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Chapter 8: Problem 78

Use graphs to determine whether there are solutions for each equation in the interval \([0,1] .\) If there are solutions, use the graphing utility to find them accurately to two decimal places. (a) \(1 /\left(\sin ^{-1} x+\cos ^{-1} x\right)=4 x^{3}\) (b) \(1 /\left(\sin ^{-1} x+\cos ^{-1} x\right)=5 x^{3}\)

Short Answer

Expert verified
Equation (a) has a solution at \( x \approx 0.56 \), but equation (b) has no solution within [0,1].

Step by step solution

01

Understand the equations

We have two equations to analyze using a graph. The equations are: (a) \( \frac{1}{\sin^{-1} x + \cos^{-1} x} = 4x^3 \) and (b) \( \frac{1}{\sin^{-1} x + \cos^{-1} x} = 5x^3 \). Both equations involve the expression \( \sin^{-1} x + \cos^{-1} x \), which is a constant value of \( \frac{\pi}{2} \) for all \( x \) in the domain [0,1]. Thus, both equations reduce to finding intersections with a constant value of \( \frac{2}{\pi} \).
02

Setup graphing functions

For equation (a), define the function \( f(x) = \frac{2}{\pi} \) and \( g(x) = 4x^3 \). For equation (b), define \( f(x) = \frac{2}{\pi} \) and \( g(x) = 5x^3 \). The goal is to determine when \( f(x) = g(x) \) within the interval \([0, 1]\).
03

Graph the equations

Plot the graphs of \( y = 4x^3 \) and \( y = \frac{2}{\pi} \) for equation (a), and \( y = 5x^3 \) and \( y = \frac{2}{\pi} \) for equation (b). The constant \( \frac{2}{\pi} \) is approximately 0.6366.
04

Analyze the graphs

For equation (a), check for intersection points between \( y = 4x^3 \) and the horizontal line \( y = \frac{2}{\pi} \) within the interval \([0,1]\). Similarly, for equation (b), check for intersections between \( y = 5x^3 \) and \( y = \frac{2}{\pi} \).
05

Find intersections (Equation a)

Graphically, observe the intersection of the graphs from equation (a). The intersection occurs where \( y = 4x^3 \) meets \( y = \frac{2}{\pi} \). Using the graphing utility find the approximation to two decimal places.
06

Find intersections (Equation b)

Do the same for equation (b) to find intersection points using a graphing utility, noting if \( y = 5x^3 \) meets \( y = \frac{2}{\pi} \) within \([0,1]\).
07

Conclude solutions

For equation (a), the intersection occurs at approximately \( x \approx 0.56 \). For equation (b), there is no intersection within the interval [0,1] since the line \( y = 5x^3 \) stays below \( y = \frac{2}{\pi} \).

