Explanation:
sec(theta) = 1/cos(theta)
cot(theta) = 1/tan(theta) = cos(theta)/sin(theta)
as
tan(theta) = sin(theta)/cos(theta)
sin²(theta) + cos²(theta) = 1
csc(theta) = 1/sin(theta)
5.
sec(theta) = sqrt(73)/8 = 1/cos(theta)
cos(theta) = 1/sec(theta) = 8/sqrt(73)
sin(theta) = sqrt(1 - (8/sqrt(73))²) = sqrt(1 - 64/73) =
= sqrt(73/73 - 64/73) = sqrt(9/73) = 3/sqrt(73)
tan(theta) = 3/sqrt(73) / 8/sqrt(73) = 3/8
cot(theta) = 1/tan(theta) = 8/3
csc(theta) = 1/sin(theta) = sqrt(73)/3
6.
cot(theta) = sqrt(3) = cos(theta)/sin(theta)
we could now say
cos(theta) = sqrt(3)
sin(theta) = 1
but sqrt(3) > 1. and that cannot be. sine and cosine must both be between -1 and +1.
so, we have to make a detour :
cos(theta)/sin(theta) = sqrt(3)
cos(theta) = sqrt(3)×sin(theta)
sin²(theta) + cos²(theta) = 1
sin²(theta) + (sqrt(3)×sin(theta))² = 1
sin²(theta) + 3×sin²(theta) = 1
4×sin²(theta) = 1
sin²(theta) = 1/4
sin(theta) = sqrt(1/4) = 1/2
cos(theta) = sqrt(3)×1/2 = sqrt(3)/2
tan(theta) = 1/cot(theta) = 1/sqrt(3)
csc(theta) = 1/sin(theta) = 1 / 1/2 = 2
sec(theta) = 1/cos(theta) = 1 / sqrt(3)/2 = 2/sqrt(3)