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Chapter 7: Problem 15

Use your GDC to find approximate solution(s) for \(0

Short Answer

Expert verified
The approximate solutions are \(x_1 \approx 1.11\) and \(x_2 \approx 4.25\).

Step by step solution

01

Set up Equation

First, we want to solve the equation \( \tan x = 2 \) within the given interval \( 0 < x < 2\pi \). We're interested in finding the values of \( x \) where the tangent of \( x \) equals 2. This is suitable for using a graphing calculator or graphing software to visualize the tangent function.
02

Graph the Tangent Function

Using your Graphical Display Calculator (GDC), graph the function \( y = \tan x \) and the horizontal line \( y = 2 \) to find their intersections in the interval \( 0 < x < 2\pi \). The tangent function is periodic and will have multiple solutions across its domain.
03

Find the Intersections

Look for the points where the line \( y = 2 \) intersects the graph of \( y = \tan x \). Using the calculator, identify the \( x \)-coordinates of these intersection points. These coordinates are the solutions to \( \tan x = 2 \).
04

Calculate Specific Solutions

Use the calculation function in your GDC to find specific values of \( x \,\text{(rad)}\) at the intersections. You'll typically find two solutions within one period (\( 0 < x < \pi \)) and additional solutions in the next period (\( \pi < x < 2\pi \)).
05

Record Accurate Values

Round the \( x \,\text{(in radians)}\) values obtained to 3 significant figures. You should find two approximate solutions within the range \( 0 < x < 2\pi \). Common solutions for this kind of problem are around the angles where tangent equals 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical Display Calculator
Using a graphical display calculator (GDC) is extremely helpful when solving trigonometric equations like \( \tan x = 2 \).It's a powerful tool that simplifies the process of visualizing and calculating complex equations. To solve our particular problem, you'll graph both \( y = \tan x \) and the horizontal line \( y = 2 \) on your GDC. Look for where these plots intersect, as the \( x \)-coordinates of these intersections are the solutions to the equation.
  • Helps visualize functions and their behaviors
  • Identifies intersection points that translate to solutions
  • Saves time compared to manual calculations
Start graphing by accessing the calculator's function mode and inputting the equations. Super simple, right? Don't worry if you don't find all the solutions immediately; just zoom in and adjust for better visibility.
Periodicity of Tangent Function
The tangent function, denoted as \( \tan x \), is known for its periodic nature. Unlike sine or cosine, which have a period of \( 2\pi \), the tangent function repeats every \( \pi \) units. This means that within any interval \( 0 < x < 2\pi \), the tangent graph will complete two full cycles.To identify solutions for \( \tan x = 2 \), you must consider this periodicity:
  • The first cycle is from \( 0 < x < \pi \)
  • The second cycle is from \( \pi < x < 2\pi \)
This periodic behavior results in multiple potential solutions of \( x \) within the specified interval (\( 0 < x < 2\pi \)). It's essential to remember this characteristic when interpreting intersections from your GDC.
Significant Figures Rounding
When working on mathematical solutions, especially with trigonometric functions, it's crucial to have precision. For the equation \( \tan x = 2 \), your solutions should be expressed to three significant figures. This rounding practice not only presents clearer data but also meets academic and practical requirements for precision.
  • Significant figures reflect measurement accuracy
  • Typically, they include the first non-zero digits
  • Rounding must be done correctly to maintain solution integrity
Being accurate yet concise is a skill, especially when using a GDC. Pay attention to your results and ensure they accurately meet the three-significant-figure requirement. It's the difference between a precise mathematical answer and an approximation.
Solution Intervals
A crucial aspect of solving \( \tan x = 2 \) is understanding the solution intervals. You are asked to find solutions in the interval \( 0 < x < 2\pi \). This range covers two periods of the tangent function, allowing you to identify potentially multiple solutions due to its periodic nature.
  • First interval: \( 0 < x < \pi \)
  • Second interval: \( \pi < x < 2\pi \)
By breaking down the intervals, you acknowledge that the function resets and repeats, a fundamental property of periodic functions. Make use of both cycles your GDC displays to comprehensively find all solutions in the given range. Remember, it's about finding the right solutions within specified constraints to understand the full scope of the equation.

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Most popular questions from this chapter

Solve each equation for \(x\) in the given interval. Give answers exactly, if possible. Otherwise, give answers accurate to three significant figures. $$2 \sin ^{2} x-\cos x=1,0 \leqslant x<2 \pi$$

Specify in which quadrant(s) an angle \(\theta\) in standard position could be given the stated conditions. $$\sin \theta<0 \text { and } \tan \theta>0$$

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Prove each identity. $$(\tan A-\sec A)^{2}=\frac{1-\sin A}{1+\sin A}$$

Solve each equation for \(x\) in the given interval. Give answers exactly, if possible. Otherwise, give answers accurate to three significant figures. $$2 \cos x+\sin 2 x=0,-180^{\circ}
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