Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ 2 \sin (3 x)=-1 $$

Step-by-Step Solution

Answer

The solutions to the equation \(2 \sin (3 x) = -1\) are \(x = \frac{1}{3} (\pi - \arcsin(0.5))\) and \(x = \frac{1}{3} (\pi + \arcsin(0.5))\).

Step 1: Solve the Equation

To start with, we need to solve the equation. This involves isolating \(x\). For \(2 \sin (3 x) = -1\), first, we can divide both sides by 2 to get \(\sin (3 x) = -0.5\). Then, we take the inverse sine (or arcsin) of both sides to isolate \(3x\). So, \(3x = \arcsin(-0.5)\). Next, since we are considering solutions on the interval \([0,2\pi]\), we have to consider the periodic nature of the sine function, which repeats every \(2\pi\), and the negative sign, which indicates a phase shift to the second and third quadrants (since the sine is negative in these quadrants). So, \(3x = \pi - \arcsin(0.5)\) or \(3x = \pi + \arcsin(0.5)\). Finally, we divide by 3 to find \(x\), thus \(x = \frac{1}{3} (\pi - \arcsin(0.5))\) and \(x = \frac{1}{3} (\pi + \arcsin(0.5))\).

Step 2: Graph the Equation

Now, to verify the solution with a graph, we plot \(y = 2 \sin (3x)\) and \(y = -1\) on the interval \([0,2\pi]\). All values of x at which the two graphs intersect are solutions to the original equation.

Step 3: Check for Intersection Points

We then check the graph for intersection points of \(y = 2 \sin (3x)\) and \(y = -1\), these are the solutions to the equation. They should correspond to the values of \(x\) that we found algebraically.

Key Concepts

Sine Function

The sine function is one of the most essential elements in trigonometry, appearing frequently in various problems and equations. It is a function that can be thought of as a wave that oscillates between -1 and 1 over the interval of [0, 2\pi]. This characteristic makes it very useful in modelling periodic phenomena such as sound and light waves. The key features of the sine function include:

• Amplitude: This is the height from the average value of the function to its peak, and for the standard sine function, the amplitude is 1.
• Period: The distance over which the wave repeats itself is 2\pi for the standard sine function.
• Vertical Shift: Sine waves can be shifted up or down along the y-axis, which changes their vertical midline.
• Phase Shift: They can also be shifted left or right along the x-axis, altering the point at which they start their new cycle.

Understanding how to manipulate the sine function through these features allows you to solve equations like \(2 \sin (3x) = -1\), where the sine function is transformed.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, play a crucial role when solving equations involving sine, cosine, and tangent. When you have an equation like \(\sin(3x) = -0.5\), applying the inverse trigonometric function allows you to solve for the angle \((3x)\).

• The inverse sine or arcsin function returns an angle whose sine is a given number. For example, \(\arcsin(-0.5)\) returns the angle whose sine is \(-0.5\).
• Arcsin is defined over the range of \([-\pi/2, \pi/2]\). This means solutions from inverse trigonometric functions are typically found in this restricted range.
• When dealing with negative results from the inverse sine function, remember the sine function is negative in the third and fourth quadrants, which impacts solutions by influencing the phase shifts.

Using inverse functions helps us find the principal solutions which can be expanded upon using the periodicity of the sine function.

Graphical Solutions

Graphical methods are a powerful tool for visually determining the solutions to trigonometric equations like \(2 \sin (3x) = -1\). The approach uses graphs to identify where two functions intersect. In this exercise, the function \(y = 2 \sin(3x)\) is graphed alongside the line \(y = -1\). The intersection points visually confirm the solutions found algebraically. Here are a few steps to apply graphical solutions effectively:

• Plot both functions within the specified interval \([0, 2\pi]\).
• Look for the x-values where the graphs meet. These intersections correspond to the solutions of the trigonometric equation.
• Consider the periodicity of the sine function, which will repeat every \(2\pi\). This characteristic helps predict additional solutions outside the immediate visible range.

This method is particularly useful to verify algebraic solutions and ensure no solutions within the given interval are overlooked.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. In trigonometry, functions like sine and cosine are inherently periodic, displaying this repetitive nature. Understanding periodic functions is crucial for analyzing and solving equations like \(2 \sin (3x) = -1\).

• The period of a function is the interval after which the function begins to repeat its pattern. For the standard sine function, this period is \(2\pi\).
• Modifying a sine function, as in \(sin(3x)\), affects its period, making it \(\frac{2\pi}{3}\), which means the function repeats its cycle more frequently.
• This decreased period causes more solutions within the same interval \([0, 2\pi]\). By comprehensively understanding periodicity, you can identify all possible solutions within a specified range.

Periodic functions are exquisite tools for modelling real-world phenomena that naturally exhibit regular and consistent patterns, such as sound waves and seasons.