ElectricPavlov

See below:

That would be where the numerator is first zero....where cos = zero    (because cot = cos/sin )

                                                                                                                   but shifted  :     pi/2 - pi/6 = pi/3

(t+4)/(t+5) = (t-5)/2t    multiply  through by    2t(t+5)

2t( t+4) = (t-5)(t+5)

2t^2 + 8t  = t^2 - 25       re-arrange

t^2 + 8t + 25 = 0       Use quadratic formula to find   t = -4 +- 3i

the PLUS  pi/6  will shift the parent function to the LEFT   

x^2 + y^2 -4 = 0     and    (x-3)^2 + (y-5)^2 - 25 = 0   equate the two and expand and simplify to get

6x + 10y-13 = 0    this is a line

10 y = -6x + 13

y = -6/10 x + 13      slope = -6/10 = -3/5    (I used desmos to look this over graphically to verify this solution)

Just as Chris found ! 

Cool.....never heard of Tebibyte before ....thought it was a typo in the question .....

 I found that Tib and TB   they are close enough in value they are often used as synonyms, though they are not EXACTLY the same value...as your answer shows

    same methodology to solution though.....  THANX !

cot = 3/5 = cos/sin       This can occur in Q I  or Q III

Remember   cos^2 + sin^2 = 1  ?

(3/x)^2  + (5/x)^2  = 1       x = sqrt 34      sin = 5/sqrt34    cos = 3/sqrt34   OR   sin = - 5/sqrt34    and cos = - 3/sqrt34

    (not sure why  did all of that ....but there it is)

arctan 5/3 =  1.03 R       AND   4.17 R

Not sure how text got that answer!

If the 6-9 thing counts   AND you cannot have leading zeroes, you have 5 choices for each position in your 7 digit positions (except there can be no leading zero)

4 x 5 x 5 x 5 x 5 x 5 x 5  =   5^6 x 4 =......

For this to be true   the discriminat   b^2 - 4ac  > 0

m^2 -4(1)(4) > 0

m^2 > 16

m <-4       m >4

(-inf,-4) U (4, + inf)

A different perspective...just an FYI

7⁵  possible numbers      4 out of 7 are odd     7⁵  x  4/7 = 9604  (as Chris calculated)