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Popular Trigonometry >

sec^2(x)<= 4/3

Solution

sec^{2} (x) \leq \frac{4}{3}

Solution

- \frac{π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn or \frac{5 π}{6} + 2 πn \leq x \leq \frac{7 π}{6} + 2 πn

Interval Notation

[- \frac{π}{6} + 2 πn, \frac{π}{6} + 2 πn] \cup [\frac{5 π}{6} + 2 πn, \frac{7 π}{6} + 2 πn]

Decimal

- 0.52359 \dots + 2 πn \leq x \leq 0.52359 \dots + 2 πn or 2.61799 \dots + 2 πn \leq x \leq 3.66519 \dots + 2 πn

Solution steps

sec^{2} (x) \leq \frac{4}{3}

Express with sin, cos

sec^{2} (x) \leq \frac{4}{3}

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

(\frac{1}{cos ( x )})^{2} \leq \frac{4}{3}

(\frac{1}{cos ( x )})^{2} \leq \frac{4}{3}

For

u^{n} \leq a

, if

n

is even then

- \frac{4}{3} \leq \frac{1}{cos ( x )} \leq \frac{4}{3}

a \leq u \leq b

then

a \leq u and u \leq b

- \frac{4}{3} \leq \frac{1}{cos ( x )} and \frac{1}{cos ( x )} \leq \frac{4}{3}

- \frac{4}{3} \leq \frac{1}{cos ( x )} : cos (x) \leq - \frac{3}{2} or cos (x) > 0

- \frac{4}{3} \leq \frac{1}{cos ( x )}

Switch sides

\frac{1}{cos ( x )} \geq - \frac{4}{3}

Rewrite in standard form

\frac{1}{cos ( x )} \geq - \frac{4}{3}

Add

\frac{4}{3}

to both sides

\frac{1}{cos ( x )} + \frac{4}{3} \geq - \frac{4}{3} + \frac{4}{3}

Simplify

\frac{1}{cos ( x )} + \frac{4}{3} \geq - \frac{4}{3} + \frac{4}{3}

Simplify

\frac{1}{cos ( x )} + \frac{4}{3} : \frac{1}{cos ( x )} + \frac{2}{3}

\frac{1}{cos ( x )} + \frac{4}{3}

\frac{4}{3} = \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

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

\frac{1}{cos ( x )} + \frac{2}{3} \geq 0

Simplify

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

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

Least Common Multiplier of

cos (x), 3 : 3 cos (x)

cos (x), 3

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

3

= 3 cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3 cos (x)

For

\frac{1}{cos ( x )} :

multiply the denominator and numerator by

3

\frac{1}{cos ( x )} = \frac{1 \cdot 3}{cos ( x ) 3} = \frac{3}{3 cos ( x )}

For

\frac{2}{3} :

multiply the denominator and numerator by

cos (x)

\frac{2}{3} = \frac{2 cos ( x )}{3 cos ( x )}

= \frac{3}{3 cos ( x )} + \frac{2 cos ( x )}{3 cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 cos ( x )}{3 cos ( x )}

Rationalize

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

\frac{3 + 2 cos ( x )}{3 cos ( x )}

Multiply by the conjugate

\frac{3}{3}

= \frac{( 3 + 2 cos ( x ) ) 3}{3 cos ( x ) 3}

3 cos (x) 3 = 3 cos (x)

3 cos (x) 3

Apply radical rule:

a a = a

33 = 3

= 3 cos (x)

= \frac{3 ( 3 + 2 cos ( x ) )}{3 cos ( x )}

= \frac{3 ( 2 cos ( x ) + 3 )}{3 cos ( x )}

\frac{3 ( 3 + 2 cos ( x ) )}{3 cos ( x )} \geq 0

Simplify

\frac{3}{3} : \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

\frac{3 ( 3 + 2 cos ( x ) )}{3 cos ( x )} \geq 0

Multiply both sides by

3

\frac{3 ( 3 + 2 cos ( x ) ) 3}{3 cos ( x )} \geq 0 \cdot 3

Simplify

\frac{3 + 2 cos ( x )}{cos ( x )} \geq 0

\frac{3 + 2 cos ( x )}{cos ( x )} \geq 0

Identify the intervals

Find the signs of the factors of

\frac{3 + 2 cos ( x )}{cos ( x )}

Find the signs of

3 + 2 cos (x)

