Evaluate each expression. Give exact values. tan² 120° - 2 cot...

Question

Evaluate each expression. Give exact values. tan² 120° - 2 cot 240°

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 three cotangent squared, 300 degrees minus tangent 150 degrees. Our answer choices are answer choice. A, the quantity of four plus square three divided by three. Answer choice B, the quantity of four minus square at three divided by three answer. Choice C, the quantity of three plus square at three divided by three and answer choice D, the quantity of three minus square at three divided by three. All right. So taking a look at our angles here, we've got 300 degrees and 150 degrees. Let's try to figure out what their reference angles are first. So 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 first for 300 degrees. That's going to have its initial side along the positive part of the X axis and its vertex at the origin. And then heading counterclockwise, it's going to go all the way around into the fourth quadrant. So it's going to head toward negative infinity for Y and positive infinity for X, a little bit closer to the posit or excuse me, the negative part of the Y axis than the positive part of the X axis. And that is our 300 degrees. Now, to find the reference angle for a quadrant four angle, we are going to take 360 degrees minus our angle. So minus 300 degrees and 360 degrees minus degrees is 60 degrees. So our reference angle is 60 degrees, we want to know what the cotangent of degrees is. And recalling from previous lessons and the tables in the textbook, we know that that has an exact value and the exact value for the cotangent of 60 degrees is the square of three divided by three. But we're talking about the cotangent squared of 300 degrees. And we need to know whether cotangent is going to be positive or negative for 300 degrees, which is in the fourth quadrant, we recall from previous lessons that cotangent, the reciprocal function of the tangent function is going to be positive in the first quadrant and it's going to be positive in the third quadrant. Therefore, in the fourth quadrant, it's going to be negative. So we've got a negative square at three divided by three for our cotangent of 300 degrees. But also noting that since it's squared, we need to square this and also not forgetting that it's being multiplied by that three in front of it either. All right, we've got an exact value for the first one. Now, let's take a look at that tangent, 150 degrees, 150 degrees also has its initial side along the positive part of the X axis. And then it has its terminal side all the way into the second quadrant. And it's going to be heading toward negative infinity for X and positive infinity for Y, a little bit closer to the negative part of the X axis than the positive part of the Y axis. There is our 150 degrees finding the reference angle for 150 degrees because it's in the second quadrant, we're going to take 180 degrees minus our angle some degrees minus 150 degrees and that is degrees. So for our reference angle, we want to know what the tangent of 30 degrees is. And recalling from those same previous lessons and tables in the textbook, the tangent of 30 degrees is equal to the square of three divided by three as well. But we're talking about the tangent of 150 degrees which is in the second quadrant. And we need to know is tangent positive or negative in the second quadrant. Well, we already said that tangent like cotangent is going to be positive in the 1st and 3rd quadrants. So in the second quadrant again, it's negative. So this is another negative square at three divided by three. Noting that this second exact value is being subtracted from the first. Now, all we have to do is simplify algebraically, let's begin with this negative square root three divided by three squared. Once we square that we're going to have taking our negative part squared, that'll be positive, taking the square out of three square, that's just going to be three and taking our three in the denominator and scoring that, that's going to be nine. So we'll have 3/9 we can just keep it as 3/ for a moment. That's going to be multiplied by the three in front of it. And then we have minus negative square at three divided by three. Well, minus and negative is going to be plus. So plus square up three divided by three. Now, we can multiply that three distribute the three to the 3/ 3 multiplied by three and the numerators is nine. So that's going to be nine ninth, which is one. But noting that we're going to need to add that to the square at three divided by three. We need a common denominator of three. So that is also equivalent to three thirds. So we'll have a three thirds plus square at three divided by three. Now that we have a common denominator, we can add our numerators. That's going to be the quantity of three plus square at three all divided by that common denominator three. And that's as simplified as this one gets. We look at our answer choices and this matches with answer choice. See, well done. We'll catch on the next one.

Evaluate each expression. Give exact values. tan² 120° - 2 cot 240°

Key Concepts

Trigonometric Functions and Their Values

Reference Angles and Quadrants

Trigonometric Identities and Simplification

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