Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\frac{1+\sin x}{\cos ^{2} x}=2$$

Step-by-Step Solution

Answer

The precise answers will vary depending on the specificities of the graphing utility used. However, using the above method and a suitable graphing utility, it is possible to approximate the solutions to the given trigonometric equation.

Step 1: Re-write the equation

First, adjust the given equation \(\frac{1+\sin x}{\cos ^{2} x}=2\) by collecting all the terms on one side. This can be achieved by subtracting 2 from both sides to get \(\frac{1+\sin x}{\cos ^{2} x}-2=0\).

Step 2: Simplify the equation

The equation can be simplified to \(1+\sin x-2\cos^{2}x=0\). This simplification was done by multiplying both sides of the previous equation by \(\cos^{2}x\), which removes the denominator.

Step 3: Graph the simplified equation

Plot the graph of the function \(1+\sin x-2\cos^{2}x\) on a graphing utility. Ensure to graph this in the interval [0, 2π).

Step 4: Find the zeroes of the function

Using the zero or root feature of the graphing utility, determine the \(x\) -intercepts or zeroes of this function. These x-intercepts will be the approximate solutions to the original trigonometric equation.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool that can help visualize mathematical equations, particularly for trigonometric and algebraic functions. It serves both as a calculator and a graph plotter, providing a visual representation that can aid in understanding and solving equations. To use a graphing utility effectively, you input your equation and specify the interval of interest, in this case \( [0, 2\pi) \).

• This makes it possible to see how the function behaves within that domain.
• Graphing utilities often offer features to adjust the scale and appearance of the plot.
• When applied to trigonometric equations, expect to see sinusoidal curves, indicating periodic behavior.

Visualizing equations can reveal key features such as intercepts, turning points, and periodicity, providing a deeper understanding of the function's nature. It also allows for the determination of approximate solutions by identifying where the function crosses the x-axis, these points where it intersects the x-axis are crucial in solving the equation.

X-Intercepts

The x-intercepts of a graph are specific points where the graph crosses the x-axis. These points hold significant importance as they represent the solutions to the equation when it is set to zero.

• For our trigonometric equation \( \frac{1+\sin x}{\cos ^{2} x}=2 \), after simplifying and setting it to zero, the x-intercepts give the values of \(x\) that solve the equation.
• By observing the graph where it touches or crosses the x-axis, these intercepts become visible.
• Each intercept corresponds to a possible solution of the equation within the given interval.

Understanding x-intercepts is important for solving equations graphically. Graphing utility displays these intercepts, offering an intuitive way to identify solutions efficiently.

Approximate Solutions

Finding exact solutions to trigonometric equations graphically can sometimes be challenging. However, a graphing utility provides approximate solutions by indicating where the graph intersects the x-axis within the specific interval.

• These are considered "approximate" because visually determining the exact point of intersection can be limited by the graphing resolution and scale.
• In the resolved exercise, using features like zoom and trace can improve the accuracy of these approximations, but they remain estimations.
• Such solutions are often expressed as decimal values rounded to a certain number of significant figures.

When precision is not the primary goal, these approximate solutions serve well, especially for practical applications or when an analytical method is cumbersome to apply.

Simplifying Equations

Simplifying equations is the foundational step in solving trigonometric problems like the one given. To simplify, you alter the equation into a form that's easier to manage, often by consolidating similar terms or eliminating denominators.

• For this exercise, simplifying involved transforming \( \frac{1+\sin x}{\cos ^{2} x}=2 \) into \(1+\sin x-2\cos^{2}x=0 \) by subtracting 2 from both sides and multiplying both sides by \( \cos^2 x \).
• Eliminating the fraction simplifies the function, making it possible to graph directly without dealing with complex expressions.
• By minimizing complexity, you reduce potential calculation errors and streamline the process of finding intercepts.

Simplifying aligns the equation to be suitable for graphing, paving the way for efficient analysis through methods such as graphing utilities.