In Exercises 55–58, use a graph to solve each equation for -2π...

Question

In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1

Accepted Answer

Welcome back. I am so glad you're here. We're asked to solve the given equation with the help of its graph in the interval from negative two pi to two pi inclusive of both negative two pi and two pi. Below the equation that we're given is the tangent of X equals negative square at three divided by three. And we're being asked to solve for X in our graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and on the graph is Y equals the tangent of X. And then we're given four horizontal lines. One, the lowest closest to negative infinity is where Y equals negative square at three. The one above that is Y equals negative square at three divided by three. And then above the X axis, we have a horizontal line Y equals square at three divided by three. And above that the last one is Y equals the square of three. And then every place where one of those four horizontal lines intersects with our graph of Y equals the tangent of X. Each point of intersection is labeled for our answer choices we have answer choice. A the solution set negative pi divided excuse me, negative five pi divided by three, negative two pi divided by three pi divided by three and four pi divided by three answer choice B has the solution set negative 11 pi divided by six, negative five pi divided by six pi divided by six and seven pi divided by six answer choice C has the solution set negative four pi divided by three negative pi divided by pi divided by three and five pi divided by three. And answer choice D has the solution set negative seven pi divided by six, negative pi divided by 65 pi divided by six and 11 pi divided by six. All right. So with this solving for X, instead of having to solve this algebraically, we can just look at our graph. So we want to know because we know that Y equals the tangent of X and because that also equals negative square at three divided by three, we also know that Y equals negative square at three divided by three. So we just need to figure out the X values when our Y equals the tangent of X intersects with the line Y equals negative square at three divided by three. So we'll create a solution set. And from left to right, the first place that our horizontal line Y equals negative square three divided by three intersects with our graph of Y equals the tangent of X. The X value is negative seven pi divided by six. The next place that those two, the graph and the line intersect are is at negative pi divided by six. The next place they intersect is five pie divided by six. And the last place they intersect. And of course, this is just from our interval from negative two pi to two pi is 11 pi divided by six. So this is a really neat way to solve this. By looking at a graph, we look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1

Key Concepts

Graph of the Tangent Function

Solving Trigonometric Equations Using Graphs

Domain and Periodicity in Trigonometric Solutions

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