythagorean identity is especially useful in this exercise:
  • \( \sin^2 x = 1 - \cos^2 x \)
Using this identity, the expression \( \sin^2 x - 1 \) transforms into a simpler form:
  • \( \sin^2 x - 1 = -(1 - \cos^2 x) = -\cos^2 x \)
Thus, the original function becomes:
  • \( f(x) = \cos x (-\cos^2 x) = -\cos^3 x \)
Understanding and applying trigonometric identities streamline the process of solving equations by reducing the complexity of expressions.
Inequalities
Inequalities in trigonometry involve understanding when certain conditions about an expression hold true.
For the function \( f(x) = -\cos^3 x \), we solve three types of inequalities:
  • When \( f(x) = 0 \), or equivalently \( -\cos^3 x = 0 \), it implies \( \cos x = 0 \).
  • When \( f(x) > 0 \), meaning \( -\cos^3 x > 0 \), it translates to \( \cos^3 x < 0 \), so \( \cos x < 0 \).
  • When \( f(x) < 0 \), which is \( -\cos^3 x < 0 \), it indicates \( \cos^3 x > 0 \), so \( \cos x > 0 \).
Each inequality guides us to solutions in different intervals based on the behavior of the cosine function over \([0, 2\pi)\). Understanding these conditions helps us find solutions within specified ranges.
Interval Notation
Interval notation is a concise way of writing subsets of the real number line. It is a handy tool in mathematics to indicate the set of solutions or domain of a function. Here鈥檚 how interval notation works in this context:
  • The interval \( [0, 2\pi) \) specifies the domain we are interested in, namely from \( 0 \) (included) to \( 2\pi \) (excluded).
  • For \( f(x) = 0 \), solutions are at the boundaries where \( \cos x = 0 \), which are \( x = \frac{\pi}{2}, \frac{3\pi}{2} \).
  • For \( f(x) > 0 \), the solutions where \( \cos x < 0 \) occur in the interval \( (\frac{\pi}{2}, \frac{3\pi}{2}) \).
  • When \( f(x) < 0 \), \( \cos x > 0 \), the intervals are \( [0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi) \).
Interval notation provides an organized approach to express solution sets and ensures clarity when describing continuous subsets of numbers.

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Most popular questions from this chapter

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Solve each problem. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of the slight variation in the frequency. This phenomenon can be seen on a graphing calculator. (a) Consider two tones with frequencies of 220 and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t \quad\) and \(\quad P_{2}=0.005 \sin 446 \pi t\) respectively. A graph of \(P_{1}+P_{2}\) as \(Y_{3}\) felt by an eardrum over the 1 -second interval \([0.15,1.15]\) is shown here. How many beats are there in 1 second? (Graph can't copy) (b) Repeat part (a) with frequencies of 220 and 216 (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.
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