Find the exact value of each expression.

sin 255°

Video transcript

Hello, today we are going to be evaluating the trigonometric expression. The expression given to us is sign of 195 degrees. Now, the tricky thing about this problem is that 195 degrees is not a standard angle on the unit circle. Instead, what we can do is we can rewrite 195 degrees to be the sum of standard angles on the unit circle. Now 60 degrees plus 135 degrees will give us 195 degrees. So we can rewrite the inside angle of the trigonometric expression as sign of 60 degrees plus 135 degrees. Furthermore, we can evaluate this function by using the sum identity for sign the sum identity for sine is given to us as sine of X plus Y is equal to sin of X multiplied by cosine of Y plus cosine of X multiplied by sine of Y. In this case here, we need to figure out our X and Y values. Well, in the expression 60 degrees is in the spot of X. So we will allow X to be 60 degrees Furthermore, 135 degrees is in the spot of Y. So Y will equal to 135 degrees. Now that we have our X and Y values, we can rewrite the trigonometric expression as sine of 60 degrees plus 135 degrees is equal to sin of X which is sine of 60 multiplied by cosine of Y which is cosine of 135 plus cosine of X which will be cosine of 60 multiplied by sine of Y which is sine of 135. Now, we need to evaluate each of the trigonometric expressions on the right hand side. In order to do that, we will need the help of a unit circle. Now, there are two main angles that we are working with. The first angle is the 60 degree angle which exists in the first quadrant of the unit circle. The corresponding terminal point is given to us as one half comma square root of three divided by two. The second angle we are working with is the 135 degree angle. This angle is located in quadrant two and the corresponding terminal point will be negative square root of two divided by two comma square root of two divided by two. Now recall that any terminal point on the unit circle can be represented as cosine comma S sign. What this means is that every cosine value will equal to the X value of the terminal points. And every sine value will equal to the Y values of the terminal points. So in order to evaluate the right side, we will start by evaluating sin of 60 degrees sine of 60 degrees will be the Y value of the terminal point at the 60 degree angle. And that value is given to us as square root of three divided by two. So sine of 60 will evaluate to give a square root of three divided by two cosine of 135 degrees will be the X value of the terminal point at the 135 degree mark. So cosine of 135 will evaluate to give us the negative square root of two divided by two. Next cosine of 60 will be the X value of the terminal point located at the 60 degree angle. This value will evaluate to give us one half and finally sign of 135 degrees is the Y value of the terminal terminal point located at the 135 degree angle. And this value will give a square root two divided by two. Now we will algebraically simplify square root of three divided by two multiplied by negative square root. Two divided by two will give us negative square root of six divided by four, one half multiplied by square root of two divided by two will give us the square root of two divided by four. Once we add the two fractions together, we will be left with the negative square root of six plus the square root of two divided by four. And this will be the final value of the original 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.