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.

cot 19𝜋
6

Accepted Answer

Welcome back. I am so glad you're here. And we are asked to evaluate the following expression manually by using reference angles. Our expression is the cotangent of 10 pi divided by three. Our answer choices are answer choice, A infinity, answer choice B one answer choice C the square out of three divided by three and answer choice D three multiplied by the square root of three. All right. So let's figure out first where this angle is. So where is 10 pi divided by three? Well, we know that that is more than one revolution. That's more than a two pi around our unit circles. So if we were to take 10 pie divided by three and subtract one full revolution, we subtract two pi but when we've got a denominator of three, how much is that? Well, if we're subtracting two pi with a denominator of three, that's going to be six pi divided by three. And now we can subtract our numerators. 10 pi minus six pi is four pi divided by three. And that we know is on our unit circle. We can draw a rough sketch of a coordinate plane. We have a horizontal X axis, a vertical y axis, they come together at the origin in the middle, overlaying that with the unit circle. And our four pi divided by three is going to be in quadrant three, we draw our initial side from the origin going to positive infinity for the X values. And our terminal side for this angle is coming down into the third quadrant and it's going to be closer to the negative Y values with the negative Y axis than it is to the negative part of the X axis. And there is our four pi divided by three. Now, if we're taking the cotangent of four pi divided by three, we still need to know what the reference angle is. So we take our four pi divided by three and subtract pi well, if we're subtracting pi, but we've got a denominator of three, then that's going to be three pi divided by three and subtracting our numerators for pi minus three pi is just one pie. So our reference angle is pi divided by three and that is up in the first quadrant that is closer to the positive part of the Y axis than the positive part of the X axis. And we recall from previous lessons that pi divided by three is 60 degrees. We can always, if we don't have that part memorized, we can always take pi divided by three, convert it to two degrees, multiply it by 180 degrees. Divided by pi pies cancel out. We have 180 degrees divided by three, which is the 60 degrees. So we're talking about the cotangent of a degree angle. Well, we can draw ourselves a right triangle and well, let me move that up a little bit. We've got, this is a very rough sketch of a 30 60 90 right triangle. The 30 degrees is going to be the smallest angle. The 90 degrees is the right angle and 60 degrees is the mid sized angle. And we recall from previous lessons that the length of the lengths of our sides from the 60 degrees to the 90 degrees, the adjacent side that's going to be a length of one from the 30 degrees to the 90 degrees, the opposite side, that's a length of the squared of three and from 60 degrees to degrees. The hypotenuse that's a length of two. For the cotangent, we are going to be taking the adjacent divided by the opposite. So the adjacent to 60 degrees is one that's divided by the opposite of 60 degrees, which is the square of three. And now we just want to make sure we don't have a square root in the denominator. So we're going to multiply this by the squared of three divided by the squared of three that simplifies two in the numerator, one multiplied by the squared of three is the squared of three divided by in the denominator the squared of three multiplied by the squared of three is just three. Now, the last question we're in the third quadrant is cotangent positive or negative in the third quadrant. And we recall from previous lessons that the third quadrant tangent and cotangent are both positive. So this remains a positive squared of three divided by three. We look at our answer choices and this matches with answer choice. C 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. cot 19𝜋 6

Key Concepts

Reference Angles

Cotangent Function

Unit Circle

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