In Exercises 8–12, draw each angle in standard position.8𝜋3

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked to sketch the given angle note that the angle should be in standard position. Our angle is 29 pi divided by six and so told standard position. While we know a couple of things from that, we know we're dealing with a rectangular coordinate plane, we can draw a rough sketch of that. We'll have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle because it's in standard position, we know that the initial side of our angle will have the vertex at the origin and then will extend through the positive side of the X axis toward positive infinity for X. So how about the terminal side? Well, that's where we need to look at that 29 pi divided by six. So it's a positive number. We're going to be moving counterclockwise from the initial side. But how much? Well, 29 pi divided by six, that's a big number and we can get some smaller numbers out of that. So we always want to look and see if there's any two pies in there. And there are because our denominator is 62 pi would be 12 pi divided by six and we can take 12. That's an 11. We can take 12 pie divided by six and add another 12 pie divided by six. We're at 24 pie here and then to continue to get to the, the rest of the way up to 29 pi we just need five more in the numerator. So five pi divided by six. So that's the same thing as our 29 pi divided by six, but much more useful as we are trying to draw our terminal side of our angle. So recalling from previous lessons, that two pi is a full rotation that's going all the way from where we started all the way around back to where we started. And we have two of these full rotations. We have two of these two pies which we see as 12 pie divided by six. So it's going around once and it's going around twice. Now, how much does it have left to go? Just the five pi divided by six? So now it can help to think of it in terms of pi, we recall from previous lessons that we know that a full rotation is two pi well half of a rotation is pi that's when we get from the positive X axis all the way over to the negative A A X axis. So in terms of pi, how much do we have well, five pi divided by six is five, six, multiplied by pi. And so we're going 5/6 of the way toward pie here. So if we divide the 1st and 2nd quadrants into sixths, so that would be taking half of it, dividing it into three, dividing it into thirds. So we have one third, two thirds, three thirds and then four thirds, five thirds or I should be talking in sixths. So we have and there is our 5/6 here. So from the origin to draw our terminal side, we're going up into the second quadrant closer to the X axis than the Y axis. We're going one third of the way up from the X axis or two thirds of the way to the left of the Y axis and drawing our terminal side heading toward negative infinity for our X values, but positive infinity for our Y values. And there is our angle. Well done. We'll catch you on the next one.

In Exercises 8–12, draw each angle in standard position.8𝜋3

Key Concepts

Standard Position of an Angle

Radians and Degrees

Drawing Angles in the Coordinate Plane

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