 Hello friends, so welcome to this session on trigonometry and we are going to start with measurement of angles. So before that, we need to know what an angle is. So an angle is a geometrical figure. So an angle is a geometrical figure in a plane. Isn't it? Figure in a plane and it is bounded by two rays. So these are the two rays two rays. And this particular measure is called angle and is usually denoted by a Greek letter theta. Okay, so now how do we measure angles? There are three system of measurement. One is called 6RJ symbol system. Another is called centesimal system, which is not that in use these days. And the most prominent one is circular measure. So let us understand one by one, each of them. So what is 6RJ symbol system? So in 6RJ symbol system, in 6RJ symbol system, we have one right angle, one right angle is equal to 90 degrees. So one right angle is measured as 90 degrees. So we know that this is 90 degrees where these two rays are perpendicular to each other. Now the further subunits of degrees, one degree is equivalent to is equal to let's say 60 minutes. And each one minute is equal to 60 seconds. These are the subunits. These are the subunits. So let us say 60 minutes will be written as 60 dash and 60 seconds will be 60 double dash and 90 degrees is written as 90 and a small dot over it, degree symbol. So this is 6RJ symbol system. Then there is another system called centesimal system. And this is much more convenient than the 6RJ symbol system because it considers one right angle to be equal to 100 grades which is written as 100 and superscript G. Now one grade was equal to 100 minutes and it is denoted as like that and one minute is equal to 100 seconds. So one grade is mentioned as 100 minutes which is denoted by 100 like this and one minute is 100 seconds like that. Now it was not popularized because you know the entire trigonometric tables which was initially done in terms of degrees would have to be changed entirely. So hence now this is not much in practice. Okay. Now third is which is found in almost all literature is circular measure, circular measure which has its genesis in the ratio of the arc length and the radius of a circle. So let us say this is the arc length and this is let's say the radius and this angle is theta such that this length, this arc length is also R same as that of radius. Then theta is defined as one radian. Okay. So this is one one radian. So now how do I define one radian once again? This is the angle of an arc of a sector such that the arc length is equal to that of radius. So that angle will be one radian. Now in the next session we will prove that this one radian is independent of the radius or the arc length because one radian is just the simple ratio of the arc length. So theta is nothing but arc length divided by the radius. So it is immaterial of the actual value of radius or the arc length because they change proportionately. So theta is defined as one radian when arc length is equal to radius. Okay. Now we know that when theta is equal to 360 degrees this is equivalent to 2 pi radians. Okay. Why? Because the angle subtended at the center is 360 degree then the arc length is 2 pi R whole circumference and then divided by R. So you will get 2 pi radians. So similarly you can now find out 180 degrees is equal to pi radians which is in short written as RAD. 90 degrees will be equal to pi by 2 radian and 45 degrees will be equal to pi by 4 radian and likewise. So for example 30 degrees will be equal to pi by 6 radian like that. This is very useful measure of the angle because the entire literature in higher order let's say in higher grades will be based on radians only. So please be familiarized with radian circular measure. So we will take a small example of converting let's say degrees into radians. So let us say I have 405 degrees and this has to be converted into let us say radians. So how many radians is 405 degrees? So this is how you will have to solve. So 360 degrees is equal to 2 pi radian. So one degree will be equal to 2 pi by 360 degree sorry 2 pi by 360 radian not this unit will not be there. So 405 degrees will be equal to 2 pi by 360 into 405. So if you see this is 45 8 times and this is 45 9 times and this is 4. So hence it is 9 pi by 4 9 times 45 degrees okay. This is how you have to convert degree into radian. Let us say we have 3 pi by 2 radian. So this is equivalent to how many degrees? Let us say we have to find this. Then we know that 2 pi radian is equal to 360 degrees is it it. So one radian is equal to 360 upon 2 pi radian. So hence sorry not radian degrees degrees. So 3 pi by 2 radian will be equal to 360 by 2 pi into 3 pi by 2. So this pi and this pi gets cancelled. This 2 and this 2 4. So this is 90. So it is basically 270 degrees. This is how an angle in given in degrees can be converted into radians and an angle given in radians can be converted back into degrees. Thank you.