Use a half-angle identity to find each exact value.
sin ...

Question

Use a half-angle identity to find each exact value.

sin 195°

Accepted Answer

Hello, today we're going to be determining the exact value of the given trigonometric expression using the half angle identity. The trigonometric expression given to us is sign of 202.5 degrees. Now, since the problem asks us to find the exact value using the half angle identity, let's quickly recall what the half angle identity for sign is. The half angle identity is given to us as sign of theta divided by two equal to the positive and negative square root of one minus cosine theta all of that divided by two. So in order to use the half angle identity, we're first going to need to figure out the angle theta that we want to plug into the half angle identity. Now we need to be careful because theta divided by two is not going to be the same as 202.5. So if we want to figure out the angle theta to plug into the half angle identity, what we can do is we can take the half angle theta divided by two and set it equal to 202.5 degrees. Once we solve our angle theta that would give us the angle that we can plug into the half angle identity to sulfur theta will multiply both sides of this equation by two. And that will give us theta is equal to 405 degrees. So now that we know that data is going to be 405 degrees. Let's go ahead and plug it in to the half angle identity. That is going to be sign of 405 degrees divided by two equal to the positive and negative square root of one minus cosine of 405 degrees. All of that divided by two. Now, there are a few things that we can simplify here. 1st 405 divided by two will simplify to give us 202.5. Furthermore cosine of 405 degrees is equivalent to the angle cosine of 45 degrees. And on the unit circle cosine of 45 degrees will evaluate to give us the square root of two divided by two. So we can simplify the half angle identity to be sign of 202.5 degrees equal to the positive and negative square root of one minus the square root of two divided by two. This entire numerator divided by two. Let's go ahead and simplify the inside of the square root inside the square root. We have one minus the square root of two divided by two, the entire thing divided by two, we can rewrite one to be 2/2. And that'll give us 2/2 minus the square root of two divided by two, all divided by two, combining the two fractions. The numerator will give us two minus the square root of two divided by two, that fraction divided by two. And if we multiply both of the denominators together, this will simplify to give us two minus the square root of two divided by four. So what we have is sign of 202.5 degrees is equal to the positive and negative square root of two minus the square root of two divided by four. Furthermore, we can simplify the denominator to two and that will give us this plus or minus square root of two minus the square root of two all divided by two. Now, currently we have two solutions to sign of 202.5 degrees. We have a positive and negative solution. But the question here is, do we take the positive solution or the negative solution? Well, in order to answer that question, we need to take a look at our angle theta. Currently, our angle theta lies in the third quadrant of the unit circle and in the third quadrant of the unit circle sine is negative. So we'll be taking the negative value of the square root. That means that sine of 202.5 degrees will equal to the negative square root of two minus the square root of two divided by two. And this is going to be the exact value of the given trigonometric 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.

Use a half-angle identity to find each exact value.
sin 195°

Key Concepts

Half-Angle Identities

Reference Angles

Quadrant Analysis

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