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. sin 240°

Accepted Answer

Hello, today we're gonna be finding the exact value of the following expression. So what we are given is sign of 210 degrees. Now, one way that we can approach this problem is to use reference angles. And in order to get the reference angle for 210 degrees, we're going to need to locate 210 degrees on the unit circle. So the question is where does 210 degrees lie on the unit circle? Well, 210 degrees is gonna be located in quadrant three of the unit circle. And now that we know the location of 210 degrees, we can go ahead and calculate the reference angle. Now, one thing to note is that the reference angle is located between our given angle and the x axis. And in order to get a reference angle in quadrant three, we're going to take our given angle which is 210 degrees and subtract 180 degrees from this measure 210 minus 180 is going to give us degrees. So that tells us that the reference angle for 210 degrees is going to be degrees. We can use this reference angle to create a right triangle in the third quadrant. And we can also state that sine of 210 degrees is going to be equivalent to sine of 30 degrees over following right triangle. So let's go ahead and draw the right triangle in the third quadrant as a bigger image. So 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 now what we need to do is we need to solve for a sign of 30 degrees of the following right triangle. In order to do this, we can use our special right triangles to label the right triangle. 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 the 90 degree angle is going to be a length of two. Now recall that this triangle is in quadrant two, any X value in quadrant three, I'm sorry, quadrant three is going to be negative. And that means that the length of the triangle is going to be negative. Furthermore, any Y value in quadrant three is going to be negative. And that means that the height of the right triangle is going to be negative as well. So what we need to do now is we need to solve for sin of 30 using the right triangle sides. Now recall that sign of any right triangle is defined as the height of the triangle, the Y value over the hypotenuse. In this case, here, the Y value or the height of the triangle is going to be negative one and the hypotenuse is going to be two. So that means the sign of 30 of the following right triangle is going to be negative 1/2. And recall that earlier, we stated that sine of 210 degrees is equivalent to sine of 30 of the right triangle. What this means is that sine of 210 degrees is equal to negative 1/2. 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 a. 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. sin 240°

Key Concepts

Unit Circle and Angle Measurement

Reference Angles

Sign of Trigonometric Functions in Quadrants

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