7.3 Polar Coordinates

[STARTS AT 15:38]

Professor: Now let's rewrite some other equations in rectangular coordinates and then identify the graph. 

We're going to still use those same identities-- x equals r cosine theta, y equals r sine theta, r squared equals x squared plus Y squared. 

Theta equals pi/3. That graph is, regardless of the radius value, the angle is pi over 3. So we actually get something that is a line. This line is going to be y equals tangent of pi over 3x. 

The slope is going to be the tangent of that angle. So that is y equals square root of 3x. 

Part B-- r equals 3. That is going to be a fixed radius of 3 cycled around. Now I'm going to go and write this as r squared equals well r squared is 9, which means x squared plus y squared equals 9. This is a circle-- a circle of radius 3. And this was a line of slope equal to square root of 3, which is the tangent of pi over 3. Now we should add in here that is centered-- centered at (0,0). 

All right r equals 6 cosine theta minus 8 sine theta. Let's go ahead and replace some things. Actually first, we want to square both sides-- or multiply b r on both sides so we get an r squared. So 6r cosine theta minus 8r sine theta. That would be x squared plus y squared. And this would be 6x minus 8y. Moving everything to one side, minus 6x. 

I'm going to leave some space, because we are going to want to complete the square for both of these. y squared minus 8y plus something equals 0 plus, and we're going to add two things. On the left side, for the x's here, that would be plus 9 so go ahead and add 9 there. 

And second would be 16. Do that in different color-- 16, 16. So we have equals 25. And then this will factor as y minus 4 squared and x minus 3 squared. So this is a circle centered-- centered at the point (3,4) with a radius of 5. 

[ENDS AT 19:36]