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Key Concepts

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

Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find angles when the values of trigonometric ratios are known. In this exercise, the expressions \( \sin^{-1} x + \cos^{-1} x \) are used, which equals a constant \( \frac{\pi}{2} \) for all \( x \) within the domain [0, 1]. This property simplifies our equations significantly, as it turns the left side of our equations into a constant value. This is critical because it helps focus our solution approach on analyzing and graphing the right side of the equation.
  • \( \sin^{-1} x \) is known as arcsine and gives an angle whose sine is \( x \).
  • \( \cos^{-1} x \) is known as arccosine and gives an angle whose cosine is \( x \).
  • The sum of arcsine and arccosine for any \( x \) in [0, 1] is \( \frac{\pi}{2} \).
Understanding these relationships is crucial for solving problems involving inverse trigonometric functions, as they often transform complex expressions into simpler forms that are easier to work with.
Interval Analysis
Interval analysis helps us examine the behavior of functions over specific intervals. In our exercise, we are particularly interested in the interval \([0,1]\). This involves determining if solutions鈥攕pecifically intersections鈥攅xist for the transformed equations over this interval.
Our task is to find out if the functions intersect within the chosen interval. If they intersect, that location will be our solution.
  • The first step is understanding the behavior of \( 4x^3 \) and \( 5x^3 \) within \([0,1]\).
  • Both functions start at zero and increase, but the rate of increase is different. \( 5x^3 \) grows faster compared to \( 4x^3 \).
  • Analyzing where approximately these functions might intersect \( y = \frac{2}{\pi} \) is crucial.
This in-depth interval analysis ensures we can verify the presence of solutions and accurately determine their location when using graphing utilities.
Graphing Utilities
Graphing utilities are powerful tools that allow us to visualize functions and find intersection points, which are potential solutions to equations. In this exercise, we use graphing to solve equations within the interval \([0,1]\) by plotting the transformed equations.Utilizing tools like graphing calculators or software helps us:
  • Accurately plot functions such as \( y = 4x^3 \) and \( y = \frac{2}{\pi} \) or \( y = 5x^3 \) and \( y = \frac{2}{\pi} \).
  • Easily manipulate the view to zoom in on intersections for more precise readings.
  • Compute intersections to two decimal places, which is often a requirement for precise mathematical solutions.
By using graphing utilities, we efficiently find the exact solutions or confirm the absence of intersections, streamlining the problem-solving process and enhancing our understanding of the behavior of these functions in their specified intervals.

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Most popular questions from this chapter

Solve the equations on the interval \([0,2 \pi]\) as follows. Graph the expression on each side of the equation and then zoom in on the intersection points until you are certain of the first three decimal places in each answer. For instance, for Exercise \(53,\) when you graph the two equations \(y=\cos x\)and \(y=0.623\) on the interval \([0,2 \pi],\) you 'll see that there are two intersection points. The \(x\) -coordinates of these points are roots of the equation \(\cos x=0.623\). $$\cos (\sin x)=\sin x$$

If \(\cos \theta=x-1\) and \(0<\theta<\pi / 2,\) express \(2 \theta-\cos 2 \theta\) as a function of \(x.\)

Prove each of the following double-angle formulas. Hint: As in the text, replace \(2 \theta\) with \(\theta+\theta,\) and use an appropriate addition formula. (a) \(\cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta\) (b) \(\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}\)

In this exercise we show that the irrational number \(\cos \frac{2 \pi}{7}\) is a root of the cubic equation $$8 x^{3}+4 x^{2}-4 x-1=0$$ (a) Prove the following two identities: $$\begin{array}{l}\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta \\\\\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1 \end{array}$$ (b) Let \(\theta=2 \pi / 7 .\) Use the reference-angle concept [not the formulas in part (a)] to explain why \(\cos 3 \theta=\cos 4 \theta\) (c) Now use the formulas in part (a) to show that if \(\theta=2 \pi / 7,\) then \(8 \cos ^{4} \theta-4 \cos ^{3} \theta-8 \cos ^{2} \theta+3 \cos \theta+1=0\) (d) Show that the equation in part (c) can be written \((\cos \theta-1)\left(8 \cos ^{3} \theta+4 \cos ^{2} \theta-4 \cos \theta-1\right)=0\) Conclude that the value \(x=\cos \frac{2 \pi}{7}\) satisfies the equation \(8 x^{3}+4 x^{2}-4 x-1=0\) (e) Use your calculator (in the radian mode) to check that \(x=\cos \frac{2 \pi}{7}\) satisfies the cubic equation \(8 x^{3}+4 x^{2}-4 x-1=0 .\) Remark: An interesting fact about the real number \(\cos \frac{2 \pi}{7}\) is that it cannot be expressed in terms of radicals within the real-number system.

Consider the equation \(\sin x \cos x=1\) (a) Square both sides and then replace \(\cos ^{2} x\) by \(1-\sin ^{2} x\) Show that the resulting equation can be written \(\sin ^{4} x-\sin ^{2} x+1=0\). (b) Show that the equation \(\sin ^{4} x-\sin ^{2} x+1=0\) has no solutions. Conclude from this that the original equation has no real-number solutions.
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