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. cos 225°

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are given is cosine of 210 degrees. Now, the first thing we're going to want to do is we're going to want to locate 210 degrees on the unit circle. 210 degrees is gonna be located in the third quadrant of the unit circle. And now we can use this to find the reference angle. The reference angle is always gonna be between the X axis and our given angle. And since we are looking for a reference angle in quadrant three, the reference angle is going to be our given angle which is 210 minus 180 degrees, 210 minus 180 is going to leave us with degrees. So what this means is that we have a reference angle of 30 degrees. And what this also states is that cosine of degrees is equivalent to cosine of degrees. So how can we get cosine of 30 degrees from this reference angle? Well, since we have the reference angle. We can use this to create a right triangle that is located in quadrant three. If we draw this triangle a little bit bigger, we end up with this image where we have a 30 degree angle located here and we have a 90 degree angle. And now what we wanna do is we want to label the side links using our special right triangles. In this case here, we're going to be using a 30 60 90 degree triangle across from 30. We're going to have a length of one across from 60. We're going to have a length of the 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 quadrant three for quadrant three, any X values or the in ca in this case, the length of the triangle is going to be negative and any Y values or the height of this triangle is going to be negative as well. So now that we have a right triangle, we can use this to solve for cosine of 30 degrees. Now recall that cosine for any right triangle is defined as the X value over the hypotenuse our X value in this case is going to be negative square root of three and our hypotenuse is going to be two. So if we plug in these values into the identity of cosine cosine of degrees is going to equal to negative square root of 3/2. So now that we have the value of cosine of 30 recall that earlier we stated that cosine of 210 is equivalent to cosine of 30. What this means is that cosine of 210 degrees is also equal to negative square root 3/2. And this is going to be the solution to this problem. 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 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cos 225°

Key Concepts

Reference Angles

Unit Circle and Quadrants

Exact Values of Common Angles

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