Given: cos(x) = sqrt(3)/2
Use the identity sin(x) = +-sqrt(1 - cos^2(x)):
Because we are given that tan(x) is a positive number, we shall drop the +-, thereby, making the sine positive, only:
sin(x) = sqrt(1 - (sqrt(3)/2)^2)
sin(x) = sqrt(4/4 - 3/4)
sin(x) = 1/2
Verify that tan(x) = sqrt(3)/3:
sin(x)/cos(x) = (1/2)/(sqrt(3)/2) = 1/sqrt(3) = sqrt(3)/3 = tan(x)
Verified.
Use the identity cot(x) = 1/tan(x):
= 1/(sqrt(3)/3)#
cot(x) = sqrt(3)
Use the identity csc(x) = 1/sin(x)
csc(x) = 1/(1/2)
csc(x) = 2
Use the identity sec(x) = 1/cos(x)
sec(x) = 1/(sqrt(3)/2)
sec(x) = 2/sqrt(3)
sec(x) = (2sqrt(3))/3
