What is sinx if x is in the interval (0,90) an cot x = 1/3?

We'll recall the definition of the cotangent
function:

cot x = cos x/sin
x

We know, from enunciation, that cot x =
1/3

We'll apply the Pythagorean
identity:

(cot x)^2 + 1 = 1/(sin
x)^2

(sin x)^2*[(cot x)^2 + 1] =
1

(sin x)^2 = 1/[(cot x)^2 +
1]

sin x = +1/sqrt[(cot x)^2 + 1] or sin x = -1/sqrt[(cot
x)^2 + 1]

sin x = 1/sqrt[(1/3)^2 +
1]

sin x = 1/sqrt [(1+9)/9] => sin x = 1/sqrt (10/9)
=> sin x = 3/sqrt 10 or sin x = -3/sqrt
10

We notice that x belongs to the 1st
quadrant, therefore, the value of the sine function is positive:sin x = 3*sqrt
10/10.