Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\tan ^{2} x-1=0$$

Step-by-Step Solution

Answer

The solutions of the equation \( \tan^2 x - 1 = 0 \) on the interval \( [0, 2\pi) \) are \( x = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4 \)

Step 1: Rearrange the equation

First, let's rearrange the original equation \( \tan^2 x - 1 = 0 \) to \( \tan^2 x = 1 \). This sets up the equation to find the tangent of what angles gives us the value of 1 or -1.

Step 2: Find solutions for x

Since we are dealing with \(\tan^2 x = 1\), \(\tan x\) could be +1 or -1. The equation \( \tan x = 1 \) has solutions of \( x = \pi/4, 5\pi/4 \) and \( \tan x = -1 \) has solutions of \( x = 3\pi/4, 7\pi/4 \) in the interval \( [0, 2\pi) \).

Step 3: Confirm the solutions

By substituting each of the solution values obtained back into the original equation, we can confirm that all these solutions satisfy the original equation. Also, we can use a graphing calculator to verify these solutions numerically via a table feature, however, this step is outside the scope of this process

Key Concepts

Tangent Function

The tangent function is one of the main functions in trigonometry, along with sine and cosine. It is defined as the ratio of the sine to the cosine of an angle. Mathematically, the tangent of an angle \( x \) can be expressed as \( \tan x = \frac{\sin x}{\cos x} \).

• This relationship means that whenever the cosine of an angle is zero, the tangent is undefined because division by zero is not possible.
• The tangent function has a periodicity of \( \pi \), which means its pattern repeats every \( \pi \) radians.
• Graphically, the tangent function has vertical asymptotes where the cosine of \( x \) is zero, causing the graph to shoot up or down to infinity.

Understanding these properties is crucial when solving trigonometric equations involving the tangent function, as seen in the exercise where we determined the values of \( x \) that satisfy \( \tan^2 x = 1 \). This involves finding angles where the tangent is either +1 or -1.

Interval [0, 2π)

In trigonometry, solving equations often requires finding solutions within a specific interval. For this exercise, we must find solutions within the interval \([0, 2\pi)\).

• This interval starts at 0 and includes all angles up to, but not including, \( 2\pi \).
• The notation \([0, 2\pi)\) means that 0 is part of the interval while \( 2\pi \) is not.
• This interval encompasses a full cycle of the unit circle, which is a circle with a radius of one, commonly used in trigonometry.

When solving an equation graphically or algebraically in this interval, it is important to list all possible solutions. For the tangent equation we examined, the relevant angles were \( \pi/4, 5\pi/4 \) for \( \tan x = 1 \) and \( 3\pi/4, 7\pi/4 \) for \( \tan x = -1 \). Each of these solutions falls within the specified interval.

Graphical Verification

Graphical verification is an important method to confirm the solutions of trigonometric equations by visual means. With tools such as graphing calculators or graphing software, you can visualize the function and its behavior over a given interval.

• When plotting \( y = \tan^2 x \), we can directly see at which points the function equals 1 by checking where the graph intersects with the line \( y = 1 \).
• This helps verify that the solutions obtained algebraically are indeed correct because they will correspond to these intersection points.
• By using the table feature of a graphing utility, you can numerically verify values of \( x \) that satisfy the equation, reducing potential errors and confirming accuracy.

Graphical verification thus serves as a valuable cross-reference to ensure algebraic solutions fall within the correct parameters of the problem, as it offers a clear visual confirmation of where the correct solutions lie.

Algebraic Manipulation

Algebraic manipulation is a crucial skill when solving trigonometric equations. It involves rearranging equations to isolate the variable of interest. In our exercise, we start with the equation \( \tan^2 x - 1 = 0 \).

• By rearranging, we solve for \( \tan^2 x = 1 \), a simpler equation to handle. This transformation is key to understanding the problem.
• Recognizing that \( \tan^2 x = 1 \) implies \( \tan x = \pm 1 \), we split the equation into two separate simpler equations: \( \tan x = 1 \) and \( \tan x = -1 \).
• From there, we can solve each equation separately and discover the solutions \( \pi/4, 5\pi/4 \) and \( 3\pi/4, 7\pi/4 \).

The process of solving these simpler equations involves recalling the values of the tangent function at key angles. Therefore, efficient algebraic manipulation enables you to systematically work through trigonometric problems and identify correct solutions comprehensively.