In Exercises 49–59, find the exact value of each expression. D...

Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)

Accepted Answer

Hello, today we're gonna be determining the value of the following expression. So what we are given is cotangent of negative 240 degrees. Now, one way we can simplify or give an expression is by using the law of signs for cotangent, the law of signs for cotangent states that if you have a negative angle in the cotangent function, you can move the negative sign to the front of cotangent and rewrite it as negative cotangent of X. If we use the law of signs on our given expression, we can rewrite cotangent of negative 240 degrees as negative cotangent of 240 degrees. So one way we can solve this is to go ahead and use reference angles. And in order to get a reference angle, we need to find the location of 240 degrees on the unit circle. So where is 240 degrees located on the unit circle while 240 degrees is gonna be located in the third quadrant of the unit circle. And now that we know the location, we're going to calculate the reference angle, the reference angle always lies between the x axis and our given angle. And in order to get a reference angle in the third quadrant, we're going to take our given angle which is 240 degrees and subtract 180 degrees from this value 240 minus 180 is going to give us degrees. So that tells us that the reference angle is going to have a measure of 60 degrees. And now that we have the reference angle, we can use this information to draw a right triangle in the third quadrant. And we can also use the reference angle to state that cotangent of 240 degrees is going to be equivalent to cotangent of 60 degrees for the following right triangle. So let's go ahead and draw a bigger version of the right triangle located in the third quadrant. So we know that this is a right triangle. We're going to have a right angle located here. We know that the reference angle is 60 degrees. So what we need to do is we need to calculate cotangent of degrees for the following right triangle. In order to do this, we're going to use our special right triangles. And in this case, we're going to be using a 30 60 90 degree triangle across from 30 is going to be a length of one across from 60 is going to be a length of the square root of three and across from 90 is going to be a length of two. Now recall that this right triangle is located in quadrant three. What this means is that any X value or the length of the triangle is going to be negative in this region and any Y value or the height of the triangle is going to be negative in the region as well. So what this means is that the length of the triangle is going to be negative one and the height of the triangle is going to be negative square root of three. Now that we have the side links of the right triangle, we can use these side links to calculate cotangent of 60. Now recall that cotangent for any right triangle is defined as the X value over the Y value or in other words, the length of the triangle over the height of the triangle, our X value is going to be negative one and our Y value is going to be negative square root of three. What this means is that cotangent of 60 degrees is going to be equal to negative one over negative square root of three. And this is going to simplify to give us positive one over the square root of three. Now we cannot leave a square root in the denominator. So we're going to need to rationalize this quantity in order to rationalize we're going to multiply the numerator and denominator by the square root of three. And we can rewrite this quantity as the square root of 3/3. So what this means is that cotangent of 60 degrees is equal to square root 3/3. And earlier, we stated that cotangent of 240 degrees is equivalent to cotangent of 60 degrees. What this means is that cotangent of 240 degrees is going to equal to the square root of 3/3. But one more thing to note is that we use the law of signs in the beginning of the problem to rewrite or given expression as negative cotangent of 240 degrees. So if we substitute square root 3/3 for cotangent of 240 we can rewrite the expression as negative one times the quantity square root 3/3. And this is going to simplify to give us negative square root 3/3. This is going to be the value for the given expression. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)

Key Concepts

Cotangent Function and Its Definition

Reference Angles and Angle Reduction

Sign of Trigonometric Functions in Different Quadrants

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