Problem 50
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. $$2 \sec ^{2} x+\tan ^{2} x=3$$
Step-by-Step Solution
Verified Answer
The solutions of the equation \(2\sec^{2}x + \tan^{2}x = 3\) are approximately the \(x\)-coordinates of the intercepts of the graph of \(3\tan^{2}x - 1 = 0\).
1Step 1: Rewrite the Equation
The first step is to collect all terms on one side of the equation to form a new equation. So the new trigonometric equation is: \(2\sec^{2}x + \tan^{2}x - 3 = 0\)
2Step 2: Simplify the Equation
It's important to simplify the equation for easier management. Since \(\sec^{2}x = 1 + \tan^{2}x\), the equation can be rewritten as: \(2(1 + \tan^{2}x) + \tan^{2}x - 3 = 0\) or \(2\tan^{2}x + 2 + \tan^{2}x - 3 = 0\). This simplifies to \(3\tan^{2}x - 1 = 0\)
3Step 3: Graph the Simplified Equation
Next, graph the simplified equation \(y = 3\tan^{2}x - 1\). This is a transformation of the graph of the basic function \(y = \tan x\), where the graph has been stretched vertically by a factor of 3, and then shifted vertically down by 1 unit.
4Step 4: Use the zero or root feature
Finally, use the zero or root feature of the graphing utility to find the approximate \(x\)-intercepts of the graph of \(y = 3\tan^{2}x - 1\). These \(x\)-intercepts are the approximate solutions of the original equation.
Key Concepts
Secant FunctionTangent FunctionGraphing UtilitiesSolving Equations
Secant Function
The secant function, denoted as \( \sec(x) \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, expressed as \( \sec(x) = \frac{1}{\cos(x)} \). This means that wherever the cosine function is zero, the secant function will be undefined. The secant function has a period of \( 2\pi \), repeating its values over every \( 2\pi \) interval. Unlike cosine, it ranges from negative to positive infinity.
When working with secant functions in trigonometric equations, it is useful because it connects to other functions like cosine and tangent, offering more strategies for solving complex equations. In our exercise, we used the identity \( \sec^{2}(x) = 1 + \tan^{2}(x) \) to simplify and rearrange the equation, allowing us to express it in terms of tangent for easier handling.
When working with secant functions in trigonometric equations, it is useful because it connects to other functions like cosine and tangent, offering more strategies for solving complex equations. In our exercise, we used the identity \( \sec^{2}(x) = 1 + \tan^{2}(x) \) to simplify and rearrange the equation, allowing us to express it in terms of tangent for easier handling.
Tangent Function
The tangent function, represented by \( \tan(x) \), is another fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function plays a crucial role due to its periodic behavior, repeating every \( \pi \), and its vertical asymptotes, which occur where the cosine function is zero.
In the solution to our exercise, the equation was simplified using the relationship \( \sec^{2}(x) = 1 + \tan^{2}(x) \), reducing it to \( 3\tan^{2}(x) - 1 = 0 \). This allowed us to focus on the tangent function, a common strategy due to its simpler behavior and periodicity that makes it easier to graph, analyze, and find solutions.
Understanding the behavior of the tangent function, such as where it increases or has asymptotes, helps in predicting the form and position of solutions when using graphing techniques.
In the solution to our exercise, the equation was simplified using the relationship \( \sec^{2}(x) = 1 + \tan^{2}(x) \), reducing it to \( 3\tan^{2}(x) - 1 = 0 \). This allowed us to focus on the tangent function, a common strategy due to its simpler behavior and periodicity that makes it easier to graph, analyze, and find solutions.
Understanding the behavior of the tangent function, such as where it increases or has asymptotes, helps in predicting the form and position of solutions when using graphing techniques.
Graphing Utilities
Graphing utilities are powerful tools for visualizing mathematical functions. They are particularly helpful for understanding complex equations by graphically showing the behavior of functions and their interactions. In trigonometric problems, graphing utilities can illustrate functions like tangent and secant across specific intervals, such as \([0, 2\pi)\).
In our exercise, graphing utilities were used after transforming the equation \( 3\tan^{2}(x) - 1 = 0 \) to visually identify where the graph crosses the x-axis. Points where the graph of the function meets the x-axis correspond to real solutions or roots of the equation. Graphing utilities provide features like the zero finder, which automatically detects these intercepts, simplifying the solution process significantly.
In our exercise, graphing utilities were used after transforming the equation \( 3\tan^{2}(x) - 1 = 0 \) to visually identify where the graph crosses the x-axis. Points where the graph of the function meets the x-axis correspond to real solutions or roots of the equation. Graphing utilities provide features like the zero finder, which automatically detects these intercepts, simplifying the solution process significantly.
- Show entire function behavior and highlights asymptotic behavior.
- Helps approximate numerical solutions visually.
- Useful in checking the correctness of algebraic manipulations.
Solving Equations
Solving trigonometric equations often involves multiple methods, from algebraic manipulation to graphical solutions. The main goal is to isolate the variable on one side of the equation, and it may require transforming identities, factoring, or using function relationships.
In our specific task, we started by rearranging the terms, applying the identity \( \sec^{2}(x) = 1 + \tan^{2}(x) \), simplifying to \( 3\tan^{2}(x) - 1 = 0 \), which set the stage for solving. This process often results in a simpler expression that retains equivalent solutions but is easier to handle, especially regarding graphing.
In our specific task, we started by rearranging the terms, applying the identity \( \sec^{2}(x) = 1 + \tan^{2}(x) \), simplifying to \( 3\tan^{2}(x) - 1 = 0 \), which set the stage for solving. This process often results in a simpler expression that retains equivalent solutions but is easier to handle, especially regarding graphing.
- Algebraic methods: Use identities and simplification to reduce equation complexity.
- Graphical methods: Use graphs to identify roots or solutions visually.
- Critical thinking: Choose the best approach based on equation form and context.
Other exercises in this chapter
Problem 50
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