11th 12th std. Cartesian and Polar coordinates

[00:00:00][music] [music] [music] Hi again friends, welcome back to Davsur's Mathematic.

[00:00:23]Today is a problem which is present in the 11th standard textbook as well as in the 12th standard topic.

[00:00:33]So let's see 11 standard colleges have just begun but your classes have already started coaching classes. So you might have come across cartesian and polar coordinates. Even if you haven't don't worry as regards 12th 12th April college it is there in trigonometric functions for 12th also. So let us straight away go to the problem. All set set cartesian coordinates or polar coordinates discuss right cartian and polar coordinates.

[00:01:22]I have taken a point on p xy on a standard circle. Standard circle means where the origin is the center. This is the x-axis. This is the y-axis. Origin this circle center is standard circle.

[00:01:44]R is the radius of the standard circle.

[00:01:47]Okay. This diagram we have seen most times in 11th standard for basic trigonometric function. Standard circle origin is the center R is the radius point P XY standard circle P M perpendicular to X-axis and this is your OM if we take this as the origin O PM so naturally by definition of X coordinate and Y coordinate X coordinate your Y coordinate Okay.

[00:02:25]Okay. Cartesian or polar coordinates is for when I take pxy usual point x coordinate ycoordinate cartesian coordinates our normal coordinates graph paper I'm plot x coordinate y coordinate these are normal cartesian coordinates x y h okay okay 10th graph paper.

[00:03:01]Okay. So, XY means cartian coordinates.

[00:03:05]11 standard or instead of cartesian coordinates instead of XY point P can be represented by R theta. This is R. This is theta.

[00:03:22]X Y R theta R means what? Radius of the standard circle. Theta means what? Angle made by the radius with the x-axis.

[00:03:35]Okay.

[00:03:37]Coordinates meaning those coordinates meaning to locate the point on the paper with reference to x and y axis.

[00:03:51]That is the meaning, that is the use, that is the purpose. So either you can do it by XY or R theta say if R is equal to 5, you draw a standard circle of radius 5. Example, R 5 cm.

[00:04:16]Draw a standard circle of radius 5 cm.

[00:04:19]And if theta is 30° to you 30 so 5 cm if you measure 30° with a protractor protractor measure exact 30° you'll get OP imagine 5 cm 30° by protector you will get the position of point P again so coordinates meaning to locate the point anywhere with reference to x and y axis.

[00:04:55]So the location can be done either by cartian coordinates or polar coordinates. Polar coordinates means rta xy radius or ang polar coordinates nothing difficult right now there are certain formula which we are going to see x is equal to r cos theta y = r sin theta how did I get this trigonometric function 11 cos theta x upon r x upon r sin theta definition y upon r theta can be anything therefore x= r cos theta cross multiply to y = r sin theta just remember this where r is equal to roo<unk> of x² + y² r value find out roo<unk> root of x² + y 2. How did I get this?

[00:06:08]Use pythogorus. In triangle OPM, if you use Pythagoras, what do you get?

[00:06:17]R² is equal to X2 + Y² right hand side.

[00:06:35]That is how I get this R H. Next formula tan theta triangle OPM tan theta basic formula 10 standard triangle OPM tan theta opposite side upon adjacent side tan theta opposite side upon adjacent side y upon x okay up till here everything is fine now comes the important part of inverse trigonometric function Inverse trigonometric function simple you just have to take tan sin on the right hand side meaning what tan theta equal to y upon x i.

[00:07:21]So if you want to find theta keep theta on the left hand side tan goes on the other side as tan inverse symbol tan to minus1 but it is read as tan inverse inverse trigonometric function chapter 11 standard okay functions inverse trigonometric function but now remember you are being introduced to what is a inverse trigonometric function. So if tan theta is equal to y upon x theta left hand side tan goes on the other side as tan inverse y upon x example tan of 45° is one to if I take tan inverse on the right hand side tan goes on the right hand side as tan inverse 1. So tan tan inverse 1 what remains 45°.

[00:08:28]Okay.

[00:08:30]Similarly tan 60 is equal to <unk>3.

[00:08:36]tan goes on the other side as tan inverse. So tan inverse roo<unk>3 equal to 60°. Simple h tan 30 is 1 upon roo<unk>3. tan goes on the other side as tan inverse. So tan inverse 1 upon <unk>3 equal to 30°.

