In Exercises 61–86, use reference angles to find the exact value ...

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.

tan (-17𝜋/6)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to evaluate the following expression manually by using reference angles. Our expression is the tangent of negative eight pi divided by three. Our answer choices are answer choice A the square red of three answer choice B the squared of three divided by two, answer choice C the square out of two and answer trace D the squared of two divided by two. All right. So where is this angle we've got here, this negative eight pi divided by three. Well, we know that that's negative. So let's get it onto our unit circle. We can add a full rotation. So a two pi but what's a full rotation when we have a denominator of three, that's going to be six pi divided by three. There's our two pi so we take negative eight pi divided by three plus six pi divided by three. That will get us to a negative two pi divided by three. Now, you might know where that is, but let's get into the positive part. We can add another six pi divided by three and that's going to get us to four pi divided by three. So we're talking about the tangent of four pi divided by three. All right. Now where is that? We draw a rough sketch of a coordinate plane, got a horizontal X axis, a vertical y axis. They come together at the origin in the middle. Now, we can overlay a unit circle. I'm not going to label the whole thing. But recalling from previous lessons, four pi divided by three is going to be in our third quadrant. So we draw our initial side from the origin heading toward the positive part of the X axis and the terminal side is going into the third quadrant and it's closer to our negative side of our Y axis than the negative side of the X axis. And that's our four pi divided by three. OK. So we need a couple of things. We need to know what our reference angle is. So we can take four pi divided by three minus pi what's pi when we have a denominator of three, that's going to be three pi divided by three and subtracting our numerators, we're going to get pi divided by three. So our reference angle is pi divided by three, which is up here in the first quadrant. Now we recall from previous lessons on the unit circle that pi divided by three is 60 degrees. But we can always check that by taking our pi divided by three, multiplied by converting to degrees with 180 degrees divided by pi pies cancel out. We take 180 degrees divided by three and that's 60 degrees. So there's our 60 degrees. Now, we can draw a 30 60 90 triangle. So the 30 degrees is the smallest. And of course, this is a very rough approximation. It's not accurate. It's just to give us the idea, we have the 30 degrees is the smallest angle. The 90 degrees is our right angle. And then the midsized one is our 60 degree angle. And recalling from previous lessons, labeling this, we've got between 60 degrees and 90 degrees. The adjacent in uh side is one the opposite side. So between 30 degrees and 90 degrees is going to be the square of three. And the hypotenuse from 60 degrees to 30 degrees is going to be a positive two. When we're talking about the tangent of 60 degrees tangent means that we're taking the opposite divided by the adjacent. And so the opposite of 60 degrees is the square it of three that's divided by the adjacent, which is one and that simplifies two, the square it of three. Now one last important question in what quadrant is our four pi divided by three, that's in the third quadrant and is tangent positive or negative in the third quadrant? Well, we recall from previous lessons that tangent is positive in the third quadrant we've got from the first quadrant, we have all of them. Second quadrant sign, third quadrant tangent and fourth quadrant cosine. So this is a positive square rut of three. We look at our answer traces and this matches with answer trace. A well done. We'll catch you on the next one.

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan (-17𝜋/6)

Key Concepts

Reference Angles

Tangent Function

Angle Coterminality

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