In Exercises 8–12, draw each angle in standard position.5𝜋6

Question

Accepted Answer

Welcome back. I am so glad you're here. And we are asked to sketch the given angle note that the angle should be in standard position. Our given angle is seven pi divided by eight. All right, let's focus on the standard position part. First that tells us that we're dealing with a coordinate plane. We can draw our vertical Y axis, a horizontal X axis. They come together at the origin in the middle because this angle is in standard position, its vertex is at the origin and its initial side extends along the positive X axis heading toward positive infinity for our X values. So how about that terminal side? Where do we draw that? And that's where we go back to our seven pi divided by eight. We note that that is positive. So because it's a positive angle, we are going to be going counterclockwise from the initial side. But how far? So the next thing we want to do is we look at our number seven pi divided by eight. If we focus on this, we know this is the same thing as seven divided by eight, multiplied by pi we focus on this 7/8 part first. Well, that's less than one. So that means we're going less than a pie of a rotation. So how much is a pie? Well, we recall from previous lessons that pie would be if we go all the way from the positive side of the X axis to the negative side of the X axis. So a straight line that's pi and so 7/8 of that. Well, we can divide the 1st and 2nd quadrants into eight parts or thinking of each one quadrant into four parts. So dividing the first quadrant into four parts. Here's about and we're heading counterclockwise. So here's about 1/4 of the way. Here's half of the way, here's 3/4 of the way and then the, the Y axis, the positive side of that is one quarter of the way around. Then continuing. We're dividing now the second quadrant into fourths, here's 1/4 of the second quadrant, half of the second quadrant, 3/4 of the second quadrant and then pi would be at full half of a rotation. So now where is 7/8 on that? Well, these are eighths, we drew it into eighths and seven of the eighths is going to be the last one before we get to pie. It's all the way in the second quadrant and it's one quarter of the way up from the X axis or it's three quarters of the way to the left of our Y axis. So from the origin drawing our terminal side, we are extending to the point that we just described. We're heading toward negative infinity for our X values and positive infinity for our Y values. And that is seven pi divided by eight. Well done. We'll catch you on the next one.

In Exercises 8–12, draw each angle in standard position.5𝜋6

Key Concepts

Standard Position of an Angle

Radian Measure

Drawing Angles

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