Use substitution to determine whether the given x-value is a s...

Question

Use substitution to determine whether the given x-value is a solution of the equation.

$\tan 2 x = - \frac{\sqrt{3}}{3},$

Accepted Answer

Hello everybody. I hope you are doing. All right. Today, today we're going to be looking at this question that states for the following trigonometric equation perform substitution to identify which of the following angles is a solution where our trigonometric equation is tangents of two X is equal to negative one. Now the answer choices provided are a five pi divided by eight B three pi divided by eight C seven pi divided by 12 and D 11 pi divided by 12. Now to solve this question, we can use, we will be using our calculator as well as recalling from our unit circle values. So let's get started to use substitution. We basically are going to be doing try and error plugging in each answer choice and see which one works at the end. So let's start with answer choice. A, if we plug in answer choice. A, we'll have tangents of open parentheses. Two multiplied by five pi divided by eight close parentheses. And that is equal to tangents of open parentheses. Five pi divided by four. Now we can plug this into our calculator, but we can also recall from our unit circle values that tangent of five pi divided by four is equal to one that is a positive one. Therefore, this would not be an answer. So we move on to answer choice B and will plug in as choice B to have tangents of open parentheses. Two multiplied by three pi divided by eight closed parentheses. And that will be equal to tangent of open parentheses. Three pi divided by four closed parentheses. And once again using our calculator or using our unit circle, this will give us an answer of negative one, which is what we are looking for. Now, we can either stop right here. But let's just try again and try the S choices C and D just to confirm that the other ones don't work. So for choice C will have tangents of open parentheses. Two multiplied by seven pi divided by 12 closed parentheses. And this will be equal to tangents of open parentheses. Seven pi divided by six closed parentheses. And this will be equal to once again using our calculator or the unit circle. This will give us the square root of three divided by three, which is not the answer we are looking for. And finally, we move on to answer choice D, we will have tangents of open parentheses. Two multiplied by 11 pi divided by 12 closed parentheses. And this will be equal to tangents of open parentheses. 11 pi divided by six closed parentheses. And this will be equal to once again with a calculator or unit circle, negative square root of three divided by three, which is not the answer that we are looking for. And we can confirm that answer choice B is the one that gives us negative one. Therefore, answer choice B will be the correct choice for this question. So as you can see, it's very important to be able to recall the unit circle values as well as know how to use them. So I hope that was helpful. And until next time.

Use substitution to determine whether the given x-value is a solution of the equation.
$\tan 2 x = - \frac{\sqrt{3}}{3},$

Key Concepts

Tangent Function and Its Properties

Substitution Method in Trigonometric Equations

Simplifying and Evaluating Trigonometric Expressions

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