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Popular >

tan^3(x)+sqrt(3)tan(x)<0

Solution

tan^{3} (x) + 3 tan (x) < 0

- \frac{π}{2} + πn < x < πn

Interval Notation

(- \frac{π}{2} + πn, πn)

Decimal Notation

- 1.57079 \dots + πn < x < πn

Solution steps

tan^{3} (x) + 3 tan (x) < 0

Let:

u = tan (x)

u^{3} + 3 u < 0

u^{3} + 3 u < 0 : u < 0

u^{3} + 3 u < 0

Factor

u^{3} + 3 u : u (u^{2} + 3)

u^{3} + 3 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= u^{2} u + 3 u

Factor out common term

u

= u (u^{2} + 3)

u (u^{2} + 3) < 0

Identify the intervals

Find the signs of the factors of

u (u^{2} + 3)

Find the signs of

u

u = 0

u < 0

u > 0

Find the signs of

u^{2} + 3 :

Positive for all real values

u^{2} + 3 > 0

Identify the intervals

Find the signs of the factors of

u^{2} + 3

Find the signs of

u^{2} + 3

Summarize in a table:

u^{2} + 3 u^{2} + 3 - \infty < u < \infty + +

Identify the intervals that satisfy the required condition:

> 0

True for all u \in R

True for all u

True for all u \in R

Summarize in a table:

u u^{2} + 3 u (u^{2} + 3) u < 0 - + - u = 0 0 + 0 u > 0 + + +

Identify the intervals that satisfy the required condition:

< 0

u < 0

u < 0

u < 0

Substitute back

u = tan (x)

tan (x) < 0

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

- \frac{π}{2} + πn < x < arctan (0) + πn

Simplify

arctan (0) : 0

arctan (0)

Use the following trivial identity:

arctan (0) = 0

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= 0

- \frac{π}{2} + πn < x < 0 + πn

Simplify

- \frac{π}{2} + πn < x < πn

sin (2 x) < \frac{3}{2}

x, sin (a x + (1 - a) y) \leq 0

2 sin (\frac{x}{2}) + 1 > 0

tan (x) > 3, 0 \leq x \leq 2 π

- 1 \leq \frac{2 + sin ( x )}{3}