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cos(x)-1/2 cos(2x)>0

Solution

cos (x) - \frac{1}{2} cos (2 x) > 0

Solution

- arccos (\frac{- 3 + 1}{2}) + 2 πn < x < arccos (\frac{- 3 + 1}{2}) + 2 πn

Interval Notation

(- arccos (\frac{- 3 + 1}{2}) + 2 πn, arccos (\frac{- 3 + 1}{2}) + 2 πn)

Decimal Notation

- 1.94553 \dots + 2 πn < x < 1.94553 \dots + 2 πn

Solution steps

cos (x) - \frac{1}{2} cos (2 x) > 0

Use the following identity:

cos (2 x) = - 1 + 2 cos^{2} (x)

cos (x) - (- 1 + 2 cos^{2} (x)) \frac{1}{2} > 0

Simplify

cos (x) - (- 1 + 2 cos^{2} (x)) \frac{1}{2} : cos (x) + \frac{1}{2} - cos^{2} (x)

cos (x) - (- 1 + 2 cos^{2} (x)) \frac{1}{2}

= cos (x) - \frac{1}{2} (- 1 + 2 cos^{2} (x))

Expand

- \frac{1}{2} (- 1 + 2 cos^{2} (x)) : \frac{1}{2} - cos^{2} (x)

- \frac{1}{2} (- 1 + 2 cos^{2} (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - \frac{1}{2}, b = - 1, c = 2 cos^{2} (x)

= - \frac{1}{2} (- 1) + (- \frac{1}{2}) \cdot 2 cos^{2} (x)

Apply minus-plus rules

- (- a) = a, + (- a) = - a

= 1 \cdot \frac{1}{2} - 2 \cdot \frac{1}{2} cos^{2} (x)

Simplify

1 \cdot \frac{1}{2} - 2 \cdot \frac{1}{2} cos^{2} (x) : \frac{1}{2} - cos^{2} (x)

1 \cdot \frac{1}{2} - 2 \cdot \frac{1}{2} cos^{2} (x)

1 \cdot \frac{1}{2} = \frac{1}{2}

1 \cdot \frac{1}{2}

Multiply:

1 \cdot \frac{1}{2} = \frac{1}{2}

= \frac{1}{2}

2 \cdot \frac{1}{2} cos^{2} (x) = cos^{2} (x)

2 \cdot \frac{1}{2} cos^{2} (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} cos^{2} (x)

Cancel the common factor:

2

= cos^{2} (x) \cdot 1

Multiply:

cos^{2} (x) \cdot 1 = cos^{2} (x)

= cos^{2} (x)

= \frac{1}{2} - cos^{2} (x)

= \frac{1}{2} - cos^{2} (x)

= cos (x) + \frac{1}{2} - cos^{2} (x)

cos (x) + \frac{1}{2} - cos^{2} (x) > 0

Let:

u = cos (x)

u + \frac{1}{2} - u^{2} > 0

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

u + \frac{1}{2} - u^{2} > 0

Rewrite in standard form

u + \frac{1}{2} - u^{2} > 0

Multiply both sides by

2

u \cdot 2 + \frac{1}{2} \cdot 2 - u^{2} \cdot 2 > 0 \cdot 2

2 u + 1 - 2 u^{2} > 0

2 u + 1 - 2 u^{2} > 0

Complete the square

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

2 u + 1 - 2 u^{2}

Write in the standard form

a x^{2} + b x + c

- 2 u^{2} + 2 u + 1

Write

- 2 u^{2} + 2 u + 1

in the form:

x^{2} + 2 a x + a^{2}

Factor out

- 2

- 2 (u^{2} - u - \frac{1}{2})

2 a = - 1 : a = - \frac{1}{2}

2 a = - 1

Divide both sides by

2

2 a = - 1

Divide both sides by

2

\frac{2 a}{2} = \frac{- 1}{2}

Simplify

a = - \frac{1}{2}

a = - \frac{1}{2}

Add and subtract

(- \frac{1}{2})^{2}

- 2 (u^{2} - u - \frac{1}{2} + (- \frac{1}{2})^{2} - (- \frac{1}{2})^{2})

x^{2} + 2 a x + a^{2} = (x + a)^{2}

u^{2} - 1 u + (- \frac{1}{2})^{2} = (u - \frac{1}{2})^{2}

- 2 ((u - \frac{1}{2})^{2} - \frac{1}{2} - (- \frac{1}{2})^{2})

Simplify

- 2 (u - \frac{1}{2})^{2} + \frac{3}{2}

- 2 (u - \frac{1}{2})^{2} + \frac{3}{2} > 0

Move

\frac{3}{2}

to the right side

- 2 (u - \frac{1}{2})^{2} + \frac{3}{2} > 0

Subtract

\frac{3}{2}

from both sides

- 2 (u - \frac{1}{2})^{2} + \frac{3}{2} - \frac{3}{2} > 0 - \frac{3}{2}

Simplify

- 2 (u - \frac{1}{2})^{2} > - \frac{3}{2}

- 2 (u - \frac{1}{2})^{2} > - \frac{3}{2}

Multiply both sides by

- 1

- 2 (u - \frac{1}{2})^{2} > - \frac{3}{2}

Multiply both sides by -1 (reverse the inequality)

(- 2 (u - \frac{1}{2})^{2}) (- 1) < (- \frac{3}{2}) (- 1)