[00:08:56]So when you take trigonometric functions on the other side, it goes as inverse function. Okay. To introduction a what is an inverse trigonometric function?

[00:09:09]Right? See you if I want to write like this.

[00:09:20]If I write like this, see tan theta is equal to y upon x. So tan goes on the other side as tan inverse y upon x theta isolate.

[00:09:39]So the procedure is to separate out the angle.

[00:09:43]Tan inverse 1 to 45° separate out.

[00:09:48]Tan inverse roo<unk>3 to 60° separate out.

[00:09:52]Separating out that angle with the help of trigonometric function is called as inverse trigonometric function. Okay.

[00:09:59]Angle separate about uh slowly you will get used to this. Okay. But now remember these are inverse trigonometric functions.

[00:10:09]Okay.

[00:10:12]Right.

[00:10:14]Now we will solve the problem.

[00:10:18]This is a new concept.

[00:10:24]We will follow everything perfectly.

[00:10:25]This question is usually asked in college exams. Find the polar coordinates of the following. H question find the polar coordinates polar coordinates find out means cartian coordinates are given x and y are given from that we have to find out r theta r theta means what polar coordinates okay that we will see we'll first start solving the problem.

[00:11:03]So my job is to convert this into R theta polar coordinates convert.

[00:11:11]Okay.

[00:11:13]So we'll begin X or Y cart.

[00:11:28]So cartian coordinates there means I will just write cartition coordinates x y right okay so when x and y are known r for roo<unk> of x² + I square 2.

[00:12:02]R is equal to roo<unk> of x² + y 2. Just substitute the values.

[00:12:17]X is - <unk>3 Uska square. Y is 1ka square.

[00:12:30]Root of a negative number is positive.

[00:12:32]And root3 k² means 3.

[00:12:38]1 2 is 1.

[00:12:43]So r is equal to 3 + 1 is 4.

[00:12:51]Therefore square root of 4 is 2.

[00:13:00]Square root is always plus minus.

[00:13:06]So plus minus actually it is plus minus but we have to neglect the negative sign. Why? Because r represents the radius of a circle. Radius is never negative. So r is always positive. Okay.

[00:13:22]So r polar coordinates r theta find out.

[00:13:29]How do you find theta?

[00:13:31]formula tan theta= y upon xt you will find it interesting so r is theta find out tan theta = y upon x therefore Just substitute the value along with the signs. Y is 1, X is minus <unk>3 minus sign.

[00:14:41]tan theta.

[00:14:44]You have to be careful. Tan theta about how will you find theta?

[00:14:51]How will you find theta? So for that you have to check now quadrant angle angle quadrant you say they minus <unk>3 1 x coordinate is negative ycoordinate is positive see 10 standard stuff x coordinate is negative side ycoordinate is positive So x is minus y is + x is minus y is plus so angle lies in the second quadrant your first quadrant second quadrant third quadrant fourth quadrant h so I'm going to say angle lies in second quadrant So, so depending on the second quadrant means over here. So, in the angle lies in the second quadrant, theta is given as 180 minus tan inverse y upon x. That is the formula.

[00:16:21]Okay, that is the rule.

[00:16:24]When the angle lies in the third quadrant, supposing if the angle was like this, angle means this angle theta be first quadrant. Second quadrant may it will be like this. Our angle is like this.

[00:16:40]Third quadrant may it will be like this.

[00:16:43]Fourth quadrant may again it will go into the fourth quadrant. Point P will go on varying according to the quadrant.

[00:16:49]So theta lies in the second quadrant Q X coordinate is negative Y coordinate is positive. So decide the quadrant first.

[00:17:05]Now when angle lies in the second quadrant up formula theta is equal to 180 minus tan inverse absolute value of y upon x. There are certain trigonometric rules which you have to remember for 11th and 12th to decide the quadrant. These are the rules. Okay. Fine. So step we'll finally write.

[00:17:49]So because angle lies in the second quadrant. Yeah. Formula theta is equal to 180 minus tan inverse x. Okay.

[00:18:06]Sorry. Tan inverse y upon x formula absolute value sign we will see why.

[00:18:26]So y upon x - 1 upon <unk>3. So theta = 180 minus tan inverse - 1 upon <unk>3 absolute value of a negative number. You have to neglect the absolute value.

