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.

sin(-240°)

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are working on is evaluating sine of negative 210 degrees. Now, one thing to note is that we are given a negative angle inside of the sine function. We can simplify this by applying the rule of signs for the sine function. The rule of signs for sine states that if we have s sign with a negative angle, we can rewrite this statement as negative sign of X. So if we apply this rule to sign of negative 210 we can rewrite sign of negative 210 as negative sign of 210. So now what we wanna do is you want to evaluate sign of by using a reference angle in order to do this, let's take a look at the unit circle. So where is 210 degrees located on the unit circle? This angle is gonna be located in quadrant three of the unit circle and the reference angle always lies between the x axis and the given angle. Now, one thing to note is that the reference angle is going to be defined as our angle minus 180 degrees because it is located in the third quadrant, our angle is 210 and we're going to be subtracting 180 from this angle and that is going to leave us with degrees. This is going to be the reference angle for our given angle. And what this means is that sine of degrees is going to be equivalent to sine of degrees. Now, one way to solve this is to use our special triangles, going back to the unit circle, we can create a right triangle by using the reference angle. So if we draw this a little bit bigger, we have a right triangle that is located in the third quadrant of the unit circle, we know that the reference angle is 30 degrees. And since this is a right triangle, we have a right angle of 90 degrees located here. Now, we can rewrite this as a special triangle. We have a special triangle which is called a 30 60 90 degree triangle. So we're going to have a 60 degree angle located at the bottom of the triangle. And across from 30 we're going to have a measure of one across from 60 we're going to have a measure of square root of three and across from the 90 degree angle, we're going to have a length of two. Now one thing to note is that this triangle is located in the third quadrant, which means that the length of the triangle or the X value in this case is going to be negative and the height of this triangle or the Y value is going to be negative as well. So what we wanna do is solve for sign of degrees for this right triangle. Now recall that sign of X is defined as our Y value over the hypotenuse for a right triangle, our Y value for this triangle is going to be negative one and our hypotenuse is going to be two. So sine of 30 degrees for the following right triangle is going to be negative 1/2. And what that means is that sine of 210 degrees is equal to negative 1/2. Now, if we replace this back into the original statement, what we get is negative, negative one half and negative one times negative half is going to give us positive half. So to summarize sine of negative is equal to a half by using reference angles. 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. sin(-240°)

Key Concepts

Reference Angles

Unit Circle

Trigonometric Function Values

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