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

Question

Use one or more of the six sum and difference identities to solve Exercises 13–54.

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

5𝝅 𝝅 5𝝅 𝝅
sin ------- cos -------- ﹣ cos -------- sin -------
12 4 12 4

Accepted Answer

Welcome back. I am so glad you're here. We're asked to express the following difference as one trigonometric function only then determine its exact value by using a unit circle. So two things to figure out here, our difference that we're given is the sign of 13 pi divided by eight, multiplied by the cosine of seven pi divided by minus the cosine of 13 pi divided by eight multiplied by the sign of seven pi divided by 24. We look at our answer choices and they are answer choice. A, the sign of four pi divided by three and negative square at three divided by two. Answer choice B the sign of 11 pi divided by six and negative one half. Answer choice C the cosine of five pi divided by six and the negative square at three divided by two and answer choice. D the cosine of seven pi divided by four and the square root of two divided by two. All right. So we look at this difference and we can kind of recognize there's a little bit of a pattern here. The first thing here is the sign of 13 pi divided by eight. And that's being multiplied by the cosine of seven pi divided by 24. But then when we're subtracting again, we've got 13 pi divided by eight. But this time, it's the cosine of 13 pi divided by eight, that's multiplied by the seven pi divided by again. But now it's the sign of seven pi divided by 24. We could say that angle A is 13 pi divided by eight and angle B is seven pi divided by 24. And then what we'd be doing is taking the sign of angle A and multiplying that by the cosine of angle B and then subtracting the cosine of angle A multiplied by the sign of angle B. And that's familiar because that is the sign of A minus B. So we know what our A is A is the 13 pi divided by eight and we know what our B is. That's seven pi divided by 24. So what we can do is we can just take the sign of 13 pie divided by eight minus seven pi divided by 24 all of that in parentheses. Now, I just need to do this subtraction. Well, we need a common denominator here and that's going to be 24. So multiplying the first fraction, the 13 pi divided by eight by three, divided by three. And so then we have 39 pi divided by 24 minus seven pi divided by 24 and 39 pi minus seven pi gives us 32 pi still divided by 24. Well, those are both divisible by eight. So that simplifies to be four pi divided by three. And hey, that is what we were looking for. The sign of four pi divided by three is the difference as one trigonometric function. So we got the first part done. Now, we need to determine its exact value using a unit circle. All right, I'm going to draw a rough sketch of a unit circle. We have a horizontal X axis, vertical y axis over lane with a unit circle. Mine is not very unity and then we have our hash marks three in each quadrant. And we recall from previous lessons that four pi divided by three is in the third quadrant. So that's where X and Y are both negative and it is going to be the hashmark that is closer to the negative part of the Y axis than the negative part of the X axis. So that's our four pi divided by three. And if you need a review on the unit circle, definitely check out previous lesson videos because I'm going through this part fairly quickly. So we know where our four pi divided by three is and we need to know what the reference angle for that is. And that's going to be our pi divided by three for our reference angle or our degrees. So we're looking at the sign of pi divided by three or the sign of 60 degrees. So if we recall our special triangles when we've got a 30 60 90 triangle, and we're taking the sign. So that's going to be opposite over hypotenuse while labeling our sides. The hypotenuse here is the squared of two, the opposite of 60 degrees is one or excuse me, it's not OK. The high pot news for our 60 degrees is going to be two, not the squared of two and the opposite of the 60 degrees is going to be the square of three. And just for kicks, the adjacent side is one. So for the sign where we want the opposite over the hypotenuse, that's the square root of three divided by two. But the last important thing here, we're in quadrant three is sine positive or negative and quadrant three, it's negative and quadrant three. So we're talking about the negative square it three divided by two and those are our two answers. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Key Concepts

Sum and Difference Identities

Trigonometric Functions

Exact Values of Trigonometric Functions

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