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Chapter 7: Problem 59

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval \([0,2 \pi)\). Express solutions to four decimal places. $$x^{2}+\sin x-x^{3}-\cos x=0$$

Short Answer

Expert verified
The solutions are approximately where the graph intersects the x-axis in \[0, 2\pi\]. Use a graphing calculator for precise values.

Step by step solution

01

Title - Enter the Equation

Enter the given equation into the graphing calculator. The function to input is \( f(x) = x^2 + \sin(x) - x^3 - \cos(x) \).
02

Title - Set the Interval

Set the interval for the x-values on the graphing calculator to \[0, 2\pi\]. This can usually be done in the settings or window options of the calculator.
03

Title - Graph the Equation

Graph the function using the graphing calculator. Look for the points where the graph intersects the x-axis. These points represent the solutions to the equation.
04

Title - Find the Intersections

Using the graphing calculator's 'zero' or 'root' function, find the exact x-values where the graph intersects the x-axis within the interval \[0, 2\pi\].
05

Title - Record the Solutions

Record the x-values found in Step 4 to four decimal places. These are the solutions to the equation within the specified interval.

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Key Concepts

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

graphing calculator
A graphing calculator is a powerful tool that allows you to visualize mathematical equations and functions. This device can graph complex functions, including polynomial and trigonometric functions, making it easier to find solutions to equations that cannot be solved algebraically. Using a graphing calculator, you can:
  • Enter the equation directly into the calculator.
  • Set the interval for x-values to focus on a specific range.
  • Graph the function and identify where it intersects the x-axis.
These intersections are the solutions to your equation. Remember, the accuracy of the solutions depends on how precisely you set your window and how refined your graphing tool is. For our problem, we set the interval from 0 to \( 2\pi \). This helps us to find the roots within one complete cycle of the trigonometric function.
trigonometric functions
Trigonometric functions, like \( \sin(x) \) and \( \cos(x) \), are periodic and oscillate between -1 and 1. These functions are integral in solving various types of equations, especially those involving periodic behavior. In the equation \( f(x) = x^2 + \sin(x) - x^3 - \cos(x) \), we combine polynomials with sine and cosine functions.
  • \( \sin(x) \) reaches its maximum value of 1 at \( \frac{\pi}{2} \) and its minimum value of -1 at \( \frac{3\pi}{2} \).
  • \( \cos(x) \) reaches its maximum value of 1 at 0 and its minimum value of -1 at \( \pi \).
Understanding these properties helps when analyzing the graph. The combination of these functions with a polynomial can create multiple intersections with the x-axis in the interval \( [0, 2\pi) \).
roots of equations
The roots of an equation are the x-values where the function equals zero. These are often referred to as the solutions of the equation. When using a graphing calculator:
  • Set up the function and graph it.
  • Identify the intersections with the x-axis, as these points are where the function equals zero.
  • Use the 'zero' or 'root' feature of the calculator to find the exact values.
In our exercise, the graph of \( x^2 + \sin(x) - x^3 - \cos(x) \) intersects the x-axis at the roots within the interval \( [0, 2\pi) \). These solutions can then be recorded to four decimal places, providing accurate answers.
interval notation
Interval notation is a way of representing a range of values. It's particularly useful in defining the domain or range over which you need to find solutions. In our exercise, we use the interval \( [0, 2\pi) \) which means:
  • We start at 0 and go up to, but do not include, \( 2\pi \).
  • Square brackets \( [ \) indicate that the endpoint is included.
  • Parentheses \( ) \) indicate that the endpoint is not included.
This notation helps us to focus on solving the equation within a specific segment of the function's graph. By setting the interval from \( [0, 2\pi) \), we specifically look for roots in one complete oscillation cycle of the trigonometric functions involved.

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Most popular questions from this chapter

The following function approximates the average monthly temperature \(y\) (in "F) in Vancouver, Canada. Here \(x\) represents the month, where \(x=1\) corresponds to January, \(x=2\) corresponds to February, and so on. $$ f(x)=14 \sin \left[\frac{\pi}{6}(x-4)\right]+50 $$ When is the average monthly temperature (a) \(64^{\circ} \mathrm{F}\) (b) \(39^{\circ} \mathrm{F}\) ?

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \sin \theta=2 \cos 2 \theta$$

Verify that each equation is an identity. $$\cot ^{2} \frac{x}{2}=\frac{(1+\cos x)^{2}}{\sin ^{2} x}$$

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\cos \left(270^{\circ}-\theta\right)$$

Verify that each equation is an identity. $$\frac{\tan (\alpha+\beta)-\tan \beta}{1+\tan (\alpha+\beta) \tan \beta}=\tan \alpha$$
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