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Chapter 5: Problem 23

In Exercises 11-24, solve the equation. \( \tan 3x (\tan x - 1) = 0 \)

Short Answer

Expert verified
The solutions for the equation are \(x = n\pi / 3\) or \(x = \pi/4 + n\pi\), where \(n\) is any integer.

Step by step solution

01

Identifying the Factors

The given equation is of the form \(ab = 0\), where \(a = \tan 3x\) and \(b = (\tan x - 1)\). An equation of this form equals zero if either \(a = 0\) or \(b = 0\).
02

Solving for 'a'

\(\tan 3x = 0\) only when \(3x = n\pi\), where \(n\) belongs to set of integers. Thus, the solutions for \(x\) will be \(x = n\pi / 3\).
03

Solving for 'b'

\(\tan x - 1 = 0\) when \(\tan x = 1\). It means \(x = \pi/4 + n\pi\), where \(n\) belongs to the set of integers.
04

Combining the solutions

The solutions for the equation are the union of solutions from steps 2 and 3. Hence, the total solutions for \(x\) are \(x = n\pi / 3\) or \(x = \pi/4 + n\pi\).

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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 six fundamental trigonometric functions. It is defined as the ratio of the sine and cosine of an angle; specifically, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This implies that the tangent function is undefined wherever \(\cos(x) = 0\), such as at \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots\).

Key properties of tangent include:
  • Periodic with a period of \(\pi\). This means \(\tan(x + \pi) = \tan(x)\) for any angle \(x\).
  • It has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\).
  • The function changes sign when the angle crosses the vertical asymptote.

Understanding these properties is crucial when solving trigonometric equations, as they help in identifying solutions over specific intervals.
Finding Angle Solutions in Trigonometric Equations
When tackling trigonometric equations, like \(\tan 3x(\tan x - 1) = 0\), the goal is often to find all possible angle solutions that satisfy the equation. Each trigonometric equation relies on known values for which the trigonometric functions attain specific results.

Here鈥檚 how it generally works:
  • For any trigonometric function, identify the key angles within one period that satisfy the equation. For example, \(\tan x = 1\) at \(x = \frac{\pi}{4}\).
  • Extend these solutions across all possible periods using the periodic properties of the functions. For the tangent function, since it's periodic with \(\pi\), any solution \(x = x_0\) can be generalized to \(x = x_0 + n\pi\) where \(n\) is an integer.
  • Consider transformations due to coefficients in the argument, like \(3x\) in \(\tan 3x\), by adjusting the periods accordingly.

Learning to manipulate and transform known solutions into general ones makes solving trigonometric equations manageable.
Exploring Zeros of Functions
Zeros of functions are the values where the function's output is zero. For trigonometric functions like tangent, zeros can be particularly interesting due to their periodic nature.

For the function \(\tan x\), zero occurs at angles where the sine of the angle is zero but the cosine is non-zero, specifically at \(x = n\pi\) for any integer \(n\).

Why are zeros important?
  • Zeros indicate points where the function intersects the x-axis.
  • In equations, they serve as crucial identifiers for potential solutions. For example, in \(\tan 3x = 0\), solving for \(3x = n\pi\) leads to angles \(x = \frac{n\pi}{3}\).
  • They represent critical points in periodic repetition and symmetry of the function.

Being able to locate these zeros accurately is vital for solving equations, graphing functions, and understanding their behavior in different contexts.

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Most popular questions from this chapter

In Exercises 125 - 128, use a graphing utility to verify the identity. Confirm that it is an identity algebraically. \( \sin 4\beta = 4 \sin \beta \cos \beta\left(1 - 2 \sin^2 \beta\right) \)

In Exercises 29-36, use a double-angle formula to rewrite the expression. \( 10 \sin^2 x - 5 \)

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{\pi}{12} \)

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( \sin 4x = - 2 \sin 2x \)

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( 75^\circ \)
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