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Popular Trigonometry >

0<2-sec(x^2)

Solution

0 < 2 - sec (x^{2})

Solution

2 πn \leq x < \frac{π}{3} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{5 π}{3} + 2 πn < x < \frac{7 π}{3} + 2 πn or \frac{5 π}{2} + 2 πn < x < \frac{7 π}{2} + 2 πn or \frac{11 π}{3} + 2 πn < x < \frac{13 π}{3} + 2 πn or \frac{3 π}{2} + 2 πn < x < \frac{11 π}{2} + 2 πn or \frac{17 π}{3} + 2 πn < x < \frac{19 π}{3} + 2 πn or \frac{13 π}{2} + 2 πn < x < \frac{15 π}{2} + 2 πn or \frac{23 π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn or \frac{17 π}{2} + 2 πn < x < \frac{19 π}{2} + 2 πn or \frac{29 π}{3} + 2 πn < x \leq 2 π + 2 πn

Interval Notation

[2 πn, \frac{π}{3} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{5 π}{3} + 2 πn, \frac{7 π}{3} + 2 πn) \cup (\frac{5 π}{2} + 2 πn, \frac{7 π}{2} + 2 πn) \cup (\frac{11 π}{3} + 2 πn, \frac{13 π}{3} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, \frac{11 π}{2} + 2 πn) \cup (\frac{17 π}{3} + 2 πn, \frac{19 π}{3} + 2 πn) \cup (\frac{13 π}{2} + 2 πn, \frac{15 π}{2} + 2 πn) \cup (\frac{23 π}{3} + 2 πn, \frac{5 π}{3} + 2 πn) \cup (\frac{17 π}{2} + 2 πn, \frac{19 π}{2} + 2 πn) \cup (\frac{29 π}{3} + 2 πn, 2 π + 2 πn]

Decimal

2 πn \leq x < 1.02332 \dots + 2 πn or 1.25331 \dots + 2 πn < x < 2.17080 \dots + 2 πn or 2.28822 \dots + 2 πn < x < 2.70746 \dots + 2 πn or 2.80249 \dots + 2 πn < x < 3.31595 \dots + 2 πn or 3.39399 \dots + 2 πn < x < 3.68965 \dots + 2 πn or 3.75994 \dots + 2 πn < x < 4.15677 \dots + 2 πn or 4.21928 \dots + 2 πn < x < 4.46057 \dots + 2 πn or 4.51888 \dots + 2 πn < x < 4.85406 \dots + 2 πn or 4.90770 \dots + 2 πn < x < 5.11663 \dots + 2 πn or 5.16754 \dots + 2 πn < x < 5.46306 \dots + 2 πn or 5.51078 \dots + 2 πn < x \leq 6.28318 \dots + 2 πn

Solution steps

0 < 2 - sec (x^{2})

Switch sides

2 - sec (x^{2}) > 0

Periodicity of

2 - sec (x^{2}) :

Not periodic

The function

2 - sec (x^{2})

is not periodic

= Not periodic

Express with sin, cos

2 - sec (x^{2}) > 0

Use the basic trigonometric identity:

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

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Find the zeroes and undifined points of

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

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

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

\frac{2 cos ( x ^{2} ) - 1}{cos ( x ^{2} )} = 0, 0 \leq x < 2 π : x = \frac{π}{3}, x = \frac{5 π}{3}, x = \frac{7 π}{3}, x = \frac{11 π}{3}, x = \frac{13 π}{3}, x = \frac{17 π}{3}, x = \frac{19 π}{3}, x = \frac{23 π}{3}, x = \frac{5 π}{3}, x = \frac{29 π}{3}

\frac{2 cos ( x ^{2} ) - 1}{cos ( x ^{2} )} = 0, 0 \leq x < 2 π

Solve by substitution

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

Let:

cos (x^{2}) = u

\frac{2 u - 1}{u} = 0

\frac{2 u - 1}{u} = 0 : u = \frac{1}{2}

\frac{2 u - 1}{u} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2 u - 1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}

Substitute back

u = cos (x^{2})

