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Popular Trigonometry >

(tan(x)+1)(tan(x)+2)+2tan(x)+2>= 0

Solution

(tan (x) + 1) (tan (x) + 2) + 2 tan (x) + 2 \geq 0

Solution

- \frac{π}{4} + πn \leq x < \frac{π}{2} + πn or - \frac{π}{2} + πn < x \leq - arctan (4) + πn

Interval Notation

[- \frac{π}{4} + πn, \frac{π}{2} + πn) \cup (- \frac{π}{2} + πn, - arctan (4) + πn]

Decimal

- 0.78539 \dots + πn \leq x < 1.57079 \dots + πn or - 1.57079 \dots + πn < x \leq - 1.32581 \dots + πn

Solution steps

(tan (x) + 1) (tan (x) + 2) + 2 tan (x) + 2 \geq 0

Let:

u = tan (x)

(u + 1) (u + 2) + 2 u + 2 \geq 0

(u + 1) (u + 2) + 2 u + 2 \geq 0 : u \leq - 4 or u \geq - 1

(u + 1) (u + 2) + 2 u + 2 \geq 0

Factor

(u + 1) (u + 2) + 2 u + 2 : (u + 1) (u + 4)

(u + 1) (u + 2) + 2 u + 2

Factor

2 u + 2 : 2 (u + 1)

2 u + 2

Factor out common term

2

= 2 (u + 1)

= (u + 1) (u + 2) + 2 (u + 1)

Factor out common term

(u + 1)

= (u + 1) (u + 2 + 2)

Simplify

u + 2 + 2 : u + 4

u + 2 + 2

Add the numbers:

2 + 2 = 4

= u + 4

= (u + 1) (u + 4)

(u + 1) (u + 4) \geq 0

Identify the intervals

Find the signs of the factors of

(u + 1) (u + 4)

Find the signs of

u + 1

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

u + 1 < 0 : u < - 1

u + 1 < 0

Move

1

to the right side

u + 1 < 0

Subtract

1

from both sides

u + 1 - 1 < 0 - 1

Simplify

u < - 1

u < - 1

u + 1 > 0 : u > - 1

u + 1 > 0

Move

1

to the right side

u + 1 > 0

Subtract

1

from both sides

u + 1 - 1 > 0 - 1

Simplify

u > - 1

u > - 1

Find the signs of

u + 4

u + 4 = 0 : u = - 4

u + 4 = 0

Move

4

to the right side

u + 4 = 0

Subtract

4

from both sides

u + 4 - 4 = 0 - 4

Simplify

u = - 4

u = - 4

u + 4 < 0 : u < - 4

u + 4 < 0

Move

4

to the right side

u + 4 < 0

Subtract

4

from both sides

u + 4 - 4 < 0 - 4

Simplify

u < - 4

u < - 4

u + 4 > 0 : u > - 4

u + 4 > 0

Move

4

to the right side

u + 4 > 0

Subtract

4

from both sides

u + 4 - 4 > 0 - 4

Simplify

u > - 4

u > - 4

Summarize in a table:

u + 1 u + 4 (u + 1) (u + 4) u < - 4 - - + u = - 4 - 00 - 4 < u < - 1 - + - u = - 1 0 + 0 u > - 1 + + +

Identify the intervals that satisfy the required condition:

\geq 0

u < - 4 or u = - 4 or u = - 1 or u > - 1

Merge Overlapping Intervals

u \leq - 4 or u = - 1 or u > - 1

The union of two intervals is the set of numbers which are in either interval

u < - 4

u = - 4

u \leq - 4

The union of two intervals is the set of numbers which are in either interval

u \leq - 4

u = - 1

u \leq - 4 or u = - 1

The union of two intervals is the set of numbers which are in either interval

u \leq - 4 or u = - 1

u > - 1

u \leq - 4 or u \geq - 1

u \leq - 4 or u \geq - 1

u \leq - 4 or u \geq - 1

u \leq - 4 or u \geq - 1

Substitute back

u = tan (x)

tan (x) \leq - 4 or tan (x) \geq - 1

tan (x) \leq - 4 : - \frac{π}{2} + πn < x \leq - arctan (4) + πn

tan (x) \leq - 4

tan (x) \leq a

then

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

- \frac{π}{2} + πn < x \leq arctan (- 4) + πn

Simplify

arctan (- 4) : - arctan (4)

arctan (- 4)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 4) = - arctan (4)

= - arctan (4)

- \frac{π}{2} + πn < x \leq - arctan (4) + πn

tan (x) \geq - 1 : - \frac{π}{4} + πn \leq x < \frac{π}{2} + πn

tan (x) \geq - 1

tan (x) \geq a

then

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

arctan (- 1) + πn \leq x < \frac{π}{2} + πn

Simplify

arctan (- 1) : - \frac{π}{4}

arctan (- 1)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

= - arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

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}

= \frac{π}{4}

= - \frac{π}{4}

- \frac{π}{4} + πn \leq x < \frac{π}{2} + πn

Combine the intervals

- \frac{π}{2} + πn < x \leq - arctan (4) + πn or - \frac{π}{4} + πn \leq x < \frac{π}{2} + πn

Merge Overlapping Intervals

- \frac{π}{4} + πn \leq x < \frac{π}{2} + πn or - \frac{π}{2} + πn < x \leq - arctan (4) + πn

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