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Popular Trigonometry >

tan^2(x)-3tan(x)+2<0

Solution

tan^{2} (x) - 3 tan (x) + 2 < 0

Solution

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

Interval Notation

(\frac{π}{4} + πn, arctan (2) + πn)

Decimal

0.78539 \dots + πn < x < 1.10714 \dots + πn

Solution steps

tan^{2} (x) - 3 tan (x) + 2 < 0

Let:

u = tan (x)

u^{2} - 3 u + 2 < 0

u^{2} - 3 u + 2 < 0 : 1 < u < 2

u^{2} - 3 u + 2 < 0

Factor

u^{2} - 3 u + 2 : (u - 1) (u - 2)

u^{2} - 3 u + 2

Break the expression into groups

u^{2} - 3 u + 2

Definition

Factors of

2 : 1, 2

2

Divisors (Factors)

Find the Prime factors of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Add 1

1

The factors of

2

1, 2

Negative factors of

2 : - 1, - 2

Multiply the factors by

- 1

to get the negative factors

- 1, - 2

For every two factors such that

u * v = 2,

check if

u + v = - 3

Check

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

FalseCheck

u = - 1, v = - 2 : u * v = 2, u + v = - 3 \Rightarrow

True

u = - 1, v = - 2

Group into

(a x^{2} + ux) + (vx + c)

(u^{2} - u) + (- 2 u + 2)

= (u^{2} - u) + (- 2 u + 2)

Factor out

u

from

u^{2} - u : u (u - 1)

u^{2} - u

Apply exponent rule:

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

u^{2} = uu

= uu - u

Factor out common term

u

= u (u - 1)

Factor out

- 2

from

- 2 u + 2 : - 2 (u - 1)

- 2 u + 2

Factor out common term

- 2

= - 2 (u - 1)

= u (u - 1) - 2 (u - 1)

Factor out common term

u - 1

= (u - 1) (u - 2)

(u - 1) (u - 2) < 0

Identify the intervals

Find the signs of the factors of

(u - 1) (u - 2)

Find the signs of

u - 1

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to 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

Add

1

to 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

Add

1

to both sides

u - 1 + 1 > 0 + 1

Simplify

u > 1

u > 1

Find the signs of

u - 2

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

u - 2 < 0 : u < 2

u - 2 < 0

Move

2

to the right side

u - 2 < 0

Add

2

to both sides

u - 2 + 2 < 0 + 2

Simplify

u < 2

u < 2

u - 2 > 0 : u > 2

u - 2 > 0

Move

2

to the right side

u - 2 > 0

Add

2

to both sides

u - 2 + 2 > 0 + 2

Simplify

u > 2

u > 2

Summarize in a table:

u - 1 u - 2 (u - 1) (u - 2) u < 1 - - + u = 1 0 - 0 1 < u < 2 + - - u = 2 + 00 u > 2 + + +

Identify the intervals that satisfy the required condition:

< 0

1 < u < 2

1 < u < 2

1 < u < 2

Substitute back

u = tan (x)

1 < tan (x) < 2

a < u < b

then

a < u and u < b

1 < tan (x) and tan (x) < 2

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

1 < tan (x)

Switch sides

tan (x) > 1

tan (x) > a

then

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

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

Simplify

arctan (1) : \frac{π}{4}

arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

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} + πn < x < \frac{π}{2} + πn

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

tan (x) < 2

tan (x) < a

then

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

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

Combine the intervals

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

Merge Overlapping Intervals

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

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