3 + 2 cos (x) = 0 : cos (x) = - \frac{3}{2}

3 + 2 cos (x) = 0

Move

3

to the right side

3 + 2 cos (x) = 0

Subtract

3

from both sides

3 + 2 cos (x) - 3 = 0 - 3

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 3}{2}

Simplify

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

3 + 2 cos (x) < 0 : cos (x) < - \frac{3}{2}

3 + 2 cos (x) < 0

Move

3

to the right side

3 + 2 cos (x) < 0

Subtract

3

from both sides

3 + 2 cos (x) - 3 < 0 - 3

Simplify

2 cos (x) < - 3

2 cos (x) < - 3

Divide both sides by

2

2 cos (x) < - 3

Divide both sides by

2

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

Simplify

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

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

3 + 2 cos (x) > 0 : cos (x) > - \frac{3}{2}

3 + 2 cos (x) > 0

Move

3

to the right side

3 + 2 cos (x) > 0

Subtract

3

from both sides

3 + 2 cos (x) - 3 > 0 - 3

Simplify

2 cos (x) > - 3

2 cos (x) > - 3

Divide both sides by

2

2 cos (x) > - 3

Divide both sides by

2

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

Simplify

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

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

Find the signs of

cos (x)

cos (x) = 0

cos (x) < 0

cos (x) > 0

Find singularity points

Find the zeros of the denominator

cos (x) : cos (x) = 0

Summarize in a table:

3 + 2 cos (x) cos (x) \frac{3 + 2 c o s ( x )}{c o s ( x )} cos (x) < - \frac{3}{2} - - + cos (x) = - \frac{3}{2} 0 - 0 - \frac{3}{2} < cos (x) < 0 + - - cos (x) = 0 + 0 Undefined cos (x) > 0 + + +

Identify the intervals that satisfy the required condition:

\geq 0

cos (x) < - \frac{3}{2} or cos (x) = - \frac{3}{2} or cos (x) > 0

Merge Overlapping Intervals

cos (x) \leq - \frac{3}{2} or cos (x) > 0

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

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

cos (x) = - \frac{3}{2}

cos (x) \leq - \frac{3}{2}

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

cos (x) \leq - \frac{3}{2}

cos (x) > 0

cos (x) \leq - \frac{3}{2} or cos (x) > 0

cos (x) \leq - \frac{3}{2} or cos (x) > 0

cos (x) \leq - \frac{3}{2} or cos (x) > 0

\frac{1}{cos ( x )} \leq \frac{4}{3} : cos (x) < 0 or cos (x) \geq \frac{3}{2}

\frac{1}{cos ( x )} \leq \frac{4}{3}

Rewrite in standard form

\frac{1}{cos ( x )} \leq \frac{4}{3}

Subtract

\frac{4}{3}

from both sides

\frac{1}{cos ( x )} - \frac{4}{3} \leq \frac{4}{3} - \frac{4}{3}

Simplify

\frac{1}{cos ( x )} - \frac{4}{3} \leq \frac{4}{3} - \frac{4}{3}

Simplify

\frac{1}{cos ( x )} - \frac{4}{3} : \frac{1}{cos ( x )} - \frac{2}{3}

\frac{1}{cos ( x )} - \frac{4}{3}

\frac{4}{3} = \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

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

\frac{1}{cos ( x )} - \frac{2}{3} \leq 0

Simplify

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

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

Least Common Multiplier of

cos (x), 3 : 3 cos (x)

cos (x), 3

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

3

= 3 cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3 cos (x)

For

\frac{1}{cos ( x )} :

multiply the denominator and numerator by

3

\frac{1}{cos ( x )} = \frac{1 \cdot 3}{cos ( x ) 3} = \frac{3}{3 cos ( x )}

For

\frac{2}{3} :

multiply the denominator and numerator by

cos (x)

