Question

Find the values of other five trigonometric functions for the following: tan $x =-\frac{5}{12}$ , x lies in second quadrant.

Answer 1

Given, tanx = -5/12 , x lies in second quadrant.

we know, in 2nd quadrant,
sin, cosec => positive.
cos, sec => negative .
tan, cot => negative.

so, tanx = -5/12 , cotx = 1/tanx = -12/5

we know, sec²x - tan²x = 1 [ from trigonometric identities ]
sec²x - (-5/12)² = 1
sec²x = 1 + (-5/12)² = 1 + 25/144
sec²x = 169/144
secx = ± 13/12 , but secx ≠ 13/12 because x lies in 2nd quadrant.
so, secx = -13/12 and cosx = 1/secx = -12/13

we also know, sin²x + cos²x = 1 [ from trigonometric identities ]

sin²x = 1 - (-12/13)² = 1 - 144/169 = 25/169
sin²x = 25/169
sinx = ± 5/13 , but sinx ≠ -5/13 because x lies in 2nd quadrant.
so, sinx = 5/13 and cosecx = 13/5