Problem 10 - 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. \()\) $$ \cos (3 x)=0.5 $$

Step-by-Step Solution

Answer

Plotting \(y = \cos(3x)\) and locating where these values intersect with \(y = 0.5\) provides a graphical verification for the solutions of the equation \(\cos(3x) = 0.5\) within the interval \([0, 2 \pi]\).

Step 1: Graphing y = cos(3x)

Plot the function \(y = \cos(3x)\) on the interval \([0, 2 \pi]\). Remember that the function \(\cos(3x)\) will oscillate between -1 and 1 and will have three full periods between 0 and \(2 \pi\).

Step 2: Graphing y = 0.5

Plot the line \(y = 0.5\) on the same graph as \(y = \cos(3x)\) to visually locate potential solutions. Since the range of \(y = \cos(3x)\) is [-1,1], there must be intersections between \(y = 0.5\) and \(y = \cos(3x)\) in the interval.

Step 3: Locate the intersection points

The locations where the graph \(y = \cos(3x)\) intersects with \(y = 0.5\) are the solutions of the equation. Those are the x-values where \(\cos(3x) = 0.5\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In terms of graphing, the cosine function can be represented as the function \(y = \text{cos}(x)\) where \(x\) is the angle in radians.

When graphing the cosine function, particularly \(y = \text{cos}(3x)\), there's a unique pattern to be aware of. The '3' inside the cosine function affects the period of the wave. Normally, the cosine function completes one full cycle over an interval of \(2\pi\) radians. However, in this case, the function will complete three full cycles within the same interval because the period of the function is \(\frac{2\pi}{3}\).

It's essential for learners to visualize that this function will oscillate or 'wave' between 1 and -1, creating a 'hill' and a 'valley' for every period. Graphing functions like these helps to understand their behavior and can make solving equations involving the cosine function much easier.

Function Intersection Points

Intersection points on the graph are where two different functions, say \(y = f(x)\) and \(y = g(x)\), cross each other. These points are of special interest because they represent the solutions to the equation \(f(x) = g(x)\).

To find the intersection points of \(y = \text{cos}(3x)\) and \(y = 0.5\), it's most efficient to look at the graph where the two functions meet. In the given exercise, the intersections between the cosine function and the horizontal line at \(y = 0.5\) indicate where the cosine function reaches the value of 0.5 within the interval \([0, 2\pi]\).

In the case of a periodic function like the cosine function, there may be multiple intersection points within a single period, or multiple intersections throughout the plotted range. Identifying and accurately plotting these points are essential skills in understanding trigonometric functions and their applications.

Trigonometric Equation Solutions

Finding the solutions to a trigonometric equation involves identifying the values of the variable that make the equation true. In the context of \(\cos(3x) = 0.5\), the solutions are the angles for which the cosine function has the value 0.5.

Since cosine values repeat themselves due to the periodicity of the function, there can be multiple solutions within a given interval. In the interval \([0, 2\pi]\), the equation \(\cos(3x) = 0.5\) has several solutions which correspond to the intersection points discussed earlier.

This type of trigonometric equation can also be solved algebraically or by using inverse trigonometric functions, but graphing provides a clear visual representation of all possible solutions and helps confirm that none are overlooked. It's important to have a strong understanding of how these functions behave graphically to effectively find all possible solutions to trigonometric equations.