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Chapter 6: Problem 139

Without actually solving the equation, describe how to solve $$ 3 \tan x-2=5 \tan x-1 $$

Short Answer

Expert verified
To solve the equation \(3 \tan x - 2 = 5 \tan x - 1\), one would first rearrange the equation to result in \(2 \tan x = 1\), then solve for \(\tan \), resulting in \(x = \tan^{-1}(\frac{1}{2})\). The actual solution is not required in this instance.

Step by step solution

01

Understand the Equation

The given equation is a trigonometric equation because it involves the tangent function. It's structured like a simple linear equation, but involves trigonometric functions.
02

Rearrange the Equation

Start by bringing like terms together to one side of the equation. This means subtracting \(3 \tan x\) from both sides of the equation: \[5 \tan x - 3 \tan x = 2 - 1\]. The equation simplifies to \[2 \tan x = 1\].
03

Isolate the variable

Now, isolate \(x\) by dividing each side by \(2\). This results in a new equation: \[\tan x = \frac{1}{2}\].
04

Write the final step without solving

The final step to actually solving the equation (which is not required here) would be to find the inverse tangent of \(1/2\) - this gives the value of \(x\). Write it as: \[x = \tan^{-1}(\frac{1}{2})\].

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Most popular questions from this chapter

Graph \(f\) and \(g\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Then solve a trigonometric equation to determine points of intersection and identify these points on your graphs. $$ f(x)=3 \sin x, g(x)=\sin x-1 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using sum and difference formulas, I can find exact values for sine, cosine, and tangent at any angle.

Will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$ \sin 60^{\circ} \sin 30^{\circ}=\frac{1}{2}\left[\cos \left(60^{\circ}-30^{\circ}\right)-\cos \left(60^{\circ}+30^{\circ}\right)\right] $$

Use words to describe the formula for each of the following: the tangent of the sum of two angles.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(\tan x=\frac{\pi}{2}\) has no solution.
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