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Chapter 7: Problem 18

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

Short Answer

Expert verified
The solutions are approximately \(x \approx 0.524, x \approx 3.665, x \approx 2.618, x \approx 5.760\).

Step by step solution

01

Rearrange the Equation

The given equation is \(3 \tan^2 x = 1\). First, divide both sides of the equation by 3 to isolate \(\tan^2 x\): \[ \tan^2 x = \frac{1}{3}\]
02

Solve for \(\tan x\)

Take the square root of both sides to solve for \(\tan x\): \( \tan x = \pm \sqrt{\frac{1}{3}} \) This gives two possible solutions for \(\tan x\): \( \tan x = \frac{1}{\sqrt{3}} \) or \( \tan x = -\frac{1}{\sqrt{3}} \).
03

Using the GDC to Find Solutions

Use your graphing calculator (GDC) to determine \(x\) for both positive and negative solutions within the interval \(0 < x < 2\pi\). Start with \( \tan x = \frac{1}{\sqrt{3}} \): - The first solution is \( x_1 \approx 0.524\) (nearest to \( \frac{\pi}{6} \)). - The second solution occurs at one full cycle of the tangent function later: \( x_2 \approx 3.665\). For \( \tan x = -\frac{1}{\sqrt{3}} \): - The first solution is \( x_3 \approx 2.618\) (nearest to \( 5\pi/6 \)).- The second solution occurs at another full cycle later: \( x_4 \approx 5.760\).
04

Verify Solutions Using GDC

Verify the approximate values obtained using the GDC by substituting them back into the equation \(3 \tan^2 x = 1\). Ensure the calculator indicates that \(3 \tan^2(0.524)\), \(3 \tan^2(3.665)\), \(3 \tan^2(2.618)\), and \(3 \tan^2(5.760)\) are all approximately equal to 1.

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Key Concepts

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

Understanding the Tangent Function
The tangent function, denoted as \( \tan x \), is one of the primary trigonometric functions. It originates from the ratio of the opposite side to the adjacent side in a right triangle. The tangent function is periodic, with a period of \( \pi \). This means that it repeats its values every \( \pi \, \text{radians} \). This periodic nature is key, as it allows for multiple solutions within a given interval, such as the interval from \( 0 \) to \( 2\pi \) in our exercise. Features of the tangent graph:
  • It has no defined value at angles where the cosine is zero, such as \( \frac{\pi}{2}, \frac{3\pi}{2} \).
  • It repeatedly goes from negative to positive and crosses zero, at each interval of \( \pi \).
  • This characteristic helps when solving equations, because with each cycle of \( \pi \), \( \tan x \) assumes the same values and patterns.
Becoming familiar with the graph and behavior of the tangent function is vital for efficiently solving trigonometric equations.
Utilizing a Graphical Calculator
A graphical calculator (GDC) is an essential tool in solving trigonometric equations, especially when approximating solutions. It is particularly useful for visualizing functions, like \( \tan x \), and finding intersections that represent solutions. When using a GDC:
  • First, graph the function vertically, setting it equal to the values solved algebraically, like \( \pm \frac{1}{\sqrt{3}} \).
  • Adjust the settings to fit the interval of interest, here within \( 0 < x < 2\pi \) radians.
  • Use the calculator鈥檚 function to find intersection points accurately, which give the approximate solutions for \( x \).
These steps make it easier to manage complex functions and obtain a precise understanding of the solutions.
Navigating Trigonometric Solutions
Finding solutions to trigonometric equations involves more than just algebraic manipulation. It includes understanding the trigonometric properties, periodicity, and using tools like a GDC. For the tangent equation \( 3 \tan^2 x = 1 \), finding solutions required:
  • Isolating \( \tan x \) by taking the square root of both sides, yielding \( \tan x = \pm \frac{1}{\sqrt{3}} \).
  • Mapping these values onto the tangent function to find intersection points within the desired interval.
The periodic nature of tangent ensures that solutions are repeated within each \( \pi \) interval, providing multiple possible angles within the range from \( 0 \) to \( 2\pi \). This repetition is crucial as it often results in more than one solution being valid.
The Square Root Method Explained
The square root method is an important algebraic technique employed in solving equations, specifically when dealing with squares of variables, such as \( \tan^2 x \). To apply it efficiently:
  • Start by isolating the squared term: here, \( \tan^2 x = \frac{1}{3} \).
  • Take the square root of both sides to solve for \( \tan x \), resulting in two values: \( \tan x = \pm \frac{1}{\sqrt{3}} \).
Why two values? Because squaring eliminates negative values, introducing the need to consider positive and negative roots when reverting to linear equations. Using this method helps in effectively narrowing down possible solutions to equations, combining mathematical rigor with logical solution seeking.

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Most popular questions from this chapter

Use the periodic properties of the sine and cosine functions to find the exact value of \(\sin x\) and \(\cos x\) $$x=\frac{13 \pi}{6}$$

Simplify each expression. $$\cos \theta-\cos \theta \sin ^{2} \theta$$

Solve for \(x\) in the indicated interval. $$\tan x \csc x=5,0

The screen in a movie cinema is 7 metres from top to bottom and is positioned 3 metres above the horizontal floor of the cinema. The first row of seats is 2.5 metres from the wall that the screen is on and the rows are each 1 metre apart. You decide to sit in the row where you get the 'best'view, that is, where the angle subtended at your eyes by the screen is a maximum. When you are sitting in one of the cinema's seats your eyes are 1.2 metres above the horizontal floor. a) Let \(x\) be the distance that you are from the wall that the screen is on, and \(\theta\) is the angle subtended at your eyes by the screen. (i) Draw a clear diagram to represent all the information given. (ii) Find a function for \(\theta\) in terms of \(x\) (iii) Sketch a graph of the function. (iv) Use your GDC to find the value of \(x\) that gives a maximum for \(\theta\). In which row should you sit? b) Suppose that, starting with the first row of seats, the floor of the cinema is sloping upwards at an angle of \(20^{\circ}\) above the horizontal. Again, the first row of seats is 2.5 metres from the wall that the screen is on and the rows are each 1 metre apart measured along the sloping floor. Let \(x\) be the distance from where the first row starts and your seat in the cinema. (i) Draw a clear diagram to represent all the information given. (ii) Find a function for \(\theta\) in terms of \(x\). (iii) Sketch a graph of the function. (iv) Use your GDC to find the value of \(x\) that gives a maximum for \(\theta\). In which row should you sit?

Prove each identity. $$\sin \left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos \theta}{2}}$$
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