 Hello and welcome to the session. In this session we will understand radiant measure of an angle as the length of the arc on unit circle subtended by an angle. And we shall also discuss how to convert radians into degrees and degrees into radians. Now let us start with unit circle and unit circle is a circle whose radius is one unit and center is at origin in the coordinate plane. This is the unit circle with radius one unit and we can see that in the coordinate plane here the center of this circle is at origin and the coordinates of the center of the circle is given by zero zero. Thus when radius of the circle is one unit that is radius r is equal to one then that circle is called unit circle. And now we shall discuss radiant measure in our unit circle. A radiant is a unit of measure of a central angle that intercepts an arc equal in length to the radius of circle. Thus in our circle of radius one centimeter a central angle whose measure is one radian intercepts an arc whose length is one centimeter. In this figure we have taken a unit circle that is radius of the circle is taken as one centimeter and central angle is one radian. So we see that the length of the arc is equal to the radius that is equal to one centimeter. Similarly in our circle of radius one centimeter a central angle of two radians intercepts an arc whose length is two centimeters. In this figure we have taken a unit circle with central angle that is two radians. So length of the arc subtended by this angle is two centimeters. Similarly if we see this figure we have taken a unit circle with central angle equal to three radians so here central angle is equal to three radians. So length of the arc subtended by this angle is three centimeters. Now let us see length of the arc when radius is two centimeters. In our circle of radius two centimeters a central angle whose measure is one radian. Now this is the central angle of measure one radian. Then this angle intercepts an arc whose length is two centimeters. Similarly a central angle of two radians will intercept an arc whose length is four centimeters and if the central angle is equal to one plus one plus one that is three radians then the length of the arc subtended by this angle is equal to six centimeters. Now let us find the general formula for finding radian measure of central angle. Now in this figure if we take theta radians as angle measure then theta will be equal to six upon two which is equal to three radians that is we have taken six that is the length of the arc and two which is the radius of the circle on dividing the length of the arc with the radius of the circle we got the value of theta as three that is we have got the value of angle measure as three. In general we can say that the radian measure theta of central angle of a circle is the length of the intercepted arc which is given by s divided by radius of the circle which is given by r. So we can write radian measure theta is equal to length of the arc divided by radius of circle or we can also write it as theta is equal to length of the arc that is s upon radius of the circle that is given by r. So we have s upon r from here we can also write s is equal to theta into r. For example if angle is given by five radians and radius is equal to four centimeters then length of the arc is given by theta into r that is s is equal to theta into r where s represents length of the arc theta is the angle measure and r is the radius of the circle which implies that s that is the length of the arc is equal to theta which is given by five radians into r that is radius of the circle which is given as four centimeters. So s is equal to five into four that is equal to twenty centimeters. Here we should note that moving in counter clockwise direction will give us positive angle and moving in clockwise direction will give us negative angle. We are now going to discuss relationship between radians and degrees. Let us draw a circle of radius r its center is given by o. Now we draw a diameter in this circle. Now we see that the upper portion of the circle forms a silly circle and this is the central angle whose measure is theta radian. Now let us find the length of the arc subtended by this angle that is we shall find the length of this arc which is subtended by this angle whose measure is theta radian and here we can see that the length of the arc is equal to circumference of semicircle and we know that circumference of a semicircle is given by one by two into two pi r that is equal to pi into r that is pi r we know that theta is given by length of arc upon radius of circle and this is equal to length of the arc is equal to pi r and radius of circle is given by r so we have got theta as pi r upon r which is equal to pi so we have got the value of theta as pi radians so the radian measure of central angle is given by pi radians and we know that in degree measure the central angle of the semicircle is 180 degrees that is this angle is equal to 180 degrees and we know that 180 degrees is equal to pi radians and if we multiply both sides of the equation by two we get 360 degrees is equal to 2 pi radians and this represents the angle shown here that is this complete angle represents 2 pi or 360 degrees now we are going to discuss how to convert degrees into radian measure and radians into degree measure and we use the relationship that is 180 degrees is equal to pi radians to convert degrees into radians and radians into degrees we have pi radians is equal to 180 degrees which implies that one radian is equal to 180 upon pi degrees let us name this equation as equation number one from this relationship we also get one degree is equal to pi upon 180 radians and we mark this equation as equation number two and we should note that angles written in radians are often in terms of pi and the term radians is omitted while writing the measure of angle in radians but we always use degrees when we write angle measure in degrees now let us convert 150 degrees in radian measure now using result from equation number two that is one degree is equal to pi upon 180 radians thus we can write 150 degrees is equal to 150 into pi upon 180 radians which implies that 150 degrees is equal to 3 into 5 is 15 and 3 into 6 is 16 so here we get 5 into pi that is 5 pi upon 6 so we say that 150 degrees is equal to 5 pi upon 6 radians now let us convert 11 pi upon 6 into degrees now using result from equation number one that is one radian is equal to 180 upon pi degrees thus we can write 11 pi upon 6 as 11 pi upon 6 into 180 upon pi degrees which implies that 11 pi upon 6 is equal to 6 into 1 is 6 and 6 into 30 is 180 so we get 11 into 30 that is 330 degrees thus we say that 11 pi upon 6 can be written as 330 degrees so in this session we have discussed radian measure of an angle as the length of the arc on unit circle subtended by the angle and also how to convert radians into degrees and degrees into radians this completes our session hope you enjoyed this session