If tan (x) = 5/12 in the third quadrant then find other trigonometric ratios

tan x = 12/5, x in quadrant III

In the third quadrant, tan x and cot x are positive. sin x, cos x, sec x, cosec x are negative.

By using the formulas,

cot x = 1/tan x

= 1/(12/5)

= 5/12

cosec x = –√(1 + cot² x)

= –√(1 + (12/5)²)

= –√(25+144)/25

= –√(169/25)

= -13/5

sin x = 1/cosec x

= 1/(-13/5)

= -5/13

cos x = – √(1 – sin² x)

= – √(1 – (-5/13)²)

= – √(169-25)/169

= – √(144/169)

= -12/13

sec x = 1/cos x

= 1/(-12/13)

= -13/12

∴ sin x = -5/13, cos x = -12/13, cot x = 5/12, cosec x = -13/5, sec x = -13/12