Problem 7 - 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. \()\) $$ 2 \sin ^{2}(2 x)=1 $$

Step-by-Step Solution

Answer

The solutions to the given equation can be found at the x-coordinates of the intersection points of the graph with the x-axis in the interval \([0, 2\pi]\). These solutions can be calculated precisely using a graphing tool or estimated visually.

Step 1: Define the Function

Let's write the given equation in terms of \(y\). The function then becomes: \[y=2\sin^2(2x) - 0.5\]

Step 2: Graph the Function

Draw the sine wave function in the interval \([0, 2\pi]\). Remember that sine wave functions have a periodic property and the function \(y= 2\sin^2(2x) - 0.5\) should also reflect this periodic characteristic.

Step 3: Find the Intersection Points

To find solutions of the equation, locate the intersection points of the function graph with the x-axis (since the value of \(y = 0\) at these points). These intersection points represent the solutions \((x)\) to the equation in the interval \([0, 2\pi]\).

Key Concepts

Graphical Solution of Equations

Understanding the graphical solution of equations is essential in visualizing how mathematical equations relate to geometric representations. In trigonometry, this involves plotting the function on a coordinate plane and identifying key features like peaks, valleys, and points of intersection.

For instance, when we are given the trigonometric equation, we can express it as a function, like so: \[ y = 2\sin^2(2x) - 0.5 \] and then plot this function over a specific interval. The graphical method enables us to see where the graph crosses the x-axis within the interval \[0, 2\pi\]. These crossing points are the visual solutions to the equation, which we can then verify algebraically or numerically. It's a handy method for ensuring that no solutions are missed, particularly within a given range, and for gaining an intuitive understanding of solution multiplicity and distribution.

Sine Wave Functions

Sine wave functions are fundamental to understanding trigonometry. They represent periodic oscillations, like the ebb and flow of tides or the cycles of a pendulum. The basic form of a sine function is \[ y = \sin(x) \], but it can be modified in various ways to shift, stretch, or reflect the wave.

In our example, the sine function is manipulated as \[ y = 2\sin^2(2x) \], meaning it has been scaled and its frequency adjusted. Different parameters inside the sine function control the wave's amplitude (height), period (width of one complete cycle), phase shift (horizontal shifting), and vertical shift. By exploring how changes to these parameters affect the wave's shape, students can develop a deeper understanding of how the sine function behaves graphically and consequently, how to anticipate the solutions to related equations.

Periodicity of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The sine function, for instance, completes a full cycle over an interval of \[2\pi\] radians. This property of periodicity implies that trigonometric equations will often have infinitely many solutions unless restricted to a specific domain.

For our exercise's function \[ y = 2\sin^2(2x) \], the period is transformed due to the coefficient in the argument of the sine function. The periodicity helps predict the number of times the graph will intersect with any horizontal line (like y=0.5) within a given interval. In this case, the function's periodicity can be thought of as the DNA of the equation - every full 'strand' or cycle holds the key information repeated throughout the graph. Grasping this concept allows students to find all possible solutions in a given interval by recognizing these repeating patterns.

Intersection Points in Trigonometry

Intersection points in trigonometry are where a trigonometric function's graph intersects a line or another graph. These points are of special interest as they often represent the solutions of trigonometric equations. In the context of our function \[ y = 2\sin^2(2x) - 0.5 \], the intersection points with the x-axis (\(y=0\)) are the 'x' values that solve the equation.

By graphing the function across its period, which in this case has been altered from the standard sine function, we can determine where these intersections occur. This visual approach complements algebraic methods and supports a comprehensive understanding of the functions involved. Recognizing these points enhances problem-solving abilities, as students learn to equate the intersections with specific angle measures or 'x' values that satisfy the original trigonometric equation within the specified domain.