English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec^3(x)+sec(x)tan^2(x)>0

Solution

sec^{3} (x) + sec (x) tan^{2} (x) > 0

Solution

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x \leq 2 π + 2 πn

Interval Notation

[2 πn, \frac{π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn]

Decimal Notation

2 πn \leq x < 1.57079 \dots + 2 πn or 4.71238 \dots + 2 πn < x \leq 6.28318 \dots + 2 πn

Solution steps

sec^{3} (x) + sec (x) tan^{2} (x) > 0

Periodicity of

sec^{3} (x) + sec (x) tan^{2} (x) : 2 π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

sec^{3} (x), sec (x) tan^{2} (x)

Periodicity of

sec^{3} (x) : 2 π

Periodicity of

sec^{n} (x) =

Periodicity of

sec (x),

if n is odd

Periodicity of

sec (x) : 2 π

Periodicity of

sec (x)

2 π

= 2 π

2 π

Periodicity of

sec (x) tan^{2} (x) : 2 π

sec (x) tan^{2} (x)

is composed of the following functions and periods:

sec (x)

with periodicity of

2 π

The compound periodicity is:

2 π

Combine periods:

2 π, 2 π

= 2 π

Express with sin, cos

sec^{3} (x) + sec (x) tan^{2} (x) > 0

Use the basic trigonometric identity:

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

(\frac{1}{cos ( x )})^{3} + \frac{1}{cos ( x )} tan^{2} (x) > 0

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

(\frac{1}{cos ( x )})^{3} + \frac{1}{cos ( x )} (\frac{sin ( x )}{cos ( x )})^{2} > 0

(\frac{1}{cos ( x )})^{3} + \frac{1}{cos ( x )} (\frac{sin ( x )}{cos ( x )})^{2} > 0

Simplify

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

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

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

(\frac{1}{cos ( x )})^{3}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{3} = 1

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

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

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

Multiply fractions:

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

= \frac{1 \cdot ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{cos ( x )}

Multiply:

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

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{cos ( x )}

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

= \frac{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{cos ( x )}

Apply the fraction rule:

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

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

cos^{2} (x) cos (x) = cos^{3} (x)

cos^{2} (x) cos (x)

Apply exponent rule:

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

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

= cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x)

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

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

Apply rule

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

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

\frac{1 + sin ^{2} ( x )}{cos ^{3} ( x )} > 0

Find the zeroes and undifined points of

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

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

\frac{1 + sin ^{2} ( x )}{cos ^{3} ( x )} = 0

\frac{1 + sin ^{2} ( x )}{cos ^{3} ( x )} = 0, 0 \leq x < 2 π :

No Solution for

x \in R

\frac{1 + sin ^{2} ( x )}{cos ^{3} ( x )} = 0, 0 \leq x < 2 π

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

1 + sin^{2} (x) = 0

Solve by substitution

1 + sin^{2} (x) = 0

Let:

sin (x) = u

1 + u^{2} = 0

1 + u^{2} = 0 : u = i, u = - i

1 + u^{2} = 0

Move

1

to the right side

1 + u^{2} = 0

Subtract

1

from both sides

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

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

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

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

Substitute back

u = sin (x)

sin (x) = i, sin (x) = - i

sin (x) = i, sin (x) = - i

sin (x) = i, 0 \leq x < 2 π :

No Solution

sin (x) = i, 0 \leq x < 2 π

Solutions for the range

0 \leq x < 2 π

No Solution

sin (x) = - i, 0 \leq x < 2 π :

No Solution

sin (x) = - i, 0 \leq x < 2 π

Solutions for the range

0 \leq x < 2 π

No Solution

Combine all the solutions

No Solution for x \in R

Find the undefined points:

x = \frac{π}{2}, x = \frac{3 π}{2}

Find the zeros of the denominator

cos^{3} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cos (x) = 0

General solutions for

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

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

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

\frac{π}{2}, \frac{3 π}{2}

Identify the intervals

0 < x < \frac{π}{2}, \frac{π}{2} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < 2 π

Summarize in a table:

1 + sin^{2} (x) cos^{3} (x) \frac{1 + s i n ^{2} ( x )}{c o s ^{3} ( x )} x = 0 + + + 0 < 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 < 2 π + + + x = 2 π + + +

Identify the intervals that satisfy the required condition:

> 0

x = 0 or 0 < x < \frac{π}{2} or \frac{3 π}{2} < x < 2 π or x = 2 π

Merge Overlapping Intervals

0 \leq x < \frac{π}{2} or \frac{3 π}{2} < x < 2 π or x = 2 π

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

x = 0

0 < x < \frac{π}{2}

0 \leq x < \frac{π}{2}

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

0 \leq x < \frac{π}{2}

\frac{3 π}{2} < x < 2 π

0 \leq x < \frac{π}{2} or \frac{3 π}{2} < x < 2 π

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

0 \leq x < \frac{π}{2} or \frac{3 π}{2} < x < 2 π

x = 2 π

0 \leq x < \frac{π}{2} or \frac{3 π}{2} < x \leq 2 π

0 \leq x < \frac{π}{2} or \frac{3 π}{2} < x \leq 2 π

Apply the periodicity of

sec^{3} (x) + sec (x) tan^{2} (x)

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x \leq 2 π + 2 πn

Popular Examples

12cos(2x)<0

12 cos (2 x) < 0

0.86<= cos^{2(9)}((68)/n)

0.86 \leq cos^{2 (9)} (\frac{68}{n})

cos(x)<= 1+sin(x)

cos (x) \leq 1 + sin (x)

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

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

cos(θ)<0,csc(θ)<0

cos (θ) < 0, csc (θ) < 0