Problem 14 - Precalculus with Limits

Question

In Exercises 11-24, solve the equation. \( \tan x + \sqrt{3} = 0 \)

Step-by-Step Solution

Answer

The solution is \(x = \pi + n \pi\) and \(x = 2 \pi + n \pi\).

Step 1: Solve the Equation

First, subtract \(\sqrt{3}\) from both sides. This results in \(\tan x = -\sqrt{3}\). This equation means that we are looking for values of x where the tangent of the angle equals negative square root of three.

Step 2: Finding Corresponding Angle

\(\tan x = -\sqrt{3}\) corresponds to \(x = \pi + n\pi\) and \(x = 2\pi + n\pi\), since in these quadrants, tangent is negative. Here, \(n\) is an integer which denotes the number of times angle x rotates around the circle.

Step 3: General Solution

So the general solution becomes \(x = \pi + n\pi\) and \(x = 2\pi + n\pi\).

Key Concepts

The Tangent Function

Understanding the tangent function is essential when solving trigonometric equations. In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to that of the adjacent side. On the unit circle, it is the y-coordinate divided by the x-coordinate at a certain angle. This can be quite abstract, so let's break it down with an example.

When you see an equation like \( \tan x = -\sqrt{3} \), you're being asked where on the unit circle (think of it as a perfectly round pizza with a radius of 1) the ratio described above equals -\(\sqrt{3}\). Remember, since the tangent function is periodic with a period of \(\pi \), it repeats its values every \(\pi \). This means there are multiple angles along the circle where tangent equals -\(\sqrt{3}\), not just one!

Equation Solutions in Radians

Radians are another way to measure angles, just like degrees. However, in mathematics, especially when dealing with trigonometry, we often prefer radians for their natural fit with the arc lengths and areas of circles. When an equation involves trigonometric functions, the solutions are very commonly expressed in radians.

For instance, in solving \(\tan x + \sqrt{3} = 0\), we've transformed the equation to \(\tan x = -\sqrt{3}\). When we find the angles that satisfy this, they'll usually be in radians. In the context of this equation, the solutions were expressed as \(x = \pi + n\pi\) and \(x = 2\pi + n\pi\), which indicate positions on the unit circle. Because circles and hence, radians, play so nicely with trigonometric functions, radians make it easier to work out, conceptualize, and generalize solutions in these contexts.

General Solution of Trigonometric Equation

A general solution of a trigonometric equation is an expression that captures all possible solutions. It's like casting a wide net to catch all the fish in the lake. For example, with the equation \(\tan x + \sqrt{3} = 0\), the general solution is found by determining the base angles where the tangent function's value is -\(\sqrt{3}\) and then taking into account the periodicity of the tangent function.

In the solution provided, \(x = \pi + n\pi\) and \(x = 2\pi + n\pi\) are general solutions because they encompass all the angles around the circle where tangent equals -\(\sqrt{3}\). Here \(n\) represents any integer, meaning that you can spin around the circle as many times as you like (full rotations), and as long as you land at those specified base angles, tangent will have the value -\(\sqrt{3}\). The concept of a general solution is powerful because it gives a comprehensive set of answers for trigonometric equations.