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 = โ(1 - cos^2 S) = โ(1 - (1/3)^2) = โ(1 - 1/9) = โ(8/9) = โ8/3
- tan S = sin S / cos S = (โ8/3) / (1/3) = โ8
- csc S = 1 / sin S = 1 / (โ8/3) = 3/โ8 = 3โ8/8
- sec S = 1 / cos S = 1 / (1/3) = 3
- cot S = 1 / tan S = 1 / โ8 = โ8/8
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 / โ(1 + tan^2 S) = (-10/7) / โ(1 + (-10/7)^2) = (-10/7) / โ(1 + 100/49) = (-10/7) / โ(49/49 + 100/49) = (-10/7) / โ(149/49) = (-10/7) / (โ149/7) = -10/โ149
- cos S = 1 / โ(1 + tan^2 S) = 1 / โ(1 + (-10/7)^2) = 1 / โ(1 + 100/49) = 1 / โ(49/49 + 100/49) = 1 / โ(149/49) = 1 / (โ149/7) = 7/โ149
- csc S = 1 / sin S = 1 / (-10/โ149) = -โ149/10
- sec S = 1 / cos S = 1 / (7/โ149) = โ149/7
- 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 = โ(1 - sin^2 S) = โ(1 - (-5/14)^2) = โ(1 - 25/196) = โ(196/196 - 25/196) = โ(171/196) = โ171/14
- tan S = sin S / cos S = (-5/14) / (โ171/14) = -5/โ171
- csc S = 1 / sin S = 1 / (-5/14) = -14/5
- sec S = 1 / cos S = 1 / (โ171/14) = 14/โ171
- cot S = 1 / tan S = 1 / (-5/โ171) = -โ171/5
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 = โ(1 - cos^2 S) = โ(1 - (11/13)^2) = โ(1 - 121/169) = โ(169/169 - 121/169) = โ(48/169) = โ48/13
- tan S = sin S / cos S = (โ48/13) / (11/13) = โ48/11
- csc S = 1 / sin S = 1 / (โ48/13) = 13/โ48 = 13โ48/48 = 13โ3/4
- cot S = 1 / tan S = 1 / (โ48/11) = 11/โ48 = 11โ48/48 = 11โ3/4
In summary, for each given case:
a) sin S = โ8/3, tan S = โ8, csc S = 3โ8/8, sec S = 3, cot S = โ8/8
b) sin S = -10/โ149, cos S = 7/โ149, csc S = -โ149/10, sec S = โ149/7, cot S = -7/10
c) cos S = โ171/14, tan S = -5/โ171, csc S = -14/5, sec S = 14/โ171, cot S = -โ171/5
d) sin S = โ48/13, cos S = 11/13, tan S = โ48/11, csc S = 13โ3/4, sec S = 13/11, cot S = 11โ3/4