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

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(-𝜋/4)

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 pie divided by six. Our answer choices are answer choice, a negative square, it three divided by three, answer choice B negative square root three divided by two answer choice C negative square at three and answer choice D negative square at two. All right. So first we've got this negative pi divided by six. You might know where that is. But let's get that to be between zero and positive two pi. So we can take our negative pi divided by six and add a full rotation so that it's on our unit circle. A full rotation is two pi. However, we've got a denominator of six. So that's going to be 12 pi divided by six. Adding these together, we'll get a positive 11 pi divided by six. Now, we can find that on our coordinated plane, we have a vertical y axis, a horizontal X axis, they come together at the origin in the middle. We can overlay this with a roughly drawn unit circle. And for our angle, the initial side, the vertex is at the origin and heads toward positive infinity for the X values along the X axis. And now for our 11 pi divided by six, the terminal side is in the fourth quadrant and it's going to be closer to the positive part of the X axis than it is to the negative part of the Y axis. That's our 11 pi divided by six. Now, what's the reference angle for this angle? When we're in the fourth quadrant, we are going to take our two pi and subtract our angle. So subtracting 11 pi divided by six. But again, two pi here, we want that with the denominator of six. So we've got 12 pi divided by six minus pi divided by six, which will leave us with just one pi divided by six. That's in our first quadrant again, closer to the positive part of the X axis. But then this time from the positive part of the Y axis. So our pi divided by six, we recall from previous lessons that that is 30 degrees. But if you ever don't remember that you can just convert it to degrees. We take our pie divide it by six and multiply it by degrees divided by pi pi is cancel out. So we have 180 degrees divided by six, which is 30 degrees. So we've got 30 degrees. We can draw a 30 60 90 triangle. Mine's a very rough sketch not to scale or anything. The right angle is the 90 degrees, you can pick whichever one looks smaller and label that 30 degrees and then the midsized one is 60 degrees. Then we recall from previous lessons that the hypotenuse for a 30 60 triangle is going to be two for 30 degrees, the opposite side. So between 60 degrees and 90 degrees is measured one and then the adjacent side between 30 degrees and 90 degrees is the square of three. Now, we're talking about the tangent of degrees and tangent is going to be the opposite over the adjacent. So opposite here, opposite is one that's divided by the adjacent, which is the square root of three. And then we don't want to have a radical in the denominator. So we're going to multiply this by the squared of three divided by the square root of three. And in the numerator that's going to be the square root of 31 multiplied by the squared of three. And in the denominator, the square root of three multiplied by the square root of three is just three. Now, the last very important question we have here is we're in the fourth quadrant. So is tangent positive or negative in the fourth quadrant and tangent is only positive in the 1st and 3rd quadrants, it's negative in the fourth. So we've got a negative squared of three divided by three. We look at our answer choices and this matches with answer choice. 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(-𝜋/4)

Key Concepts

Reference Angles

Tangent Function and Its Properties

Exact Values of Special Angles

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