In Exercises 14–19, use a sum or difference formula to find th...

Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

Accepted Answer

Hello, everyone. We are asked to use a sum or different formula to determine the exact value of the given trigonometric expression. The expression we are given is the sign of degrees. There are four answer choices to begin with. I have to decide if I'm going to be working with a sum or a difference. So I'm asking myself, are there numbers that I already have in a unit circle or anywhere else that I know the exact value of that either add up to 255 or I can subtract and get 255. And though I can't come up with any that are exact, I think working with 100 80 degrees plus degrees might be our best bet to get to degrees with minimal extra work. So we are gonna treat this as the sign of degrees plus 75 degrees from there. Recall that when we have a sum, the sign of a sum. So the sign of A plus B equals the sign of a multiplied by the cosine of B plus the cosine of A multiplied by the sign of B. So in our example, we're gonna say 180 degrees is A 75 degrees is B. So we're gonna start filling this out. So this equals the sign of A. So sine of 180 degrees multiplied by the cosine of B. So the cosine of 75 degrees plus the cosine of A. So cosine of 100 80 degrees multiplied by the sign of B. So the sign of 75 degrees. Well, we easily from a unit circle can find that the sign of 100 80 degrees equals zero and the cosine of 180 degrees equals negative one, we will plug those in. But unfortunately, 75 is not on our unit circle. So to find the sign and the cosine of 75 degrees, we need to use another sum formula. So starting with the cosine of degrees. I'm gonna treat this as the cosine of degrees plus 30 degrees. And now I have another sum formula I need to work with. And the cosine sum formula states that the cosine of A plus B equals the cosine of A multiplied by the cosine of B minus the sign of A multiplied by the sign of B. So we do need to use this formula in order to find out what the cosine of 75 is to go back to our original problem. What is helpful is using a unit circle. We can find that the cosine of 45 degrees is radical three oh I'm sorry, radical two divided by two. And what else do we need here? We need to know the cosine of 30 degrees, which is a radical three divided by two. Um Also for this formula, we need to know the sign of 45 degrees. So radical two divided by two and we need to know the sign of 30 degrees, which is one half. So let's plug those into our cosine uh sum formula. And when we do so we get the cosine of 45 degrees multiplied by the cosine of degrees minus the sign of degrees multiplied by the sign of 30 degrees. So putting the exact values in there, I have the cosine of 45 degrees. So square root of two divided by two multiplied by the square to three divided by two minus the sine of 45 degrees. So the squared of two divided by two multiplied by the sine of 30 degrees, one half multiplying straight across my first term, I have the square root of six divided by four minus. And this will be the square root of two divided by four, which I can write as one single fraction. And that is the square root of six minus the square root of two divided by four. And that is what the cosine of 75 degrees equals. We still need to find the sign of 75 degrees, however, to be able to finish this original formula. So we will do so right here. So the sine of 75 degrees, we again are gonna treat this as if it's the sign of degrees plus the sign of 30 degrees. So we're using the same formula we have up of top in red. So it'll be the sign of 45 degrees multiplied by the cosine of 30 degrees plus the cosine of 45 degrees multiplied by the sign of degrees. Good news is we've already looked up what those values are in our unit circle. So it's one less step. So the sign of 45 degrees was radical two divided by two multiplied by the cosine of 30 degrees, which is radical three divided by two plus the cosine of 45 degrees. So radical two divided by two multiplied by the sine of 30 degrees one half. All right. So this gives me the square root of six divided by two plus the square root of two. Oops, I'm sorry, it's a squared of six divided by four and then plus the squared of two divided by four, we can rewrite this as one single fraction and it would be the squared of six plus the square root of two divided by four. And that is what the sign of 75 degrees equals. So now I think we finally have all of our values. So let's go back and plug them all into our original sum formula. So the sign of 180 degrees was zero. And we're gonna multiply that by the cosine to 75 degrees. So square to six minus the square to two divided by four plus the cosine of 180 degrees, which is negative one multiplied by our sign of 75 degrees. So the square is six plus the square to two divided by four. Anything multiplied by zero is zero. So I guess we really didn't have to do the cosine of 75 degrees in the end. But the more practice the better, right? And then one multiplied by or sorry, negative one multiplied by the square to six plus the square to two divided by four will give us negative radical six plus radical two divided by four. And that is our exact value of the sign of degrees. And answer choice B have a nice day.

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

Key Concepts

Sum and Difference Formulas for Sine

Reference Angles and Angle Decomposition

Exact Values of Sine and Cosine for Special Angles

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