Question

For what values of A in the first quadrant, the expression $\frac{cot^{3} A - 3 cot A}{3 cot^{2} A - 1}$ is positive?

Answer 1

Let cotA = x

then, $\frac{cot^{3} A - 3 cot A}{3 cot^{2} A - 1}$ = (x³ - 3x)/(3x² - 1)

a/c to question, we have to find value of A for which $\frac{cot^{3} A - 3 cot A}{3 cot^{2} A - 1}$ ≥ 0

take, (x³ - 3x)/(3x² - 1) ≥ 0

or, x(x - √3)(x + √3)/(√3x + 1)(√3x - 1) ≥ 0

put in number line and use inequality concepts
then, -√3 ≤ x ≤ -1/√3 or, 0 ≤ x ≤ 1/√3 or, x ≥ √3
so, -√3 ≤ cotA ≤ -1/√3 or, 0 ≤ cotA ≤ 1/√3 or, cotA ≥ √3
but A lies in the first quadrant.
so, -√3 ≤ cotA ≤ -1/√3 is not included in the solution because in first quadrant cotA can't be negative.

solve 0 ≤ cotA ≤ 1/√3 , cotA ≥ √3

we get, nπ ≤ A ≤ nπ + π/6 , $n\in\mathbb{Z}$
nπ - 2π/3 ≤ A ≤ nπ - π/2 , $n\in\mathbb{Z}$
nπ - 4π/3 ≤ A ≤ nπ - 7π/6 , $n\in\mathbb{Z}$

Answer 2

Answer:

Step-by-step explanation: