00:01
This question we got to solve for theta which is sign 3 theta is equal to sign 6 theta between 0 and 2 pi right so whenever we are given that sign of 3 theta is equal to sign of 6 theta this means that 3 theta will be equal to 2k pi in fact we can write it as because this is applicable in quadrant 2 as in quadrant uh sorry quadrant one as well as in quadrant two so if this is in quadrant one this is as simple as six theta plus two k pi and this is in quadrant two then uh we write that three theta is two pi uh sorry no two pi minus six theta plus two k pi over here because pi minus six theta would be its corresponding angle in the quadrant two right so they are basically same so if we solve this, the negative 3 theta is equal to 2k pi, which means that the value of theta is negative 2 over 3k pi.
01:10
So definitely k we have to substitute as negative because overall theta should be positive.
01:16
So if k, let's start from 0.
01:18
If k is 0, then theta is 0, which is acceptable solution.
01:21
If k is negative 1, then theta is 2 pi over 3, which is acceptable solution.
01:27
And if k is minus 2 if k is minus 2 then theta is 4 over 3 pi 4 over 3 pi this is also an acceptable solution and if k is equal to negative 3 then theta is equal to i think 2 pi which is not acceptable because remember that at 2 pi we have an open interval so from the left side's equation we have three solutions over here and let's talk about the right side side now.
01:59
So over here we add 6 theta both sides to get 9 theta.
02:04
So 9 theta is equal to pi plus 2k pi.
02:07
So the value of theta will be 1 9th of 2k pi plus pi.
02:13
So if k is 0, then the value of theta will be pi over 9, which absolutely works.
02:20
If k is 1, k cannot be negative by the way, because if it is negative, then this number will lead this number and theta will come out as negative which is not allowed.
02:30
If k is one, then theta will become one ninth of three pi, which is nothing but pi over three...