Point R has cylindrical coordinates of 5 comma pi over 6 comma 4. Plot R and describe its location in space using rectangular or Cartesian coordinates.

I'm going to begin by drawing an xyz coordinate set of axes. And 5, pi over 6, 4-- that would indicate that I have a radius of 5. So in my x, y, and z, I have a radius of 5 from the origin. So I'm 5 away from the origin.

The angle with the positive x-axis is pi over 6. And then I have a z component of 4. So from that point, I will go directly up 4 units. And that is my point R.

Now let's see if this matches, this description, this drawing matches once I convert these coordinates. So beginning with my cylindrical coordinates that are r, theta, z, to convert this into an ordered pair of x, y, z in Cartesian coordinates, there are a couple of equations I can use. Two that are most useful at the moment are r equals-- or sorry, x equals r cosine theta and y equals r sine theta.

Because I know what theta and r both are, I can say that x is equal to r value of 5 multiplied by cosine of pi over 6, and y is equal to 5 sine of pi over 6, which means, respectively, my x is equal to 5 root 3 over 2, because cosine of power 6 is root 3 over 2.

And because sine of pi over 6 is equal to 1/2, my y value is 5 over 2. And of course, z is equal to 4 because that doesn't change. So my new coordinate for this would be 5 root 3 over 2 comma 5 over 2 comma 4.

And if I see where that goes-- so a positive x of 5 root 3 over 2, wherever that is, a positive y, and then a positive z of 4? Yes, that should get me to the exact same point, which is the point.