Simplify

2 (u - \frac{1}{2})^{2} < \frac{3}{2}

2 (u - \frac{1}{2})^{2} < \frac{3}{2}

Divide both sides by

2

2 (u - \frac{1}{2})^{2} < \frac{3}{2}

Divide both sides by

2

\frac{2 ( u - \frac{1}{2} ) ^{2}}{2} < \frac{\frac{3}{2}}{2}

Simplify

\frac{2 ( u - \frac{1}{2} ) ^{2}}{2} < \frac{\frac{3}{2}}{2}

Simplify

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

\frac{2 ( u - \frac{1}{2} ) ^{2}}{2}

Divide the numbers:

\frac{2}{2} = 1

= (u - \frac{1}{2})^{2}

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{4}

\frac{\frac{3}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

(u - \frac{1}{2})^{2} < \frac{3}{4}

(u - \frac{1}{2})^{2} < \frac{3}{4}

(u - \frac{1}{2})^{2} < \frac{3}{4}

For

u^{n} < a

, if

n

is even then

- \frac{3}{4} < u - \frac{1}{2} < \frac{3}{4}

a < u < b

then

a < u and u < b

- \frac{3}{4} < u - \frac{1}{2} and u - \frac{1}{2} < \frac{3}{4}

- \frac{3}{4} < u - \frac{1}{2} : u > \frac{- 3 + 1}{2}

- \frac{3}{4} < u - \frac{1}{2}

Switch sides

u - \frac{1}{2} > - \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

u - \frac{1}{2} > - \frac{3}{2}

Move

\frac{1}{2}

to the right side

u - \frac{1}{2} > - \frac{3}{2}

Add

\frac{1}{2}

to both sides

u - \frac{1}{2} + \frac{1}{2} > - \frac{3}{2} + \frac{1}{2}

Simplify

u - \frac{1}{2} + \frac{1}{2} > - \frac{3}{2} + \frac{1}{2}

Simplify

u - \frac{1}{2} + \frac{1}{2} : u

u - \frac{1}{2} + \frac{1}{2}

Add similar elements:

- \frac{1}{2} + \frac{1}{2} > 0

= u

Simplify

- \frac{3}{2} + \frac{1}{2} : \frac{- 3 + 1}{2}

- \frac{3}{2} + \frac{1}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 3 + 1}{2}

u > \frac{- 3 + 1}{2}

u > \frac{- 3 + 1}{2}

u > \frac{- 3 + 1}{2}

u - \frac{1}{2} < \frac{3}{4} : u < \frac{3 + 1}{2}

u - \frac{1}{2} < \frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

u - \frac{1}{2} < \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u - \frac{1}{2} < \frac{3}{2}

Move

\frac{1}{2}

to the right side

u - \frac{1}{2} < \frac{3}{2}

Add

\frac{1}{2}

to both sides

u - \frac{1}{2} + \frac{1}{2} < \frac{3}{2} + \frac{1}{2}

Simplify

u - \frac{1}{2} + \frac{1}{2} < \frac{3}{2} + \frac{1}{2}

Simplify

u - \frac{1}{2} + \frac{1}{2} : u

u - \frac{1}{2} + \frac{1}{2}

Add similar elements:

- \frac{1}{2} + \frac{1}{2} < 0

= u

Simplify

\frac{3}{2} + \frac{1}{2} : \frac{3 + 1}{2}

\frac{3}{2} + \frac{1}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 + 1}{2}

u < \frac{3 + 1}{2}

u < \frac{3 + 1}{2}

u < \frac{3 + 1}{2}

Combine the intervals

u > \frac{- 3 + 1}{2} and u < \frac{3 + 1}{2}

Merge Overlapping Intervals

u > \frac{- 3 + 1}{2} and u < \frac{3 + 1}{2}

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

u > \frac{- 3 + 1}{2}

and

u < \frac{3 + 1}{2}

\frac{- 3 + 1}{2} < u < \frac{3 + 1}{2}

\frac{- 3 + 1}{2} < u < \frac{3 + 1}{2}

\frac{- 3 + 1}{2} < u < \frac{3 + 1}{2}

Substitute back

u = cos (x)

\frac{- 3 + 1}{2} < cos (x) < \frac{3 + 1}{2}

a < u < b

then

a < u and u < b

\frac{- 3 + 1}{2} < cos (x) and cos (x) < \frac{3 + 1}{2}

\frac{- 3 + 1}{2} < cos (x) : - arccos (\frac{- 3 + 1}{2}) + 2 πn < x < arccos (\frac{- 3 + 1}{2}) + 2 πn

\frac{- 3 + 1}{2} < cos (x)

Switch sides

cos (x) > \frac{- 3 + 1}{2}

For

cos (x) > a

, if

- 1 \leq a < 1

then

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

- arccos (\frac{- 3 + 1}{2}) + 2 πn < x < arccos (\frac{- 3 + 1}{2}) + 2 πn

cos (x) < \frac{3 + 1}{2} :

True for all

x \in R

cos (x) < \frac{3 + 1}{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) < \frac{3 + 1}{2} and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

y < \frac{3 + 1}{2} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < \frac{3 + 1}{2} and - 1 \leq y \leq 1

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

y < \frac{3 + 1}{2}

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

- arccos (\frac{- 3 + 1}{2}) + 2 πn < x < arccos (\frac{- 3 + 1}{2}) + 2 πn and True for all x \in R

Merge Overlapping Intervals

- arccos (\frac{- 3 + 1}{2}) + 2 πn < x < arccos (\frac{- 3 + 1}{2}) + 2 πn

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