How can csc and sec be determined from tan(x) = (3/2)^(1/3) without using a calculator?

Question

Or even just x.

Answer 1

You know the identity sec² x=1+tan² x. So compute tan² x: we obtain (3/2)^(2/3). Thus sec² x = 1+(3/2)^(2/3) and sec x = ±√(1+(3/2)^(2/3)). csc x may be obtained similarly from the identity csc² x = cot² x + 1 (knowing that cot x=1/tan x). Thus csc x = ±√(1+(2/3)^(2/3)). Further, note that since tan x = sec x/csc x, and tan x is positive, the same sign must be taken in both cases. However, you cannot tell without additional information (e.g. whether x is in quadrant I or III) whether to take the positive or negative sign, so this is as far as you can go.

Answer 2

sec x = sqrt(tan²x + 1)

csc x = sqrt(1 + cot²x)

Plug in the value of tan x and simplify as much as possible.