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. cot(7𝜋/4)

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are given is cotangent of five pi over three. Now, if you're unsure where a radon is located on the unit circle, you can always convert the radiant into a degree. In order to do this, we take our given angle which is five pi over three and we multiply it by 1 80 over pi multiplying this value by 1 80 over pi is going to cancel pi from the equation and that leaves us to five times which is going to give us 900 over three times one, which is three and 900 divided by three is going to give us 300. So we can rewrite the given angle as cotangent of 300 degrees. Now where is 300 degrees located on the unit circle while 300 degrees is gonna be located in quadrant four of the unit circle. And now what we wanna do is we want to look for the reference a angle. Well, the reference angle is going to lie between the x axis and our given angle. And since our reference angle is located in quadrant four. In order to get the reference angle, we're going to be taking 360 degrees and subtracting our given angle which is 303 160 minus 300 is going to leave us with 60 degrees. So that means that the reference angle of code tension 300 degrees is going to be 60 degrees. And what this means is that cotangent of 300 degrees is going to be equivalent to cotangent of 60 degrees. So how do we go about finding the value by using the reference angle while we can go ahead and create a right triangle using our reference angle. So if we create a right triangle that's located in quadruple four, the reference angle is gonna be located at the center of the triangle which in this case is degrees and we have a 90 degree mark right here, we can use the method of special right triangles to solve for this value. And in this case, we're going to be using a 30 60 90 degree triangle across from 30 is going to be the value of one across from 60 is going to be the value of the square root of three and across from the 90 degree angle is going to be the value of two. Now, one thing to note is that because this triangle is located in quadrant four, any X value or the length of the triangle is going to be positive and any height or the Y value of the triangle is going to be negative. So the height of the triangle is going to be negative square root of three. And now we want what we want to solve is cotangent of degrees. Now recall that cotangent is the inverse of tangent and is defined as the X value over the Y value. So our X value for this triangle is going to be positive one. And the Y value for this triangle is going to be negative square root of three. So if we plug this into the definition of cotangent, cotangent of 60 degrees is going to equal to negative one over the square root of three, now we cannot leave a square root in the denominator of our answer. So we're going to need to rationalize negative one over the square root of three. In order to rationalize this value, we're gonna multiply the numerator and denominator by the square root of three. And doing so is gonna leave us with the value of negative square root of 3/3. That means cotangent of 60 is equal to negative square root 3/3. And since cotangent of 300 or cotangent of five pi over three is equivalent to this value. That means that cotangent of five pi over three is equal to negative square root 3/3. And with that being said, the answer to this problem is going to be c 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 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cot(7𝜋/4)

Key Concepts

Reference Angles

Cotangent Function

Angle Reduction and Quadrants

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