Find the values of the other five trigonometric functions for angle S.

To find the values of the other trigonometric functions for angle S, we will use the given information and the definitions of the trigonometric functions.

a) Given cos S = 1/3 and the terminal side of angle S is in Q1.
Since cos S is positive in Q1, we know that S is an acute angle.
To find the values of the other trigonometric functions, we can use the following definitions:
- sin S = √(1 - cos^2 S) = √(1 - (1/3)^2) = √(1 - 1/9) = √(8/9) = √8/3
- tan S = sin S / cos S = (√8/3) / (1/3) = √8
- csc S = 1 / sin S = 1 / (√8/3) = 3/√8 = 3√8/8
- sec S = 1 / cos S = 1 / (1/3) = 3
- cot S = 1 / tan S = 1 / √8 = √8/8

b) Given tan S = -10/7 and the terminal side of S is in Q2.
Since tan S is negative in Q2, we know that S is an obtuse angle.
To find the values of the other trigonometric functions, we can use the following definitions:
- sin S = tan S / √(1 + tan^2 S) = (-10/7) / √(1 + (-10/7)^2) = (-10/7) / √(1 + 100/49) = (-10/7) / √(49/49 + 100/49) = (-10/7) / √(149/49) = (-10/7) / (√149/7) = -10/√149
- cos S = 1 / √(1 + tan^2 S) = 1 / √(1 + (-10/7)^2) = 1 / √(1 + 100/49) = 1 / √(49/49 + 100/49) = 1 / √(149/49) = 1 / (√149/7) = 7/√149
- csc S = 1 / sin S = 1 / (-10/√149) = -√149/10
- sec S = 1 / cos S = 1 / (7/√149) = √149/7
- cot S = 1 / tan S = 1 / (-10/7) = -7/10

c) Given sin S = -5/14 and the terminal side of S is in Q3.
Since sin S is negative in Q3, we know that S is an obtuse angle.
To find the values of the other trigonometric functions, we can use the following definitions:
- cos S = √(1 - sin^2 S) = √(1 - (-5/14)^2) = √(1 - 25/196) = √(196/196 - 25/196) = √(171/196) = √171/14
- tan S = sin S / cos S = (-5/14) / (√171/14) = -5/√171
- csc S = 1 / sin S = 1 / (-5/14) = -14/5
- sec S = 1 / cos S = 1 / (√171/14) = 14/√171
- cot S = 1 / tan S = 1 / (-5/√171) = -√171/5

d) Given sec S = 13/11 and the terminal side of S is in Q4.
Since sec S is positive in Q4, we know that S is an acute angle.
To find the values of the other trigonometric functions, we can use the following definitions:
- cos S = 1 / sec S = 1 / (13/11) = 11/13
- sin S = √(1 - cos^2 S) = √(1 - (11/13)^2) = √(1 - 121/169) = √(169/169 - 121/169) = √(48/169) = √48/13
- tan S = sin S / cos S = (√48/13) / (11/13) = √48/11
- csc S = 1 / sin S = 1 / (√48/13) = 13/√48 = 13√48/48 = 13√3/4
- cot S = 1 / tan S = 1 / (√48/11) = 11/√48 = 11√48/48 = 11√3/4

In summary, for each given case:
a) sin S = √8/3, tan S = √8, csc S = 3√8/8, sec S = 3, cot S = √8/8
b) sin S = -10/√149, cos S = 7/√149, csc S = -√149/10, sec S = √149/7, cot S = -7/10
c) cos S = √171/14, tan S = -5/√171, csc S = -14/5, sec S = 14/√171, cot S = -√171/5
d) sin S = √48/13, cos S = 11/13, tan S = √48/11, csc S = 13√3/4, sec S = 13/11, cot S = 11√3/4