Find the exact value of each expression. See Example 1.

sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .

Video transcript

Hey, everyone in this problem, we're asked to determine the exact value the following trigonometric expression. And we're given sine of three pi divided by four multiplied by cosine of pi divided by 12 minus cosine of three pi divided by four multiplied by sine of pi divided by 12. We have four answer choices. Option A, the square root of three divided by two. Option B negative square root of three divided by two, option C one half and option D one. So first instinct might be to go through and evaluate each of these terms individually and simplify. OK. And that's an OK strategy. The problem here is that we don't know an exact value for cosine of pi divided by 12 and sine of pi divided by 12. OK. So we can't just evaluate them in this case and get an exact value. However, we can use a trigonometric identity. OK. So recall that if we have sign of A minus B, we can write this as sign of a multiplied by cosine of V minus cosine of a multiplied by sign of B. Now this expression on the right hand side is exactly what we have in the expression we were given. OK? Where the A value is equal to three pi divided by four and the B value is equal to pi divided by 12. OK. All right. So we can take our expression sign A three pi divided by four, multiplied by cosine of pie, divided by 12 harness cosign three pie divided by four, multiplied by sign of pi divided by 12. And we can write this as sign of three P divided by four minus pi divided by 12. All right. So now we've simplified this into a single trigonometric expression and we just have one trig function sign evaluated it at some value. What's simplifying the brackets? And we're gonna hope that the thing inside the brackets is a value that we can evaluate sign at exactly. So we can find a common denominator by multiplying the first value by three in the numerator three in the denominator that gives us nine pi divided by 12. And then we are subtracting pi divided by 12. OK? This is gonna be equivalent to assign of eight pi divided by 12. And we can divide numerator and denominator by four to get sign of two pi divided by three. Now the value of sine of two pi divided by three might be something you already remember. If not, we can think about this graphically. So if we draw a little graph, OK, two pi divided by three is gonna be in quadrant two we can go ahead and draw that out and we know that the angle measured from the positive X axis is going to be two pi divided by three. Now, our reference angle, the prime is going to be that same angle we had. But now we're gonna measure down to the negative x axis. That's gonna give us that angle between zero and 90 degrees or between zero and pi divided by two. OK. So our reference angle theta prime is going to be equal to pi minus two pi divided by three. And because if we go all the way from the positive X axis to the negative X axis, we know that that's pi and then we have to subtract that two pi divided by three that we're already going to on the right hand side. OK. This gives us that our reference angle theta prime is going to be pi divided by three. So we have a reference angle so we can evaluate sign at a reference angle. The only other thing to figure out is the sign now sign is positive in quadrant two. OK. We're in quadrant two sin is positive. So what this means is that sine of two pi divided by three is just going to be equal to the value of S at that reference angle of pi divided by three. Now sine of pi divided by three. That's one of those values that we see over a note. OK. So sine of pi divided by three is a value that you should remember or that you can use your unit circle to find or your, your um triangles, identity triangles, any of those things that you need to remember sign of pi divided by three. And that is gonna give us the square root of three divided by two. OK. All right. So we started with this messy expression. We used our trig identity to simplify it down. And we found that it's equivalent to the square root of three divided by two which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.