English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

(4cos^2(x)-3)(1-tan^2(x))<= 0

Solution

(4 cos^{2} (x) - 3) (1 - tan^{2} (x)) \leq 0

Solution

\frac{π}{6} + πn \leq x \leq \frac{π}{4} + πn or \frac{3 π}{4} + πn \leq x \leq \frac{5 π}{6} + πn

Interval Notation

[\frac{π}{6} + πn, \frac{π}{4} + πn] \cup [\frac{3 π}{4} + πn, \frac{5 π}{6} + πn]

Decimal

0.52359 \dots + πn \leq x \leq 0.78539 \dots + πn or 2.35619 \dots + πn \leq x \leq 2.61799 \dots + πn

Solution steps

(4 cos^{2} (x) - 3) (1 - tan^{2} (x)) \leq 0

Use the following identity:

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

Therefore

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

(4 (1 - sin^{2} (x)) - 3) (1 - tan^{2} (x)) \leq 0

Simplify

(4 (1 - sin^{2} (x)) - 3) (1 - tan^{2} (x)) : (- 4 sin^{2} (x) + 1) (1 - tan^{2} (x))

(4 (1 - sin^{2} (x)) - 3) (1 - tan^{2} (x))

Expand

4 (1 - sin^{2} (x)) - 3 : - 4 sin^{2} (x) + 1

4 (1 - sin^{2} (x)) - 3

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= 4 - 4 sin^{2} (x) - 3

Simplify

4 - 4 sin^{2} (x) - 3 : - 4 sin^{2} (x) + 1

4 - 4 sin^{2} (x) - 3

Group like terms

= - 4 sin^{2} (x) + 4 - 3

Add/Subtract the numbers:

4 - 3 = 1

= - 4 sin^{2} (x) + 1

= - 4 sin^{2} (x) + 1

= (- 4 sin^{2} (x) + 1) (- tan^{2} (x) + 1)

= (- 4 sin^{2} (x) + 1) (1 - tan^{2} (x))

(- 4 sin^{2} (x) + 1) (1 - tan^{2} (x)) \leq 0

Periodicity of

(- 4 sin^{2} (x) + 1) (1 - tan^{2} (x)) : π

(- 4 sin^{2} (x) + 1) (1 - tan^{2} (x))

is composed of the following functions and periods:

sin (x)

with periodicity of

2 π

The compound periodicity is:

= π

Express with sin, cos

(- 4 sin^{2} (x) + 1) (1 - tan^{2} (x)) \leq 0

Use the basic trigonometric identity:

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

(- 4 sin^{2} (x) + 1) (1 - (\frac{sin ( x )}{cos ( x )})^{2}) \leq 0

(- 4 sin^{2} (x) + 1) (1 - (\frac{sin ( x )}{cos ( x )})^{2}) \leq 0

Simplify

(- 4 sin^{2} (x) + 1) (1 - (\frac{sin ( x )}{cos ( x )})^{2}) : \frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )}

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

Apply exponent rule:

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

= (- 4 sin^{2} (x) + 1) (- \frac{sin ^{2} ( x )}{cos ^{2} ( x )} + 1)

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

= \frac{cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )} (- 4 sin^{2} (x) + 1)

Multiply fractions:

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

= \frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )}

\frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )} \leq 0

Find the zeroes and undifined points of

\frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )}

for

0 \leq x < π

To find the zeroes, set the inequality to zero

\frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )} = 0

\frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )} = 0, 0 \leq x < π : x = \frac{π}{4}, x = \frac{3 π}{4}, x = \frac{π}{6}, x = \frac{5 π}{6}

\frac{( cos ^{2} ( x ) - sin ^{2} ( x ) ) ( - 4 sin ^{2} ( x ) + 1 )}{cos ^{2} ( x )} = 0, 0 \leq x < π