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

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

cos (x^{2}) = \frac{1}{2}, 0 \leq x < 2 π : x = \frac{π}{3}, x = \frac{5 π}{3}, x = \frac{7 π}{3}, x = \frac{11 π}{3}, x = \frac{13 π}{3}, x = \frac{17 π}{3}, x = \frac{19 π}{3}, x = \frac{23 π}{3}, x = \frac{5 π}{3}, x = \frac{29 π}{3}

cos (x^{2}) = \frac{1}{2}, 0 \leq x < 2 π

General solutions for

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

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x^{2} = \frac{π}{3} + 2 πn, x^{2} = \frac{5 π}{3} + 2 πn

x^{2} = \frac{π}{3} + 2 πn, x^{2} = \frac{5 π}{3} + 2 πn

Solve

x^{2} = \frac{π}{3} + 2 πn : x = \frac{π + 6 πn}{3}, x = - \frac{π + 6 πn}{3}

x^{2} = \frac{π}{3} + 2 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{π}{3} + 2 πn, x = - \frac{π}{3} + 2 πn

Simplify

\frac{π}{3} + 2 πn : \frac{π + 6 πn}{3}

\frac{π}{3} + 2 πn

Join

\frac{π}{3} + 2 πn : \frac{π + 6 πn}{3}

\frac{π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 6 πn}{3}

= \frac{π + 6 πn}{3}

Simplify

- \frac{π}{3} + 2 πn : - \frac{π + 6 πn}{3}

- \frac{π}{3} + 2 πn

Join

\frac{π}{3} + 2 πn : \frac{π + 6 πn}{3}

\frac{π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 6 πn}{3}

= - \frac{π + 6 πn}{3}

x = \frac{π + 6 πn}{3}, x = - \frac{π + 6 πn}{3}

Solve

x^{2} = \frac{5 π}{3} + 2 πn : x = \frac{5 π + 6 πn}{3}, x = - \frac{5 π + 6 πn}{3}

x^{2} = \frac{5 π}{3} + 2 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{5 π}{3} + 2 πn, x = - \frac{5 π}{3} + 2 πn

Simplify

\frac{5 π}{3} + 2 πn : \frac{5 π + 6 πn}{3}

\frac{5 π}{3} + 2 πn

Join

\frac{5 π}{3} + 2 πn : \frac{5 π + 6 πn}{3}

\frac{5 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{5 π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{5 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 π + 6 πn}{3}

= \frac{5 π + 6 πn}{3}

Simplify

- \frac{5 π}{3} + 2 πn : - \frac{5 π + 6 πn}{3}

- \frac{5 π}{3} + 2 πn

Join

\frac{5 π}{3} + 2 πn : \frac{5 π + 6 πn}{3}

\frac{5 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{5 π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{5 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 π + 6 πn}{3}

= - \frac{5 π + 6 πn}{3}

x = \frac{5 π + 6 πn}{3}, x = - \frac{5 π + 6 πn}{3}

x = \frac{π + 6 πn}{3}, x = - \frac{π + 6 πn}{3}, x = \frac{5 π + 6 πn}{3}, x = - \frac{5 π + 6 πn}{3}

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{3}, x = \frac{5 π}{3}, x = \frac{7 π}{3}, x = \frac{11 π}{3}, x = \frac{13 π}{3}, x = \frac{17 π}{3}, x = \frac{19 π}{3}, x = \frac{23 π}{3}, x = \frac{5 π}{3}, x = \frac{29 π}{3}

Combine all the solutions

x = \frac{π}{3}, x = \frac{5 π}{3}, x = \frac{7 π}{3}, x = \frac{11 π}{3}, x = \frac{13 π}{3}, x = \frac{17 π}{3}, x = \frac{19 π}{3}, x = \frac{23 π}{3}, x = \frac{5 π}{3}, x = \frac{29 π}{3}

Find the undefined points:

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{5 π}{2}, x = \frac{7 π}{2}, x = \frac{3 π}{2}, x = \frac{11 π}{2}, x = \frac{13 π}{2}, x = \frac{15 π}{2}, x = \frac{17 π}{2}, x = \frac{19 π}{2}

