Evaluating trigonometric functions Without using a calculator,...

Question

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cot (-17π/3)

Accepted Answer

Welcome back, everyone. In this problem, we want to solve for the exact value of the trigonometric expression of the cotangent of negative 23 quarters of pi. For our answer choices, a is negative 1, b is 1, c is the square root of 3, and d is the negative of the square root of 3. Now, what do we know here? Well, we have a trigonometric expression. And if we look at this angle, we'll notice that it's greater than 2 pi. So when an angle is greater than 2 pi, that means it has an a corresponding angle. So if we can figure out what that corresponding angle is, then we'll be able to solve. We also know that we're finding it for the cotangent and recall I should put that in a different color here. Recall that the cotangent of an angle is the reciprocal of the tangent of that angle. So whenever we figure out what that angle is, then we can apply this idea to help us to get a definite value, the exact value, because we already know the exact values for certain angles of the tangent. So, first, let's see if we can find the corresponding angle to negative 23 fourths of Pi. So, to do that, let's get some help with the unit circle. So, I'm going to put it right here. Gonna try it as big as I can. And we're dealing with the angle negative 23 fourths of pi. Now remember, as we said, this angle is greater than 2 pi. So, to find the corresponding angle, we need to do the revolutions until we are in that space between 0 and 2pi. Since it's a negative angle, that means we'll be going clockwise. Okay? And, since we're dealing with 4ths, it would make sense to rewrite 2 pi as 8 4ths of pi. Because now, it will be easier to track our rotation as we go. So let's go ahead and give it a go. So we have the y axis and the x axis. And now starting from the center, we are going around one time, so that would be 8 pi. Then we go around another time, that would be 16 pi. And now the next one would be 24 pi, but that's too far. Just before we get to 24 pi, we'd get to 23 pi. So this rotation is the angle negative I should negative 23 fourths of pi. Now, if that's the case, that means my angle starting here would have been this small section that's left before the rotation and that would be a 4th of Pi because we have a 4th of Pi to go until we make the 3rd revolution. So, what does this tell us? This tells us that the cotangent of negative 23 fourths of Pi is the same thing as the cotangent of a 4th of Pi. Now, we have our reference angle so we can go ahead and try to figure out what that is. Now, again, recall that the cotangent is the reciprocal of the tangent. So, the cotangent of a 4th of Pi is the same thing as 1 divided by the tangent of a 4th of Pi. Recall as well that the tangent of a 4th of Pi by definition equals 1. So, that tells us then that our cotangent of negative 23 fourths of Pi would be equal to 1 divided by 1 which would just be equal to 1. So, that would be our final answer and when we look in our answer choices that would have been answer choice b. Thanks a lot for watching everyone. I hope this video helped.

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cot (-17π/3)

Key Concepts

Cotangent Function

Periodic Properties of Trigonometric Functions

Angle Reduction

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