Find the exact value of each real number y. Do not use a calcu...

Question

Find the exact value of each real number y. Do not use a calculator.

y = arccot (―1)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the given inverse trigonometric function without using a calculator. Our given function is that Y equals the inverse cotangent of negative square root three divided by three plus the inverse cotangent of negative square root three. Our answer choices are answer choice A Y equals two pi divided by three answer choice by equals three pi divided by two answer choice cy equals seven pi divided by three and answer choice. Dy equals seven pi divided by six. All right, we recall from previous lessons that when we're talking about the inverse trigonometric function, here's the inverse cotangent. What we're doing is we're asking ourselves what angle are we do? We have when we would have the cotangent of that angle would be equal to what's in the parenthesis. So in this case, the first one, it's negative square root three divided by three. And then over for the second one, we'd have that cotangent of what angle is equal to negative square root three. So we can just substitute in the angles four hour inverse trigonometric functions and then add those together So when we have the cotangent of an angle equals negative square root three divided by three, that looks a lot like a special angle, which is why we don't get to use a calculator. For this one, we can recall those tables in the textbook and from previous lessons, and we would start off by saying, OK, we know the cotangent of an angle equals a positive square at three divided by three. When that angle is equal to looking at those tables that's going to be 60 degrees, but that's not equal to a negative. So what would it be when it's equal to a negative square at three divided by three? Well, from tables in the textbook, we recall that the range of our inverse cotangent function is going to be from zero to P. So in other words, we're in the first and second quadrants, we draw a really quick sketch of our coordinate plane. We have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. We're talking about quadrants one up into the right of the origin and quadrant two up into the left of the origin. If 60 degrees in the first quadrant is our reference angle, then what's our second quadrant angle here? That will give us our negative square root three divided by three. And so we take 180 degrees minus 60 degrees and that is going to give us 120 degrees. So our theta, when cotangent theta equals negative square root three divided by three is going to be 120 degrees. But we notice our answer choices are not in degrees, they're in radiance. So if you look at a unit circle, 120 degrees is the same as two pi divided by three. So we can go back up to our original problem. And instead of the inverse cotangent of negative square root three divided by three, that is two pi divided by three. Now we just need to do the same process for our cotangent, our inverse cotangent of negative square root three. So the cotangent of theta equals the square root of three, the positive square root of three looking at those charts when theta equals 30 degrees. So we're still do dealing with cotangent or inverse cotangent. So we're still in just quadrants one and two. So if our reference angle in quadrant one is 30 degrees, again, we take 180 degrees minus 30 degrees and that will get us 150 degrees in quadrant two. So our cotangent of theta equals a negative square root three when theta equals 150 degrees or in radiance, if you look at the unit circle, 150 degrees is five pi divided by six. So we can plug five pi divided by six in for the inverse cotangent of negative square root three. So we're adding two pi divided by three plus five pi divided by six. If you get a common denominator in there of six, that'll be four pi divided by six plus five pi divided by six, which is nine pi divided by six. And that simplifies to be two pi or excuse me, not two by three pi divided by two. And that is our solution for why we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Find the exact value of each real number y. Do not use a calculator.
y = arccot (―1)

Key Concepts

Inverse Trigonometric Functions

Cotangent Function

Quadrants of the Unit Circle

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