\frac{2}{3} = \frac{2 cos ( x )}{3 cos ( x )}

= \frac{3}{3 cos ( x )} - \frac{2 cos ( x )}{3 cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{3 - 2 cos ( x )}{3 cos ( x )}

Rationalize

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

\frac{3 - 2 cos ( x )}{3 cos ( x )}

Multiply by the conjugate

\frac{3}{3}

= \frac{( 3 - 2 cos ( x ) ) 3}{3 cos ( x ) 3}

3 cos (x) 3 = 3 cos (x)

3 cos (x) 3

Apply radical rule:

a a = a

33 = 3

= 3 cos (x)

= \frac{3 ( 3 - 2 cos ( x ) )}{3 cos ( x )}

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

\frac{3 ( 3 - 2 cos ( x ) )}{3 cos ( x )} \leq 0

Simplify

\frac{3}{3} : \frac{1}{3}

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}}}{3}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{1}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{1}{3}

\frac{3 ( 3 - 2 cos ( x ) )}{3 cos ( x )} \leq 0

Multiply both sides by

3

\frac{3 ( 3 - 2 cos ( x ) ) 3}{3 cos ( x )} \leq 0 \cdot 3

Simplify

\frac{3 - 2 cos ( x )}{cos ( x )} \leq 0

\frac{3 - 2 cos ( x )}{cos ( x )} \leq 0

Identify the intervals

Find the signs of the factors of

\frac{3 - 2 cos ( x )}{cos ( x )}

Find the signs of

3 - 2 cos (x)

3 - 2 cos (x) = 0 : cos (x) = \frac{3}{2}

3 - 2 cos (x) = 0

Move

3

to the right side

3 - 2 cos (x) = 0

Subtract

3

from both sides

3 - 2 cos (x) - 3 = 0 - 3

Simplify

- 2 cos (x) = - 3

- 2 cos (x) = - 3

Divide both sides by

- 2

- 2 cos (x) = - 3

Divide both sides by

- 2

\frac{- 2 cos ( x )}{- 2} = \frac{- 3}{- 2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

3 - 2 cos (x) < 0 : cos (x) > \frac{3}{2}

3 - 2 cos (x) < 0

Move

3

to the right side

3 - 2 cos (x) < 0

Subtract

3

from both sides

3 - 2 cos (x) - 3 < 0 - 3

Simplify

- 2 cos (x) < - 3

- 2 cos (x) < - 3

Multiply both sides by

- 1

- 2 cos (x) < - 3

Multiply both sides by -1 (reverse the inequality)

(- 2 cos (x)) (- 1) > (- 3) (- 1)

Simplify

2 cos (x) > 3

2 cos (x) > 3

Divide both sides by

2

2 cos (x) > 3

Divide both sides by

2

\frac{2 cos ( x )}{2} > \frac{3}{2}

Simplify

cos (x) > \frac{3}{2}

cos (x) > \frac{3}{2}

3 - 2 cos (x) > 0 : cos (x) < \frac{3}{2}

3 - 2 cos (x) > 0

Move

3

to the right side

3 - 2 cos (x) > 0

Subtract

3

from both sides

3 - 2 cos (x) - 3 > 0 - 3

Simplify

- 2 cos (x) > - 3

- 2 cos (x) > - 3

Multiply both sides by

- 1

- 2 cos (x) > - 3

Multiply both sides by -1 (reverse the inequality)

(- 2 cos (x)) (- 1) < (- 3) (- 1)

Simplify

2 cos (x) < 3

2 cos (x) < 3

Divide both sides by

2

2 cos (x) < 3

Divide both sides by

2

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

Simplify

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

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

Find the signs of

cos (x)

cos (x) = 0

cos (x) < 0

cos (x) > 0

Find singularity points

Find the zeros of the denominator

cos (x) : cos (x) = 0

Summarize in a table:

