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Popular Trigonometry >

cos^2(x)-cos(x)-2>= 0

Solution

cos^{2} (x) - cos (x) - 2 \geq 0

Solution

x = π + 2 πn

Decimal

x = 3.14159 \dots + 2 πn

Solution steps

cos^{2} (x) - cos (x) - 2 \geq 0

Let:

u = cos (x)

u^{2} - u - 2 \geq 0

u^{2} - u - 2 \geq 0 : u \leq - 1 or u \geq 2

u^{2} - u - 2 \geq 0

Factor

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

u^{2} - u - 2

Break the expression into groups

u^{2} - 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 = - 1

Check

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

TrueCheck

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

False

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) \geq 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

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 - 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:

\geq 0

u < - 1 or u = - 1 or u = 2 or u > 2

Merge Overlapping Intervals

u \leq - 1 or u = 2 or u > 2

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

u < - 1

u = - 1

u \leq - 1

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

u \leq - 1

u = 2

u \leq - 1 or u = 2

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

u \leq - 1 or u = 2

u > 2

u \leq - 1 or u \geq 2

u \leq - 1 or u \geq 2

u \leq - 1 or u \geq 2

u \leq - 1 or u \geq 2

Substitute back

u = cos (x)

cos (x) \leq - 1 or cos (x) \geq 2

cos (x) \leq - 1 : x = π + 2 πn

cos (x) \leq - 1

For

cos (x) \leq a

, if

- 1 < a < 1

then

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (- 1) + 2 πn \leq x \leq 2 π - arccos (- 1) + 2 πn

Simplify

arccos (- 1) : π

arccos (- 1)

Use the following trivial identity:

arccos (- 1) = π

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= π

Simplify

2 π - arccos (- 1) : π

2 π - arccos (- 1)

Use the following trivial identity:

arccos (- 1) = π

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - π

Add similar elements:

2 π - π = π

= π

π + 2 πn \leq x \leq π + 2 πn

Simplify

x = π + 2 πn

cos (x) \geq 2 :

False for all

x \in R

cos (x) \geq 2

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) \geq 2 and - 1 \leq cos (x) \leq 1 :

False

Let

y = cos (x)

Combine the intervals

y \geq 2 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \geq 2 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \geq 2

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

Combine the intervals

x = π + 2 πn or False for all x \in R

Merge Overlapping Intervals

x = π + 2 πn

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