In Exercises 61–86, use reference angles to find the exact value ...

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.

sec 240°

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are evaluating is second of degrees. So the first thing we're gonna want to do is we're gonna wanna locate 210 degrees on the unit circle. 210 degrees is gonna be located in the third quadrant of the unit circle. And now, what we need to do is we need to find a reference angle. Now, the reference angle is always gonna lie between the X axis and our given angle. And in order to get the reference angle for a quadrant three, we're going to have to take our given angle which is 210 and subtract 180 from this value 210 minus 180 is going to leave us with degrees. So that means the reference angle is going to have a measure of degrees. And what this also means is that second of degrees is equivalent to see of 30 degrees. So how do we solve for seeking of 30 degrees? Well, since we have the reference angle, we can use this to create a right triangle for our given angle. So what we have is a right triangle that is located in quadrant three. So let's go ahead and draw a bigger picture for that triangle. Now we know that the reference angle is 30 degrees. And since this is a right triangle, we have a 90 degree angle located here. So we can solve for seeking of 30 by using our special right triangle. And in this case here, we're going to be using a 30 60 90 degree triangle across from 30 is going to be the measure of one across from 60 is going to be the measure of the square root of three. And across from the 90 degree mark is going to be a measure of two. Now, one thing to note is that this triangle is located in quadrant three. What this means is that the X values or the length of the triangle is going to be negative in this quadrant and the height of the triangle or the Y values is going to be negative as well. So what we need to do is solve for seeking of 30 for this right triangle. Now recall that the second for a right triangle is defined as our hypotenuse over our X value our hypotenuse in this case, is going to be two and the X value or the length of the triangle is going to be negative square root of three. So if we plug this into the identity of sequins of is going to be two over negative square root of three. Now keep in mind that we cannot leave a, a square root in the denominator. So we're going to need to rationalize this value and we can rationalize by multiplying the value by the square root of three over the square root of three. And doing this is going to leave us with seeking of 30 is equal to negative two times the square root of 3/3. So this is going to be the value for seeking of 30. But keep in mind earlier, we stated that seeking of 210 is equivalent to seeking of 30. What this means is that sin of 210 degrees is equal to negative two times the square root of 3/3. And this is going to be the solution to the problem. And with that being said, the answer to this problem is going to be b 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. sec 240°

Key Concepts

Reference Angles

Secant Function

Unit Circle

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