SOLUTION: Please help. Thank you in advance. Given tan 𝛼 =7/24, 𝛼 in the third quadrant, sin 𝛽 =2/√13, 𝛽 in the second quadrant, find: a) the quadrant containing 𝛼 + �

Notice that 7^2 + 24^ = 625 = 25^2.


So, you may think that 

    tan(a) =  in the third quadrant is   = .


In other words, you may think that sin(a) = ,  cos(a) = .

    The signs at  sin(a)  and  cos(a)  do agree that  "a"  is in the third quadrant.



Next, you are given that sin(b) =  in the second quadrant; so, you can calculate 

    cos(b) =  =  =  =  = .

    The sign at  cos(b)  does agree that  "b"  is in the second quadrant.



Now, as you know  sin(a) ,  cos(a) = , sin(b) = ,  cos(b) = ,  you can calculate


    sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b) =  +  =  -  = 

             =  = ,


    cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b) =  -  =  -  = 

             =  = .


Thus, sin(a+b) is a negative real number;  cos(a+b) is a positive real number.


It means that angle a+b is in fourth quadrant.


ANSWER.  Angle a+b is in fourth quadrant.