Problem 139 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

By re-arranging and simplifying the equation, you can see that \(\tan x = 1/2\). Now, to solve for x, one should use either the tangent function's inverse (aka arctan or atan) or look up in a table or calculator, keeping in mind the periodicity of tangent function to find all possible solutions.

Step 1: Rearrange Equation

Rearrange the formula to get all terms involving \(\tan x\) on one side and constants on the other. Subtract \(3 \tan x\) from both sides to get: \(5 \tan x - 3 \tan x = 2 - 1\)

Step 2: Simplify Equation

Simplify the equation on both sides to get: \(2 \tan x = 1\)

Step 3: Solve for Tangent of x

Divide both sides by 2 to solve for \(\tan x\) which gives \(\tan x = 1/2\)

Key Concepts

Tangent Function

The tangent function, often symbolized by \( \tan \), is a fundamental trigonometric function that relates the angle of a right triangle to the ratios of the triangle's sides. Specifically, for an angle \( x \) in a right triangle, the tangent of \( x \) is the ratio of the opposite side to the adjacent side. This can be mathematically expressed as:

• \( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \)

The tangent function is periodic, with a period of \(180^\circ\) or \(\pi \) radians. This means that \( \tan(x) \) repeats its values every \(\pi\) radians. Its graph has vertical asymptotes where it is undefined, mainly at odd multiples of \(\frac{\pi}{2}\). It is crucial to understand the behavior of \( \tan(x) \) when solving trigonometric equations, as it can influence solutions, especially where the function is undefined.

Solving Equations

Solving equations involves finding the value(s) of the variable that makes the equation true. In the context of trigonometric equations involving the tangent function, we often look for angles \( x \) that satisfy the equation.When solving a trigonometric equation such as \( 2 \tan x = 1 \), the primary step is to isolate the tangent function, leading to \( \tan x = \frac{1}{2} \). The next step involves determining the angles \( x \) that produce a tangent value of \(\frac{1}{2}\). Using a calculator or trigonometric tables is helpful for this step; however, it's critical to remember the periodic nature of tangent. You should ensure to consider additional solutions \( x + n\pi \) for integer \( n \), due to the periodicity of \( \tan \). It is this periodicity that can sometimes create infinite numbers of solutions within given constraints, such as a specific interval.

Rearranging Equations

Rearranging equations is a key step in solving them, particularly when dealing with trigonometric equations. The goal is to isolate terms involving the variable of interest. Here's how you can do it:

• Identify all terms involving the variable \( \tan x \).
• Move these terms to one side of the equation by subtracting or adding terms.
• Place constant terms on the opposite side.

In the original problem, this strategy was applied by moving \( 3 \tan x \) across the equality sign, adjusting the equation to consolidate all tangent terms on one side and numbers on the other. This simplifies the equation, allowing for easier further reduction to isolate \( \tan x \). Rearranging is fundamental because it sets up the equation in a way that makes the solving process straightforward, especially when dealing with more complex trigonometric relationships.