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. csc(-2𝜋/3)

Accepted Answer

Hello, today we're going to be calculating the value of the following expression. So what we are given is sequent of negative five pi over six. Now, one thing we can do is we can simplify the expression by using the rule of science for sequent the rule of signs for coin states that if you have a negative angle inside of the coin function, you can move the negative sign to the front of the coin function and rewrite cos as negative cosin of X. If we use the law of signs on our given expression, we can rewrite sequent of negative five pi over six as negative sequent of five pi over six. So now what we wanna do is we want to evaluate the value of cos in phi pi over six. And one way we can do this is to use reference angles. In order to use the reference angles, we need to find the location of five pi over six on the unit circle. Now, if you're ever unsure of where radiant is located on the unit circle, you can always convert the radiant into degrees by multiplying or given angle by the quantity 180 over pine multiplying by this value allows us to reduce the pi in the numerator and denominator to one. And what we are left with is five times 180 which gives us the value of 900 over six times one which is six, 900 divided by six, gives us the value of 150. So what this means is that the angle five pi over six is equivalent to 150 degrees. So now the question is where is 150 degrees located on the unit circle? While 150 degrees is gonna be located in quadrant two of the unit circle. And now that we have the location of the angle, we can use this to calculate the reference angle. Now, the reference angle always lies between our given angle and the x axis. And in order to get a reference angle in quadrant two, we're going to take the angle measure of 180 degrees and subtract our given angle which is 150 degrees, 180 minus 150 is going to give us the value of 30 degrees. So what this means is that the reference angle for five pi over six is going to be 30 degrees. Now that we have the reference angle, we can use this to draw a right triangle in quadrant two. And this also allows us to state that co sequent a five pi over six is going to be equivalent to co sequent of 30 degrees of the following right triangle. So let's go ahead and draw a bigger image of the right triangle in quadrant two. So we know that the reference angle is 30 degrees. And since this is a right triangle, we have a right angle located here. So what we want to do now is we want to solve for co seeking of 30 for the following right triangle. In order to do this, we can go ahead and label the triangle by using our special right triangles. And 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, one thing to note is that this right triangle is located in quadrant two, any X value located in quadrant two is going to be negative and any Y value located in quadrant two is going to be positive. So what this means is that the length of the triangle is going to be negative and the height of the triangle is going to be positive. So now that we have the side links of our right triangle, we can use this to solve for co seeking of 30. Now recall that cos of any right triangle is defined as our hypo news over the height of the right triangle, which is the Y value, the Y value in this case is going to be one and the hypotenuse is going to be two. So what this means is that co sequent of 30 is going to be 2/1 which simplifies to give us two. And recall that earlier, we stated that co sequent of five pi over six is equivalent to cos seeking of 30. So what this means is that sequence of five pi over six is equivalent to two. But also keep in mind that we use the law of science for coin to rewrite our given expression. So currently we have negative sequent a five pie over six, we know that sequent of phi pi over six is equal to two. So if we replace this value, we get negative one times negative two and that is going to simplify to give us negative two. So what this means is that the value of sequent five pi over six is going to be negative too. And 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. csc(-2𝜋/3)

Key Concepts

Understanding the Cosecant Function

Evaluating Trigonometric Functions at Negative Angles

Reference Angles and Unit Circle Values

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