Find the zeros of the denominator

cos (x^{2}) = 0

General solutions for

cos (x^{2}) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x^{2} = \frac{π}{2} + 2 πn, x^{2} = \frac{3 π}{2} + 2 πn

x^{2} = \frac{π}{2} + 2 πn, x^{2} = \frac{3 π}{2} + 2 πn

Solve

x^{2} = \frac{π}{2} + 2 πn : x = \frac{π + 4 πn}{2}, x = - \frac{π + 4 πn}{2}

x^{2} = \frac{π}{2} + 2 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{π}{2} + 2 πn, x = - \frac{π}{2} + 2 πn

Simplify

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Join

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{π + 4 πn}{2}

Simplify

- \frac{π}{2} + 2 πn : - \frac{π + 4 πn}{2}

- \frac{π}{2} + 2 πn

Join

\frac{π}{2} + 2 πn : \frac{π + 4 πn}{2}

\frac{π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= - \frac{π + 4 πn}{2}

x = \frac{π + 4 πn}{2}, x = - \frac{π + 4 πn}{2}

Solve

x^{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π + 4 πn}{2}, x = - \frac{3 π + 4 πn}{2}

x^{2} = \frac{3 π}{2} + 2 πn

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{3 π}{2} + 2 πn, x = - \frac{3 π}{2} + 2 πn

Simplify

\frac{3 π}{2} + 2 πn : \frac{3 π + 4 πn}{2}

\frac{3 π}{2} + 2 πn

Join

\frac{3 π}{2} + 2 πn : \frac{3 π + 4 πn}{2}

\frac{3 π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{3 π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{2}

= \frac{3 π + 4 πn}{2}

Simplify

- \frac{3 π}{2} + 2 πn : - \frac{3 π + 4 πn}{2}

- \frac{3 π}{2} + 2 πn

Join

\frac{3 π}{2} + 2 πn : \frac{3 π + 4 πn}{2}

\frac{3 π}{2} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 2}{2}

= \frac{3 π}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{3 π + 2 πn \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{2}

= - \frac{3 π + 4 πn}{2}

x = \frac{3 π + 4 πn}{2}, x = - \frac{3 π + 4 πn}{2}

x = \frac{π + 4 πn}{2}, x = - \frac{π + 4 πn}{2}, x = \frac{3 π + 4 πn}{2}, x = - \frac{3 π + 4 πn}{2}

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{5 π}{2}, x = \frac{7 π}{2}, x = \frac{3 π}{2}, x = \frac{11 π}{2}, x = \frac{13 π}{2}, x = \frac{15 π}{2}, x = \frac{17 π}{2}, x = \frac{19 π}{2}

\frac{π}{3}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{3}, \frac{7 π}{3}, \frac{5 π}{2}, \frac{7 π}{2}, \frac{11 π}{3}, \frac{13 π}{3}, \frac{3 π}{2}, \frac{11 π}{2}, \frac{17 π}{3}, \frac{19 π}{3}, \frac{13 π}{2}, \frac{15 π}{2}, \frac{23 π}{3}, \frac{5 π}{3}, \frac{17 π}{2}, \frac{19 π}{2}, \frac{29 π}{3}

Identify the intervals

0 < x < \frac{π}{3}, \frac{π}{3} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < \frac{5 π}{3}, \frac{5 π}{3} < x < \frac{7 π}{3}, \frac{7 π}{3} < x < \frac{5 π}{2}, \frac{5 π}{2} < x < \frac{7 π}{2}, \frac{7 π}{2} < x < \frac{11 π}{3}, \frac{11 π}{3} < x < \frac{13 π}{3}, \frac{13 π}{3} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < \frac{11 π}{2}, \frac{11 π}{2} < x < \frac{17 π}{3}, \frac{17 π}{3} < x < \frac{19 π}{3}, \frac{19 π}{3} < x < \frac{13 π}{2}, \frac{13 π}{2} < x < \frac{15 π}{2}, \frac{15 π}{2} < x < \frac{23 π}{3}, \frac{23 π}{3} < x < \frac{5 π}{3}, \frac{5 π}{3} < x < \frac{17 π}{2}, \frac{17 π}{2} < x < \frac{19 π}{2}, \frac{19 π}{2} < x < \frac{29 π}{3}, \frac{29 π}{3} < x < 2 π

