 Hello and welcome to the session. In this session, we will explain how the unit circle in the coordinate plane enables the expansion of trigonometric functions to all real numbers interpreted as radial measures of angles promised counterclockwise around the unit circle. Now in our early session, we have discussed about radial measures of an angle and how to convert radials into degrees and degrees into radials. Now let us recall one degree is equal to pi by 180 radials. Now by using this result and convert any degree measure into radial measure, let us see some important radial measures. Now 30 degrees is equal to 30 into and this is equal to pi by 6. Then 35 degrees is equal to 35 into pi by 180 which is equal to pi by 4, which is equal to 19 into which is equal to pi by 2 equal to pi is equal to 3000 degrees is equal to 2 pi of a unit circle with center having coordinates 00. Now let us radius OP from center O and this horizontal line that is the x axis. Now OP is 1 unit which means circle with radius 1 unit. Now moving in anti-platforms the subtendent and then 30 degrees the point P by 30 degrees. We are taking 30 degrees in clockwise direction. So here it has minus pi minus 30 degrees. Now moving minus 30 degrees in clockwise direction minus 30 degrees which is equal to 30 degrees currently. Now we know that 360 degrees is equal to 2 pi radials minus 30 degrees that is pi by 6 radials. Now 2 pi minus pi by 3 30 degrees that is pi by 6 radials. Now let us see how to use radials to find trigonometric function values. As we find coordinates of any point of a unit circle in terms of trigonometric ratios. Now at this point Vm is equal to 1 unit where P is an x axis so it is coordinate. Now from this point we have here let us draw a coordinate that every September is V theta. Now from the point N we draw a perpendicular. The perpendicular means as hyper to use equal to radius of the circle that is 1 unit. Now let O n be x be y and y units vertical. Now we have taken O n is equal to x is equal to radius of the circle that is 1 unit. Now we have cos theta is equal to x or we can write x is equal to this triangle O nm perpendicular upon hyper to use. Now here perpendicular is mn upon hyper to use is equal to y or we can write y is equal to sin theta. So we have and y is equal to sin theta. Now in triangle O nm we have tan theta is equal to perpendicular upon adjacent side which is equal to cos theta. So this is equal to time theta upon cos theta. Now we know that secant theta is equal to 1 upon cos theta. Here secant theta will be 1 upon x secant theta is equal to 1 upon sin theta. I theta is equal to y. So secant theta will be 1 upon y that is cos theta is equal to 4 upon sin theta is equal to x upon y. Now let us see this unit circle when 0 then when sin theta. Now we have cos 0 is equal to 1 sin 0 is equal to 0 equal to pi by 2. So on y let's say at 0, 1 the coordinates cos theta sin theta over 2 is equal to 0. Theta is equal to pi then the plate p will lie on negative x axis and its coordinates are minus 1, 0. Cos theta sin theta we have cos pi is equal to minus 1 and sin pi is equal to 0. Note that the signs of the trigonometric ratios will depend on the signs of x and y according to the coordinate in which the y trigonometric ratios for angle 34 is equal to 135 degrees which lies between 90 degrees, 180 degrees and this circle with coordinates 00. Now the angle is given to us as 135 degrees. The terminal coordinate which makes 135 degrees angle in n to clockwise direction is equal to 1. So we have drawn the perpendicular from point M on x axis and this perpendicular meets x axis at point M is 180 degrees. Now let this point be p. So we have angle m of p is equal to 135 degrees. That is angle equal to 180 degrees minus this angle that is angle m of p which is 135 degrees so 35 degrees. In second quadrant theta will be negative so y is equal to same second quadrant. We have reference is equal to 45 degrees f over is equal to 1 by root 2 cos 45 degrees is equal to 1 by root 2 will be negative. 1 by root 2 this means y is equal to 1 by root 2 x y is equal to odd repair minus 1 over root 2 1 over root 2. Now here the given angle x is equal to cos 3 by root 4 and y is equal to sin 3 by root 4 is equal to odd repair minus 1 by root 2 is equal to cos 3 by root 4 is equal to minus 1 by root 2. 5 by 4 is equal to 1 by root 2 is equal to y upon x which is equal to 1 upon root 2 whole upon minus 1 upon root 2. This is equal to minus 1 now cos 3 by by 4 is equal to x upon y which is equal to minus 1 upon root 2 whole upon 1 upon root 2. And this is equal to minus 1 then equal to 1 upon x which is equal to 1 is equal to minus 4 is equal to 1 upon y which is equal to 1 by root 2. And this is equal to root 2 we have discussed how to find trigonometric function values using radians. And this completes our session. Hope you all have enjoyed the session.