3 - 2 cos (x) cos (x) \frac{3 - 2 c o s ( x )}{c o s ( x )} cos (x) < 0 + - - cos (x) = 0 + 0 Undefined 0 < cos (x) < \frac{3}{2} + + + cos (x) = \frac{3}{2} 0 + 0 cos (x) > \frac{3}{2} - + -

Identify the intervals that satisfy the required condition:

\leq 0

cos (x) < 0 or cos (x) = \frac{3}{2} or cos (x) > \frac{3}{2}

Merge Overlapping Intervals

cos (x) < 0 or cos (x) = \frac{3}{2} or cos (x) > \frac{3}{2}

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

cos (x) < 0

cos (x) = \frac{3}{2}

cos (x) < 0 or cos (x) = \frac{3}{2}

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

cos (x) < 0 or cos (x) = \frac{3}{2}

cos (x) > \frac{3}{2}

cos (x) < 0 or cos (x) \geq \frac{3}{2}

cos (x) < 0 or cos (x) \geq \frac{3}{2}

cos (x) < 0 or cos (x) \geq \frac{3}{2}

Combine the intervals

(cos (x) \leq - \frac{3}{2} or cos (x) > 0) and (cos (x) < 0 or cos (x) \geq \frac{3}{2})

Merge Overlapping Intervals

cos (x) \leq - \frac{3}{2} or cos (x) > 0 and cos (x) < 0 or cos (x) \geq \frac{3}{2}

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

cos (x) \leq - \frac{3}{2} or cos (x) > 0

and

cos (x) < 0 or cos (x) \geq \frac{3}{2}

cos (x) \leq - \frac{3}{2} or cos (x) \geq \frac{3}{2}

cos (x) \leq - \frac{3}{2} or cos (x) \geq \frac{3}{2}

cos (x) \leq - \frac{3}{2} : \frac{5 π}{6} + 2 πn \leq x \leq \frac{7 π}{6} + 2 πn

cos (x) \leq - \frac{3}{2}

For

cos (x) \leq a

, if

- 1 < a < 1

then

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

arccos (- \frac{3}{2}) + 2 πn \leq x \leq 2 π - arccos (- \frac{3}{2}) + 2 πn

Simplify

arccos (- \frac{3}{2}) : \frac{5 π}{6}

arccos (- \frac{3}{2})

Use the following trivial identity:

arccos (- \frac{3}{2}) = \frac{5 π}{6}

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}

= \frac{5 π}{6}

Simplify

2 π - arccos (- \frac{3}{2}) : \frac{7 π}{6}

2 π - arccos (- \frac{3}{2})

Use the following trivial identity:

arccos (- \frac{3}{2}) = \frac{5 π}{6}

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 - \frac{5 π}{6}

Simplify

2 π - \frac{5 π}{6}

Convert element to fraction:

2 π = \frac{2 π 6}{6}

= \frac{2 π 6}{6} - \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 6 - 5 π}{6}

2 π 6 - 5 π = 7 π

2 π 6 - 5 π

Multiply the numbers:

2 \cdot 6 = 12

= 12 π - 5 π

Add similar elements:

12 π - 5 π = 7 π

= 7 π

= \frac{7 π}{6}

= \frac{7 π}{6}

\frac{5 π}{6} + 2 πn \leq x \leq \frac{7 π}{6} + 2 πn

cos (x) \geq \frac{3}{2} : - \frac{π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn

cos (x) \geq \frac{3}{2}

For

cos (x) \geq a

, if

- 1 < a < 1

then

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

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

Simplify

- arccos (\frac{3}{2}) : - \frac{π}{6}

- arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

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}

= - \frac{π}{6}

Simplify

arccos (\frac{3}{2}) : \frac{π}{6}

arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

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}

= \frac{π}{6}

- \frac{π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn

Combine the intervals

\frac{5 π}{6} + 2 πn \leq x \leq \frac{7 π}{6} + 2 πn or - \frac{π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn

Merge Overlapping Intervals

- \frac{π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn or \frac{5 π}{6} + 2 πn \leq x \leq \frac{7 π}{6} + 2 πn

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