Summarize in a table:

2 cos (x^{2}) - 1 cos (x^{2}) \frac{2 c o s ( x ^{2} ) - 1}{c o s ( x ^{2} )} x = 0 + + + 0 < x < \frac{π}{3} + + + x = \frac{π}{3} 0 + 0 \frac{π}{3} < x < \frac{π}{2} - + - x = \frac{π}{2} - 0 Undefined \frac{π}{2} < x < \frac{3 π}{2} - - + x = \frac{3 π}{2} - 0 Undefined \frac{3 π}{2} < x < \frac{5 π}{3} - + - x = \frac{5 π}{3} 0 + 0 \frac{5 π}{3} < x < \frac{7 π}{3} + + + x = \frac{7 π}{3} 0 + 0 \frac{7 π}{3} < x < \frac{5 π}{2} - + - x = \frac{5 π}{2} - 0 Undefined \frac{5 π}{2} < x < \frac{7 π}{2} - - + x = \frac{7 π}{2} - 0 Undefined \frac{7 π}{2} < x < \frac{11 π}{3} - + - x = \frac{11 π}{3} 0 + 0 \frac{11 π}{3} < x < \frac{13 π}{3} + + + x = \frac{13 π}{3} 0 + 0 \frac{13 π}{3} < x < \frac{3 π}{2} - + - x = \frac{3 π}{2} - 0 Undefined \frac{3 π}{2} < x < \frac{11 π}{2} - - + x = \frac{11 π}{2} - 0 Undefined \frac{11 π}{2} < x < \frac{17 π}{3} - + - x = \frac{17 π}{3} 0 + 0 \frac{17 π}{3} < x < \frac{19 π}{3} + + + x = \frac{19 π}{3} 0 + 0 \frac{19 π}{3} < x < \frac{13 π}{2} - + - x = \frac{13 π}{2} - 0 Undefined \frac{13 π}{2} < x < \frac{15 π}{2} - - + x = \frac{15 π}{2} - 0 Undefined \frac{15 π}{2} < x < \frac{23 π}{3} - + - x = \frac{23 π}{3} 0 + 0 \frac{23 π}{3} < x < \frac{5 π}{3} + + + x = \frac{5 π}{3} 0 + 0 \frac{5 π}{3} < x < \frac{17 π}{2} - + - x = \frac{17 π}{2} - 0 Undefined \frac{17 π}{2} < x < \frac{19 π}{2} - - + x = \frac{19 π}{2} - 0 Undefined \frac{19 π}{2} < x < \frac{29 π}{3} - + - x = \frac{29 π}{3} 0 + 0 \frac{29 π}{3} < x < 2 π - - + x = 2 π - - +

Identify the intervals that satisfy the required condition:

> 0

x = 0 or 0 < x < \frac{π}{3} or \frac{π}{2} < x < \frac{3 π}{2} or \frac{5 π}{3} < x < \frac{7 π}{3} or \frac{5 π}{2} < x < \frac{7 π}{2} or \frac{11 π}{3} < x < \frac{13 π}{3} or \frac{3 π}{2} < x < \frac{11 π}{2} or \frac{17 π}{3} < x < \frac{19 π}{3} or \frac{13 π}{2} < x < \frac{15 π}{2} or \frac{23 π}{3} < x < \frac{5 π}{3} or \frac{17 π}{2} < x < \frac{19 π}{2} or \frac{29 π}{3} < x < 2 π or x = 2 π

Merge Overlapping Intervals

0 \leq x < \frac{π}{3} or \frac{π}{2} < x < \frac{3 π}{2} or \frac{5 π}{3} < x < \frac{7 π}{3} or \frac{5 π}{2} < x < \frac{7 π}{2} or \frac{11 π}{3} < x < \frac{13 π}{3} or \frac{3 π}{2} < x < \frac{11 π}{2} or \frac{17 π}{3} < x < \frac{19 π}{3} or \frac{13 π}{2} < x < \frac{15 π}{2} or \frac{23 π}{3} < x < \frac{5 π}{3} or \frac{17 π}{2} < x < \frac{19 π}{2} or \frac{29 π}{3} < x < 2 π or x = 2 π