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

(cos^{2} (x) - sin^{2} (x)) (- 4 sin^{2} (x) + 1) = 0

Solving each part separately

cos^{2} (x) - sin^{2} (x) = 0 or - 4 sin^{2} (x) + 1 = 0

cos^{2} (x) - sin^{2} (x) = 0, 0 \leq x < π : x = \frac{π}{4}, x = \frac{3 π}{4}

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

Rewrite using trig identities

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

Use the Double Angle identity:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

= cos (2 x)

cos (2 x) = 0

General solutions for

cos (2 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}

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

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

Solve

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

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Solve

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

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

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

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

Solutions for the range

0 \leq x < π

x = \frac{π}{4}, x = \frac{3 π}{4}

- 4 sin^{2} (x) + 1 = 0, 0 \leq x < π : x = \frac{π}{6}, x = \frac{5 π}{6}

- 4 sin^{2} (x) + 1 = 0, 0 \leq x < π

Solve by substitution

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

Let:

sin (x) = u

- 4 u^{2} + 1 = 0

- 4 u^{2} + 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 4 u^{2} + 1 = 0

Move

1

to the right side

- 4 u^{2} + 1 = 0

Subtract

1

from both sides

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

Simplify

- 4 u^{2} = - 1

- 4 u^{2} = - 1

Divide both sides by

- 4

- 4 u^{2} = - 1

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 1}{- 4}

Simplify

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

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

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, 0 \leq x < π : x = \frac{π}{6}, x = \frac{5 π}{6}

sin (x) = \frac{1}{2}, 0 \leq x < π

General solutions for

sin (x) = \frac{1}{2}

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solutions for the range

0 \leq x < π

x = \frac{π}{6}, x = \frac{5 π}{6}

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

No Solution

sin (x) = - \frac{1}{2}, 0 \leq x < π

General solutions for

sin (x) = - \frac{1}{2}

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

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solutions for the range

0 \leq x < π

No Solution

Combine all the solutions

x = \frac{π}{6}, x = \frac{5 π}{6}

Combine all the solutions

x = \frac{π}{4}, x = \frac{3 π}{4}, x = \frac{π}{6}, x = \frac{5 π}{6}

Find the undefined points:

x = \frac{π}{2}

Find the zeros of the denominator

cos^{2} (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 < π

x = \frac{π}{2}

\frac{π}{6}, \frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}, \frac{5 π}{6}

Identify the intervals

0 < x < \frac{π}{6}, \frac{π}{6} < x < \frac{π}{4}, \frac{π}{4} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{3 π}{4}, \frac{3 π}{4} < x < \frac{5 π}{6}, \frac{5 π}{6} < x < π

Summarize in a table:

cos^{2} (x) - sin^{2} (x) - 4 sin^{2} (x) + 1 cos^{2} (x) \frac{( c o s ^{2} ( x ) - s i n ^{2} ( x ) ) ( - 4 s i n ^{2} ( x ) + 1 )}{c o s ^{2} ( x )} x = 0 + + + + 0 < x < \frac{π}{6} + + + + x = \frac{π}{6} + 0 + 0 \frac{π}{6} < x < \frac{π}{4} + - + - x = \frac{π}{4} 0 - + 0 \frac{π}{4} < x < \frac{π}{2} - - + + x = \frac{π}{2} - - 0 Undefined \frac{π}{2} < x < \frac{3 π}{4} - - + + x = \frac{3 π}{4} 0 - + 0 \frac{3 π}{4} < x < \frac{5 π}{6} + - + - x = \frac{5 π}{6} + 0 + 0 \frac{5 π}{6} < x < π + + + + x = π + + + +

Identify the intervals that satisfy the required condition:

\leq 0

x = \frac{π}{6} or \frac{π}{6} < x < \frac{π}{4} or x = \frac{π}{4} or x = \frac{3 π}{4} or \frac{3 π}{4} < x < \frac{5 π}{6} or x = \frac{5 π}{6}

Merge Overlapping Intervals

\frac{π}{6} \leq x \leq \frac{π}{4} or \frac{3 π}{4} \leq x < \frac{5 π}{6} or x = \frac{5 π}{6}