Evaluate each expression. Give exact values. sec² 300° - 2 cos...

Question

Evaluate each expression. Give exact values. sec² 300° - 2 cos² 150°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to find the value of the given expression. Our given expression is six cosine squared, 240 degrees minus C 150 degrees. Our answer choices are answer choice. A, the quantity of nine minus four square at three divided by six. Answer choice B the quantity of nine plus four square at three divided by six answer choice C the quantity of nine plus two square at three divided by six and answer choice D the quantity of nine plus four square at three divided by three. All right, looking at our angles here, we've got 240 degrees and 150 degrees. Let's find out what their reference angles are for quadrant one, we'll draw a rough sketch of a coordinate plane. We have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle for 240 degrees. It's got its initial side along the positive part of the X axis vertex at the origin and then 240 degrees moving counterclockwise heads all the way down into the third quadrant heading toward negative infinity for X and Y a little bit closer to the negative part of the Y axis than the negative part of the X axis. So there's our quadrant three, 240 degrees. And so our reference angle for a quadrant three angle, we're going to subtract 180 degrees and degrees minus 180 degrees is 60 degrees. So our reference angle is 60 degrees. We can look at the tables in the textbook or recall from previous lessons that for 60 degrees, we've got an exact value for the cosine of that. So for the cosine of degrees, that's equal to the exact value of one half. But we're talking about the cosine of 240 degrees, which is in quadrant three now is cosine positive or negative in quadrant three. Well, we recall from previous lessons that cosine is going to be positive in quadrants one and quadrant four. So it's negative in quadrant three. That means that for the cosine of 240 degrees, we are talking about a negative one half. Now, the other thing to note is that cosine here is being squared. So we're going to need to square our negative one half and also not forgetting that that's being multiplied by a six in front of it. All right, we've got an exact value for that first trigonometric function. Now let's look at 150 degrees, let's figure out that reference angle. So for 150 degrees again, that's got an initial side along the positive part of the X axis. And then it's got a terminal side that's going to be in the second quadrant heading toward negative infinity for X and heading toward positive infinity for Y a little bit closer to the negative part of the X axis than the positive part of the Y axis. So there's our quadrant two, 150 degrees to find that reference angle. For a quadrant two angle, we are going to take 180 degrees minus our angle. So minus 150 degrees, 180 degrees minus 150 degrees is 30 degrees. So our reference angle here is 30 degrees. So we can look at those tables in the textbook or recall from previous lessons that if we're talking about the sea can of 30 degrees that's equal to the exact value of two square at three divided by three. But again, we are talking about C can. And so we need to ask ourselves if the sea can of 150 degrees is positive or negative in the second quadrant. And we recall from those same previous lessons that C can is the reciprocal function of cosine. So it's also going to be positive in the 1st and 4th quadrants negative in the second quadrant. So again, we've got a negative value here for this exact value, it's going to be negative two square at three divided by three. And we note that that's being subtracted from the first value. Now, all we need to do is simplify algebraically. So let's start off with this negative one half squared, negative one half squared is going to be a positive 1/4. We're still multiplying that by six, then subtracting we'll just bring this down negative two square at three divided by three. Now we can distribute the 66 multiplied by 1/4 is six fourths which simplifies 23 halves. And then we have a minus a negative. Well, that's just going to be a positive. So we'll have a three halves plus two square at three divided by three. We want to have a common denominator and that's going to be six. So we're going to multiply the three halves by three thirds to get our common denominator of six. So that will be a nine sixth plus for the two square at three, divided by three to get to our common denominator of six. We're going to multiply that by two halves. And that is going to simplify to be four square at three divided by six. Now that we have a common denominator, we can combine the numerators that will be the quantity of nine plus four square at three, all divided by the common denominator of six. And that's as simplified as this one gets. So we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Evaluate each expression. Give exact values. sec² 300° - 2 cos² 150°

Key Concepts

Trigonometric Functions and Their Values

Unit Circle and Angle Reference

Pythagorean Identities and Simplification

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