Use one or more of the six sum and difference identities to so...

Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the following expression by using the difference identity. Our expression is the sign of 315 degrees minus degrees. Our answer choices are answer choice A negative square at six divided by four. Answer choice B, the quantity of the square rate of two minus the square rate of six divided by four answer to ac the square it of three divided by four. And answer choice D, the quantity of the square of two plus the square of six divided by two. All right. So we are finding the exact value of this expression using the difference identity. We recall from previous lessons that the difference identity for sign, when we have the sign of angle A minus angle B that equals the sign of A multiplied by the cosine of B minus the cosine of A multiplied by the sign of B. And we see here that 315 degrees is our angle A and 120 degrees is our angle B. So what we want to do here is find the exact values of our sign of angle A, our cosine of angle B etcetera and then plug those in and then simplify. And we do that, we can get a unit circle draw on a very rough sketch. We have a horizontal X axis, a vertical y axis. They come together at the origin in the middle, we overlay that with a rough unit circle having three hash marks in each quadrant. And we'll start off by working on angle A which is 315 degrees. 315 degrees is in the fourth quadrant. It's halfway between the positive part of the X axis and the negative part of the Y axis. And so the recalling from previous lessons, the reference angle for 315 degrees is 45 degrees. We can draw a rough sketch of a 45 45 triangle, 90 degrees is the right angle and the other two angles are degrees. The hypotenuse for this angle is the square root of two and the two other side lengths are equal to one. So now if we're talking about and the sign of our reference angle 45 degrees, that's going to be the opposite, divided by the hypotenuse, the opposite here is one opposite of 45 degrees is one and the hypotenuse is the square it of two. But we don't want that radical in the denominator. So we rationalize it by multiplying both the numerator and the denominator by the square it of two. And then this simplifies two, the square it of two divided by two. So now for the sign of our original angle 315 degrees, we're in the fourth quadrant is sine positive or negative in the fourth quadrant. Well sine is negative. So we're talking about the negative square at two divided by two for the sign of angle A. Now let's keep working on angle A. Now we'll find the cosine of angle A. So let's go back to our reference angle, the cosine of 45 degrees. Very similarly, this one's the adjacent divided by the hypo and here the adjacent I can't write hypotenuse. OK. There adjacent divided by hypotenuse adjacent is also one. So it's again one divided by the square of two. So again, this is the square of two divided by two. But this time its cosine is cosine positive or negative in the fourth quadrant and cosine is positive. So the cosine of angle A is going to be the positive square root of two divided by two. All right. Now let's work on angle B. Going back to our unit circle angle B, 120 degrees is going to be in the second quadrant and 120 degrees is the hash mark that is closer to the positive part of the Y axis than the negative part of the X axis. And recalling from previous lessons, the reference angle for 120 degrees is 60 degrees. So if we draw a rough right triangle for a 30 60 90 which mine looks a lot like my 45 90. But I'll label it differently. The right angle is still 90 degrees. The smaller one will call 30 degrees and then the one that's left 60 degrees, labeling all of our sides. The hypotenuse is the square re of two. No, it's not. That was the 45 45 90. The hypo news is just two and we're talking about 60 degrees. The opposite of that is the square of three and the adjacent to that is one. So if we are working on the sign, we can do the sign first of 120 degrees. But we'll look at the reference angle sine of 60 degrees is equal to, again, sign is opposite divided by hypotenuse, opposite of 60 degrees is the square of three. Hypotenuse is two. Now, we're talking about the sign of 120 degrees in the second quadrant and in the second quadrant sine is positive. So for the sign of 120 degrees, we have the positive square it of three divided by two. Now let's work on its cosine, we'll do the reference angle. The cosine of 60 degrees which will be adjacent, divided by hypo again, adjacent to 60 degrees is one divided by hypotenuse is two cosine in the second quadrant. Is that positive or negative. And that here is negative. So we've got the cosine of angle B is negative one half. And now we've got this all plugged in and now we can just do the multiplication and subtraction. So the first multiplication is the negative square it of two divided by two multiplied by negative one half. And that's going to be a positive square root of two divided by four bring down minus. And now we can do the second bit of multiplication, taking the square it of two divided by two, multiplied by the square it of three divided by two. That will be the square of six divided by four. We have a common denominator. So we can combine our numerators but they're not like terms. So it's just the quantity of the square it of two minus the square it of six all divided by four. And that is our exact value. We look at our answer choices and this matches with answer choice B well done. We'll catch you on the next one.

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

Key Concepts

Sum and Difference Identities

Exact Values of Common Angles

Angle Measurement in Degrees

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