00:01
Hi, from the question given that let us assume that alpha beta gamma are the angles in quadratic 1, 2, 3 respectively such that tan alpha is equal to 3 and tan beta is equal to minus 5 by 2 and tan gamma not tan here it is given that cos gamma is equal to minus 4 by 5.
00:26
So also assume that 0 is less than alpha beta gamma is less than 2 pi, 0 is less than alpha beta gamma which is less than 2 pi.
00:39
Now here we need to evaluate the following.
00:41
So first one we need to evaluate cos gamma by 2.
00:47
In general we know that cos theta by 2 is equal to plus or minus square root of 1 plus cos theta divided by 2.
00:57
Here it is given that the cos gamma is equal to minus 4 by 5.
01:03
So cos gamma by 2 which is equal to plus or minus square root of by using the formula square root of 1 plus cos gamma divided by 2.
01:22
So which is equal to plus or minus square root of 1 minus 4 by 5 divided by 2.
01:27
So simplify this we obtain the results could be equal to plus or minus root 10 divided by 10.
01:35
Since gamma is in the third quadrant so where cos is negative so that cos gamma by 2 is equal to negative root 10 by 10.
01:51
Now we need to evaluate tan second one.
01:55
Second one is tan gamma by 2.
02:00
In general we know that tan theta by 2 is equal to plus or minus square root of 1 minus cos theta divided by 1 plus cos theta.
02:10
So it is equal to 1 plus or minus under root of here tan gamma by 2.
02:18
1 minus cos gamma divided by 1 plus cos gamma.
02:23
So from the given information it will be equal to plus or minus under root of 1 plus 4 by 5 divided by 1 minus 4 by 5.
02:36
So simplify this we obtain plus or minus 9.
02:42
So it is equal to plus or minus 3 but gamma is in the third quadrant so here the cos is negative so tan gamma by 2 is equal to negative 3.
02:54
Now let us move on to the third part.
02:57
So in the third part we need to evaluate sin beta by 2.
03:02
So off of the angle formula again sin theta by 2 is equal to plus or minus under root of 1 minus cos theta divided by 2...