 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that, find all the 6 trigonometric ratios for the angle 4 pi by 3. We know that on unit circle in coordinate plane the ordered pair x, y can be found using trigonometric ratios as cos of theta, sin of theta, where angle measure is known. Here we have a unit circle whose radius is 1 unit and this angle is theta. Then, coordinates of point P which is given by the ordered pair x, y can be found using trigonometric ratios as the ordered pair cos of theta, sin of theta. So we have cos of theta is equal to x, sin of theta is equal to y, tan of theta is equal to y upon x, secant of theta is equal to 1 upon x, cos secant of theta is equal to 1 upon y, cot of theta is equal to x upon y. The sign of the trigonometric ratios will depend on the sin of x and y according to the quadrant in which they lie. With this key idea let us proceed to the solution. We have to find all the 6 trigonometric ratios for the angle that is 4 pi by 3. Now we are given this angle in radians and to convert this angle in degree measure we multiply it by 180 upon pi and we get 4 pi by 3 as 240 degrees and here we see that 240 degrees lies between 180 degrees and 270 degrees. Now let us draw a unit circle with center O. Here we should note that the radius of this unit circle that is OA is given by 1 unit. Now the terminal side of the angle is in the third quadrant which makes 240 degrees angle in anticlockwise direction. Let this point be m such that O m is equal to 1. Now we draw perpendicular from m on x axis. Let this point be n. This is the point where this perpendicular meets the x axis. So now we have the right angle triangle that is triangle O n m and we need to find this angle theta and we know that this total angle is 240 degrees and this angle till here is 180 degrees. So we can say that the remaining angle theta is equal to 240 degrees minus 180 degrees which is equal to 60 degrees or we can also write it in radian measure as pi by 3. We know that 60 degrees can be written as 60 into pi upon 180 radians which is equal to pi by 3 radians. So we have got the angle theta as 60 degrees or pi by 3 radians. Now we know that in the third quadrant x is negative and y is also negative. So here cos theta will be negative and sine theta will also be negative and we have sine theta that is sine pi by 3 and this is equal to sine of 60 degrees which is equal to square root of 3 by 2 and similarly cos of pi by 3 is equal to cos of 60 degrees which is equal to 1 by 2. From the key idea we know that on unit circle in coordinate plane the ordered pair x y can be found using trigonometric ratios as ordered pair cos of theta sine of theta where angle measure is known. So we have cos of theta is equal to x sine of theta is equal to y tan of theta is equal to y upon x secant of theta is equal to 1 upon x cosecant of theta is equal to 1 upon y and cot of theta is equal to x upon y. Now using the key idea we have on unit circle the ordered pair x y which is equal to the ordered pair cos of theta sine of theta so in third quadrant we have the ordered pair minus of x minus of y and this is equal to minus of cos of theta that is minus of 1 by 2 and then minus of sine of theta which is equal to minus of square root of 3 by 2 so ordered pair that is minus of x minus of y is equal to minus of 1 by 2 minus of square root of 3 by 2 that is the ordered pair minus of 1 by 2 minus of square root of 3 by 2 using this we shall find all these six trigonometric ratios for the angle 4 pi upon 3 and thus we write x is equal to cos of 4 pi by 3 which is equal to minus of 1 by 2 y is equal to sine of 4 pi by 3 which is equal to minus of square root of 3 by 2 similarly tan of 4 pi by 3 will be equal to y upon x that is minus of square root of 3 by 2 whole upon x that is minus of 1 by 2 and this can be written as square root of 3 then cot of 4 pi by 3 is equal to x upon y which is equal to minus of 1 upon 2 whole upon minus of square root of 3 by 2 and this is equal to 1 upon square root of 3 then secant of 4 pi by 3 is equal to 1 upon x that is 1 upon minus of 1 by 2 and this is equal to minus of 2 similarly cosecant of 4 pi by 3 is equal to 1 upon y that is 1 upon minus of square root of 3 by 2 and is equal to minus of 2 upon square root of 3 thus we have got sine of 4 pi by 3 is equal to minus of square root of 3 by 2 cos of 4 pi by 3 is equal to minus of 1 by 2 tan of 4 pi by 3 is equal to square root of 3 cos secant of 4 pi by 3 is equal to minus 2 upon square root of 3 secant of 4 pi by 3 is equal to minus 2 and cot of 4 pi by 3 is equal to 1 upon square root of 3 which is the required answer this completes our session hope you enjoyed this session