Find the values of other five trigonometric functions is

(a) cot x = 3/4, x lies in third quadrant.

(b) sec x = 13/5, x lies in fourth quadrant.

(a) cot x = 3/4, tan x = 1 / cot x = 1 / (3/4) = 4/3

1 + tan²x = sec²x

⇒ 1 + (4/3)² = sec²x = 1 + 16/9 = 25/9

⇒ sec x = ±5/3

Since x lies in the 3^rd quadrant, sec x is negative.

∴ sec x = −5/3

cos x = 1 / sec x = 1 / (−5/3) = −3/5

tan x = sin x / cos x

⇒ 4/3 = sin x / (−3/5)

⇒ sin x = (4/3) × (−3/5) = −4/5

cosec x = 1 / sin x = −5/4

(b) sec x = 13/5

cos x = 1 / sec x = 5/13

sin²x + cos²x = 1

⇒ sin²x = 1 − cos²x

= sin²x = 1 − (5/13)²

⇒ sin²x = 1 − 25/169 = 144/169

⇒ sin x = ±12/13

Since x lies in the 4^th quadrant, sin x is negative.

∴ sin x = −12/13

cosec x = 1 / sin x = −13/12

tan x = sin x / cos x = (−12/13) / (5/13) = −12/5

cot x = 1 / tan x = −5/12