In Exercises 87–92, find the exact value of each expression. W...

Question

In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin 𝜋/3 cos 𝜋 - cos 𝜋/3 sin 3𝜋/2

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the following expression. Without the help of a calculator. Write your answer as a single fraction. Our expression is the sign of three pi divided by two, multiplied by the cosine of pi divided by three minus the cosine of pi divided by six, multiplied by the sign of three pi divided by four. Our answer choices are answer choice A in the numerator, the quantity of two plus the squared of six all divided by four. Answer choice B in the numerator, the quantity of negative two minus the squared of six all divided by four answer choice C in the numerator, the quantity of two minus squared of six all divided by four and answer choice D in the numerator, the quantity of negative two plus the squared of six all divided by four. All right. So where are all of these angles? Let's draw a rough sketch of a coordinate plane. We can visualize where these angles are. We have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. Let's label our quadrants too. So starting from the positive part of the X axis heading counterclockwise, we have our first quadrant, then our second quadrant, then our third quadrant and our fourth quadrant. So we our first angle here is three pi divided by two, well, three pi divided by two. That's 3/4 of the way toward a full rotation at two pi three pi divided by two is the negative part of the y axis That is a quadrant angle. And by definition, we recall from previous lessons that the sign of a quadruple angle of three pi divided by two is equal to negative one. So that is that first one simplified now that's being multiplied by the cosine of pi divided by three. Where's pi divided by three. Well, we know from previous lessons that halfway from a full rotation of two pi is going to be just pi that's the negative part of the X axis. So pi divided by three is a third of the way two pi which is still going to be in the first quadrant just closer to the positive part of the Y axis than the positive part of the X axis. And so for our cosine of pi divided by three, we recall from previous lessons by definition that is positive one half now is our cosign positive in the first quadrant. Yes, it is, they're all positive in the first quadrant. So yes, that one stays positive. Now we subtract and then we're taking the cosine of pi divided by six. Where is that one at all? Right, pi divided by six is 1/6 of the way to pi. So that's in the first quadrant as well. And it is going to be closer to the positive part of the X axis than the positive part of the Y axis. And so we recall that by definition, the cosine of pi divided by six is going to be the square of three divided by two. And as we said before, that's positive, they're all positive in the first quadrant. Now, how about the sign of three pi divided by four? Where's three pi divided by four? That one is 3/4 of the way to pie. So that one's in the second quadrant directly halfway between the positive part of the Y axis and the negative part of the X axis. And so we know that the reference angle for three pi divided by four in the second quadrant is going to be pi divided by four. So we're talking about the sign of pi divided by four and we know that the sign of pi divided by four by definition is going to be the square root of two divided by two. Is that positive or negative? Well, the sign in the second quadrant is going to be positive. So that one stays the square root of two divided by two. Now it's just some algebra to simplify following the order of operations, we'll do the multiplication first, negative one multiplied by one half is negative one half and subtracting. And now multiplying the square root of three divided by two by the squared of two divided by two, multiplying our numerators. That's the squared of six divided by our denominators multiplied together two multiplied by two is four. So in order to subtract these two fractions, we need a common denominator that's going to be four. So we multiply negative one half by two, divided by two. So we'll have a negative two divided by four minus 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 they just stay the quantity of negative two minus the squared of six in the numerator all divided by our common denominator four. And that's as simplified as that one gets. So there is uh the exact value of our expression. We take a look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin 𝜋/3 cos 𝜋 - cos 𝜋/3 sin 3𝜋/2

Key Concepts

Unit Circle and Special Angles

Trigonometric Identities and Angle Multiplication

Algebraic Simplification of Trigonometric Expressions

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