[00:18:59]Absolute value meaning negative sign neglect though directly you can write 1 upon <unk>3 minus sign neglect because actually I wrote what is tan inverse y upon x Okay, tan theta value right to you absolute value meaning minus sign goes and you have to neglect the negative sign okay uh one more thing how did I get tan inverse 1 upon <unk>3 what is the formula formula theta is equal to tan inverse y upon upon x. So if tan theta is equal to minus. So if I take tan on the other side what do I get?

[00:20:10]tan inverse or is my absolute value convert.

[00:20:20]So tan goes on the other side as tan inverse. today's new formula to be remembered.

[00:20:26]Okay. So this is the value or absolute value cube because formula absolute value tan inverse y upon x this is the sign of absolute value.

[00:20:38]Okay. So because this is the sign of absolute value minus sign 1 upon <unk>3 some things I have to repeat because these are new things you should know.

[00:20:55]So theta is equal to 180 minus discuss what is tan inverse 1 upon <unk>3 minus sign to neglect minus sign neglect because formula there is no negative sign. So tan inverse 1 upon <unk>3°.

[00:21:18]So this is nothing but 30°.

[00:21:23]Now 180 minus 30 50°.

[00:21:31]So that's theta.

[00:21:34]Okay. Answer.

[00:21:37]Okay. The final answer. How am I going to write?

[00:21:41]Final answer.

[00:21:56]polar coordinate equal to P equal to R theta. Polar coordinate meaning R theta R 2 theta 50° uh sorry 180 - 30 150° okay 180° - 30° 150° so 2 comma 150° this is our these are our polar coordinates right so much so because of these polar coordinates we can locate the point anywhere it's meaning 2 cm radius or 150° point locate clear example simple example just discuss instead of minus <unk>3 1 supposing you are given plus <unk>3 1 find the polar coordinates signs change minus plus so basic remains the same is x y r is first thing to find is r so r= roo<unk> x² + y² that is r value to same roo<unk>3 squar is 3 1 square is 1 3 3 + 1 4 roo<unk> of 4 is 2 so r = 2 minus plus <unk>3 polar coordinates change okay so r is equal to 2 why do you find r because I require r theta should be our answer to r Same with tan theta tan theta equal to y upon x.

[00:24:27]Okay. So y is 1, x is <unk>3. So tan theta is equal to 1 upon <unk>3. So tan side. So theta is equal to tan inverse 1 upon <unk>3 tan goes on the other side as tan inverse definition inverse trigonometric tan right x is positive Y is positive. So X positive A Y positive. So angle lies in the first quadrant. Yeah. For in the earlier case angle was lying in the second quadrant quadrant. Okay. So quadrant we'll write down angle lies in the [clears throat] first quadrant.

[00:25:49]So when the angle lies in the first quadrant quadrant there is no change theta is equal to just tan inverse y upon x answer second quadant 180 minus third quadrant 180 plus whatever answer we get and fourth quadrant 360 minus these are rules of trigonometric functions but now you don't have to worry angle lies in the first quadrant therefore Or theta is equal to tan inverse y upon x your basic formula that means I answer tan inverse 1 upon <unk>3 same answer tan inverse 1 upon roo<unk>3 tan inverse 1 upon roo<unk>3 3 is 30°.

[00:26:56]So, theta equal to 30° theta r 180us 30 because angle was not lying in the second quadrant followed that is the rule. So r theta that's your answer I'll write over here now answer I have some place over here polar coordinates p corresponds to r theta corresponds to r value to two same theta 30° Right. That means 2 cm radius or theta 30° to 30° protractor measure to automatically first quadrant first because both X and Y are positive.

[00:28:04]I'm drilling this into your mind again and again. followed.

[00:28:09]Right. There are few more problems in the textbook. Decide the quadrant and then use I hope you have followed these are new things which doubt you can ask me fourth quadrant angle 360us 30 360 minus tan inverse y upon x that is tan inverse 1 upon <unk>3 which is 30 so 360 minus 30 will be 330° these are the differences so decide the quadrant properly follow.

[00:28:57]Okay. Get used to 11th standard for 12th standard students. It is there in trigonometra how to find the polar coordinates equally important for 12th standard. Okay.

[00:29:10]Right. That will be all for today. We have revised everything. Inverse trigonometric function to revision.

[00:29:19]So we will meet you in another day or two. Bye. See you.

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