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. csc(7𝜋/6)

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are given is co of four pi over three. Now, in order to evaluate this value, we need to first locate four pi over three on the unit circle. If you're ever unsure where a radiant 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 four pi over three and multiply it by 180 over pi doing so is going to allow us to simplify pie out of the expression. And what we are left with is four times 180 which is going to give us 720 over three times one, which is going to give us three and 720 divided by three is going to give us the angle of 240 degrees. So where is 240 degrees located on the unit circle? While 240 is gonna be located in the third quadrant of the unit circle. Now, what we wanna do is you want to get the reference angle, the reference angle is always located between the x axis and our angle given to us. And since the angle is located in quadrant three, in order to get the reference angle, we're going to be taking our given angle which is 240 degrees and subtracting 180 degrees from the spell 240 subtract 180 is going to leave us with 60 degrees. And what this means is that the reference angle is going to be 60 degrees for our given expression. Furthermore, what this states is that co sequent of four pi over three is equivalent to co of 60 degrees. So what we wanna do now is you want to make a right triangle in order to solve for this value. So let's go ahead and draw a right triangle that is located in the fourth quadrant of the unit circle. So we know that the reference angle is 60 degrees. And since this is the right triangle, we're going to have a 90 degree angle located here. The easiest way to solve for coking of 60 is to go ahead and use a special right triangle. Now, in this case here, we're going to be using a 30 60 90 degree triangle across from the 30 degree mark is going to be the value of one across from 60 is going to be the square root of three and across from 90 is going to be the value of two. Now, one thing to note is that this triangle is located in the third quadrant that means any X value or the length of the triangle is going to be negative and any height of the triangle or Y value is going to be negative as well. So what we need a sulfur is cos of 60 degrees. Now recall that sequent for a right triangle is defined as our hypotenuse over the Y value. Our Y value for this right triangle is going to be negative square root of three. And our hypotenuse is going to be positive too. So if we plug this in co seeking of 60 degrees is going to equal to negative two over the square root of three. But keep in mind that we cannot keep a radical in the denominator of our answer. So we're going to need to rationalize this value by multiplying the numerator and denominator by the square root of three. And doing this is going to give us the value cos seeking of is equal to negative two times the square root of 3/3. Now keep in mind that coin of four pi over three is equivalent to coin of 60. And since coking of 60 is equal to negative two times square root of 3/3, that means that cos of four pi over three is equal to negative two times the square root of 3/3, this is going to be the answer to this problem. 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 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. csc(7𝜋/6)

Key Concepts

Reference Angles

Trigonometric Functions and Their Signs in Different Quadrants

Exact Values of Common Angles

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