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

Solution

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

Solution

πn \leq x \leq 0.61547 \dots + πn or - 0.61547 \dots + π + πn \leq x \leq π + πn

Interval Notation

[πn, 0.61547 \dots + πn] \cup [- 0.61547 \dots + π + πn, π + πn]

Decimal Notation

πn \leq x \leq 0.61547 \dots + πn or 2.52611 \dots + πn \leq x \leq 3.14159 \dots + πn

Solution steps

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

Move

sin^{2} (x)

to the left side

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

Subtract

sin^{2} (x)

from both sides

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

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

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

Use the following identity:

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

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

Simplify

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

Periodicity of

cos^{2} (x) - 2 sin^{2} (x) : π

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

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

Periodicity of

cos^{2} (x) : π

Periodicity of

cos^{n} (x) = \frac{Periodicity of cos ( x )}{2},

if n is even

Periodicity of

cos (x) : 2 π

Periodicity of

cos (x)

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Periodicity of

2 sin^{2} (x) : π

Periodicity of

sin^{n} (x) = \frac{Periodicity of sin ( x )}{2},

if n is even

Periodicity of

sin (x) : 2 π

Periodicity of

sin (x)

2 π

= 2 π

\frac{2 π}{2}

Simplify

π

Combine periods:

π, π

= π

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (cos (x) + 2 sin (x)) (cos (x) - 2 sin (x))

(cos (x) + 2 sin (x)) (cos (x) - 2 sin (x)) \geq - 2

To find the zeroes, set the inequality to zero

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

Solve

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

for

0 \leq x < π

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

Solving each part separately

cos (x) + 2 sin (x) = 0 : x = - 0.61547 \dots + π

cos (x) + 2 sin (x) = 0, 0 \leq x < π

Rewrite using trig identities

cos (x) + 2 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 + 2 tan (x) = 0

1 + 2 tan (x) = 0

Move

1

to the right side

1 + 2 tan (x) = 0

Subtract

1

from both sides

1 + 2 tan (x) - 1 = 0 - 1

Simplify

2 tan (x) = - 1

2 tan (x) = - 1

Divide both sides by

2

2 tan (x) = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

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

\frac{- 1}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

tan (x) = - \frac{2}{2}

tan (x) = - \frac{2}{2}

tan (x) = - \frac{2}{2}

Apply trig inverse properties

tan (x) = - \frac{2}{2}

General solutions for

tan (x) = - \frac{2}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{2}{2}) + πn

x = arctan (- \frac{2}{2}) + πn

Solutions for the range

0 \leq x < π

x = - arctan (\frac{2}{2}) + π

Show solutions in decimal form

x = - 0.61547 \dots + π

cos (x) - 2 sin (x) = 0 : x = 0.61547 \dots

cos (x) - 2 sin (x) = 0, 0 \leq x < π

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 - 2 tan (x) = 0

1 - 2 tan (x) = 0

Move

1

to the right side

1 - 2 tan (x) = 0

Subtract

1

from both sides

1 - 2 tan (x) - 1 = 0 - 1

Simplify

- 2 tan (x) = - 1

- 2 tan (x) = - 1

Divide both sides by

- 2

- 2 tan (x) = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 tan ( x )}{- 2} : tan (x)

\frac{- 2 tan ( x )}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{2 tan ( x )}{2}

Cancel the common factor:

2

= tan (x)

Simplify

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

\frac{- 1}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{2}

Rationalize

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

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

tan (x) = \frac{2}{2}

tan (x) = \frac{2}{2}

tan (x) = \frac{2}{2}

Apply trig inverse properties

tan (x) = \frac{2}{2}

General solutions for

tan (x) = \frac{2}{2}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{2}{2}) + πn

x = arctan (\frac{2}{2}) + πn

Solutions for the range

0 \leq x < π

x = arctan (\frac{2}{2})

Show solutions in decimal form

x = 0.61547 \dots

Combine all the solutions

0.61547 \dots or - 0.61547 \dots + π

The intervals between the zeros

0 < x < 0.61547 \dots, 0.61547 \dots < x < - 0.61547 \dots + π, - 0.61547 \dots + π < x < π

Summarize in a table:

cos (x) + 2 sin (x) cos (x) - 2 sin (x) (cos (x) + 2 sin (x)) (cos (x) - 2 sin (x)) x = 0 + + + 0 < x < 0.61547 \dots + + + x = 0.61547 \dots + 00 0.61547 \dots < x < - 0.61547 \dots + π + - - x = - 0.61547 \dots + π 0 - 0 - 0.61547 \dots + π < x < π - - + x = π - - +

Identify the intervals that satisfy the required condition:

\geq 0

x = 0 or 0 < x < 0.61547 \dots or x = 0.61547 \dots or x = - 0.61547 \dots + π or - 0.61547 \dots + π < x < π or x = π

Merge Overlapping Intervals

0 \leq x \leq 0.61547 \dots or - 0.61547 \dots + π \leq x < π or x = π