Match each expression in Column I with its value in Column II....

Question

Match each expression in Column I with its value in Column II.

8. tan (-π/8)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the tangent of negative seven pi divided by 12. Our answer choices are answer choice A two minus square root three, answer choice B two plus square root three, answer choice C two plus two, square root three and answer choice D square root three minus two. All right. So first thing here we recall from previous lessons, the tangent is an odd function which means that the tangent of a value will say a negative X is equal to a negative tangent of that positive X. So here where we have our tangent of negative seven pi divided by 12, that's equal to a negative tangent of a positive seven pi divided by 12. Well, that is nice, but that doesn't get us our exact value. So how do we do that? Well, seven pi divided by 12 isn't on the unit circle. So it's not so easy to identify its exact value. However, seven pi divided by 12 is really similar to something that is on the unit circle. If we consider that seven pi divided by 12 would be half of an angle, we'll say a divided by two. If we cross multiply here to solve for a, we have 14 pi equals 12. A dividing both sides by 12, we'll have a equals 14 pi divided by 12, which simplifies to be seven pi divided by six, which is on the unit circle. Well, if our seven pi divided by 12 is half of an angle, we can use a half angle identity to get an exact value. So we recall from previous lessons, the half angle identity where the tangent of A divided by two is equal to. We've got options here. We can use the one where we have one minus the cosine of A all divided by the sine of A. We know that our A is equal to seven pi divided by six. And we can use that unit circle to find the cosine of seven pi divided by six and the sine of seven pi divided by six. And then we're just simplifying. Now, one very important thing here to note it is that we're not dealing with a positive tangent of seven pi divided by six, divided by two. We're dealing with a negative. So we need to be sure that we keep both sides of our equation negative. So we have a negative tangent of our A is seven pi divided by six and that's divided by two equals on the right. This whole thing is negative we have in the numerator one minus the cosine of a seven pi divided by six. And that's divided by the sine of a which is seven pi divided by six. Recalling from previous lessons, figuring out the cosine and the sine of seven pi divided by six from the unit circle. We've got a vertical y axis, a horizontal x axis, they come together at the origin. We'll put a unit circle over that every point on the unit circle is one unit away from the origin, which is not at all true. Of what I drew seven pi divided by six is an angle in the third quadrant. It's closer to the negative part of the X axis than the negative part of the Y axis. And its XY coordinates are negative square three divided by two negative one half, which is important because we recall from previous lessons that the cosine of an angle on the unit circle is equal to the X value. So here the cosine is going to be a negative three square A three divided by two. And the sign of an angle on the unit circle is going to be equal to the Y value. So the sine of seven pi divided by six is going to be negative on half. Now we can just plug those in to our equation. So we have the negative tangent of and seven pi divided by six divided by two simplifies to be seven pi divided by 12. What we started with equals a negative. Then in the numerator we have one minus. Well, the cosine of seven pi divided by six is going to be that X value that's negative square root three divided by two. That's all divided by the sine of seven pi divided by six is that Y value that's a negative one half. And now it's now it's just simplification. So let's take a look at our numerator first. If we have one minus negative square root three, divided by two, that's the same as one plus square root three divided by two, we need a common denominator. That'll be two. So we'll multiply our one by two halves. So two halves plus square root three divided by two is two plus square root three divided by two. That's in the numerator all over a negative one half. And this whole thing is still negative. Simplifying this a little bit. Let's move this to a different part of our screen for a moment, give a little more room. So if we rewrite this with a fraction divided by a fraction, if we have remembering the whole thing is negative, we have a two plus square at three divided by two divided by, with a division symbol negative one half. That's the same as multiplying by the reciprocal of the second fraction. So this is equal to a whole thing is negative. We have the first fraction, two plus square root three divided by two multiplied by the reciprocal of negative one half, which is negative two. Now, here are twos cancel out and this will be a negative negative two plus the square of three and those negatives cancel out. So this is just going to be a positive two plus square root three. And we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Match each expression in Column I with its value in Column II.
8. tan (-π/8)

Key Concepts

Tangent Function and Its Properties

Reference Angles and Angle Reduction

Exact Values of Special Angles

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