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Popular >

1-2cos(2x)>sin^2(x)

Solution

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

Solution

arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn or - π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

Interval Notation

(arcsin (\frac{1}{3}) + 2 πn, π - arcsin (\frac{1}{3}) + 2 πn) \cup (- π + arcsin (\frac{1}{3}) + 2 πn, - arcsin (\frac{1}{3}) + 2 πn)

Decimal Notation

0.61547 \dots + 2 πn < x < 2.52611 \dots + 2 πn or - 2.52611 \dots + 2 πn < x < - 0.61547 \dots + 2 πn

Solution steps

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

Move

sin^{2} (x)

to the left side

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

Subtract

sin^{2} (x)

from both sides

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

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

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

Use the following identity:

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

1 - sin^{2} (x) - 2 (1 - 2 sin^{2} (x)) > 0

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Add/Subtract the numbers:

1 - 2 = - 1

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

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

3 sin^{2} (x) - 1 > 0

Move

1

to the right side

3 sin^{2} (x) - 1 > 0

Add

1

to both sides

3 sin^{2} (x) - 1 + 1 > 0 + 1

Simplify

3 sin^{2} (x) > 1

3 sin^{2} (x) > 1

Divide both sides by

3

3 sin^{2} (x) > 1

Divide both sides by

3

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

Simplify

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

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

For

u^{n} > a

, if

n

is even then

sin (x) < - \frac{1}{3} or sin (x) > \frac{1}{3}

sin (x) < - \frac{1}{3} : - π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

sin (x) < - \frac{1}{3}

For

sin (x) < a

, if

- 1 < a \leq 1

then

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (- \frac{1}{3}) + 2 πn < x < arcsin (- \frac{1}{3}) + 2 πn

Simplify

- π - arcsin (- \frac{1}{3}) : - π + arcsin (\frac{1}{3})

- π - arcsin (- \frac{1}{3})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - π - (- arcsin (\frac{1}{3}))

Apply rule

- (- a) = a

= - π + arcsin (\frac{1}{3})

Simplify

arcsin (- \frac{1}{3}) : - arcsin (\frac{1}{3})

arcsin (- \frac{1}{3})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3})

- π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

sin (x) > \frac{1}{3} : arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn

sin (x) > \frac{1}{3}

For

sin (x) > a

, if

- 1 \leq a < 1

then

arcsin (a) + 2 πn < x < π - arcsin (a) + 2 πn

arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn

Combine the intervals

- π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn or arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn

Merge Overlapping Intervals

arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn or - π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

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