Given that sin B = -5/13 and angle B is in quadrant III, we can use the Pythagorean identity to find the value of cos B:cos² B + sin² B = 1
cos² B + (-5/13)² = 1
cos² B = 1 - 25/169
cos² B = 144/169
cos B = -12/13 (since cos B is negative in quadrant III)a) To find csc B, we can use the reciprocal identity:
csc B = 1/sin B
csc B = 1/(-5/13)
csc B = -13/5b) To find cot B, we can use the quotient identity:
cot B = cos B/sin B
cot B = (-12/13)/(-5/13)
cot B = 12/5c) To find cos B, we have already calculated it above:
cos B = -12/13d) To find sec B, we can use the reciprocal identity:
sec B = 1/cos B
sec B = 1/(-12/13)
sec B = -13/12e) To find tan B, we can use the ratio identity:
tan B = sin B/cos B
tan B = (-5/13)/(-12/13)
tan B = 5/12Therefore, the values of the trigonometric functions for angle B in standard position with terminal side in quadrant III and sin B = -5/13 are:
a) csc B = -13/5
b) cot B = 12/5
c) cos B = -12/13
d) sec B = -13/12
e) tan B = 5/12
Explanation: