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

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 3𝜋 tan (-15𝜋/4) - cos (-5𝜋/3) 
2

Accepted Answer

Welcome back. I am so glad you're here. We're 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 four, multiplied by the tangent of negative 13 pi divided by four minus the cosine of negative seven pi divided by six. And our answer choices are answer choice A in the numerator, the quantity of negative squared of two minus the squared of three all divided by two. Answer choice B in the numerator, the quantity of the squared of two plus the square root of three all divided by two answer choice C in the numerator, the negative square it of two plus the square, it of three all divided by two. And answer choice D in the numerator, the quantity of the squared of two minus the squared of three all divided by two. All right. So let's figure out where our angles are. I'm going to draw a rough sketch of a coordinate plane. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle and starting from the positive part of the X axis. As we head toward our quadrants, we have quadrant one. If we're going counterclockwise up from the X axis, then continuing counterclockwise quadrant two, then quadrant three and quadrant four. So our first angle three pi divided by four. Well, that's 3/4 of the way toward pi pi is the negative part of the X axis and we're heading counterclockwise. So 3/4 of the way there is approximately halfway, exactly halfway in between the positive part of the Y axis and the negative part of the X axis, we are talking about the second quadrant. And so the reference angle for our three pi divided by four is going to be pi divided by four. So we want to figure out the sign of pi divided by four. Well, we recall from previous lessons that by definition, with this special right angle, the sign of pi divided by four is the square root of two divided by two. Now, in the second quadrant, we need to know if sine is positive or negative and it is positive in the second quadrant. So that's what we've got for that first one. Now let's start with breaking down our tangent of negative 13 pi divided by four. So where is that angle? Well, negative 13 pi divided by four, that's got a full rotation in it. So if we're talking about a denominator of four, our two pi our full rotation is going to be eight pi divided by four. We can add that to, we take negative 13 pi divided by four plus eight pi divided by four. And that will bring us to a negative five pi divided by four, negative five pi divided by four is actually from our initial side heading all the way around. That's the exact same place as our three pi over four in the second quadrant. So this is the same angle here that we're talking about. And again, we're going to have that same reference angle of pi divided by four. So we're multiplying this by the tangent of pi divided by four. And by definition, we know that the tangent of pi divided by four is going to be one. Now, here's the thing is tangent positive or negative in the second quadrant. Well, it's negative. So this is being multiplied by a negative one, we could bring these together a little bit. So we have the squared of two divided by two, multiplied by negative one. Got the first two done. Now let's take a look at our minus the cosine of negative seven pi divided by six. So let's figure out where negative seven pi divided by six is. So from our initial side, negative seven pi divided by six is going to be again in the second quadrant, it's just passed hour while I can draw a straight line here just past our pie. So just above the negative part of the x axis in the second quadrant. And we know that that the reference angle for a negative seven pi divided by six, for that angle right there is going to be pi divided by six. So we're looking at minus the cosine of pi divided by six. And we know that by definition, the cosine of pi divided by six is going to be the square root of three divided by two. And so that one's being subtracted. But the question again is, is cosine positive or negative in the second quadrant and cosine is negative. So we have minus a negative or in other words, that's going to be plus. So now just a little bit of algebra taking negative one multiplied by the squared of two divided by two is a negative squared of two divided by two plus the squared of three divided by two. We have a common denominator. So we can combine our numerators in the numerator. We can put the quantity of negative square it two plus the squared of three all divided by two. And there is our exact value of the expression solved. Without the help of a calculator. We look at our answer choices and this matches with answer choice. C 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 3𝜋 tan (-15𝜋/4) - cos (-5𝜋/3) 2

Key Concepts

Trigonometric Functions

Angle Reduction and Reference Angles

Combining Trigonometric Values

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