Find the value of the other five trigonometric functions, given tanx= 4/11, secx<0.

Question

This is for my trig class, thanks for the help!

Answer 1

tan(x) = 4/11, sec(x) < 0
tan > 0 in quadrants I & III.
sec(x) = 1/cos(x) < 0 in II & III

So x lies in quadrant III:

............ -4
.....  _______
..... | ........../
..... | ........ /
..... | ....... /
..... | ...... /
..... | ..... / sqrt(137)
-11 | ...../
..... | .../
..... | ../
..... | ./
..... | /
..... |/

sin(x) = -11/sqrt(137)
cos(x) = -4/sqrt(137)
tan(x) = sin(x)/cos(x) = -11/(-4) = 11/4
cot(x) = 1/tan(x) = 4/11
sec(x) = 1/cos(x) = -sqrt(137)/4
csc(x) = 1/sin(x) = -sqrt(137)/11

Answer 2

sec² x = 1+tan² x = 1+(4/11)² = 137/121.

 

sec x = (-1/11)√137

 

cot x = 1/tan x = 11/4

 

csc² x = 1+cot² x = 137/16

 

csc x = (-1/4)√137

 

sin x = 1/csc x = -4/√137

 

cos x = 1/sec x = -11/√137