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Popular >

cos^2(x)-cos(2x)>0

Solution

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

Solution

2 πn < x < π + 2 πn or - π + 2 πn < x < 2 πn

+2

Interval Notation

(2 πn, π + 2 πn) \cup (- π + 2 πn, 2 πn)

Decimal Notation

2 πn < x < 3.14159 \dots + 2 πn or - 3.14159 \dots + 2 πn < x < 2 πn

Solution steps

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

Use the following identity:

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

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

Simplify

sin^{2} (x) > 0

For

u^{n} > 0

, if

n

is even then

u < 0

or

u > 0

sin (x) < 0 or sin (x) > 0

sin (x) < 0 : - π + 2 πn < x < 2 πn

sin (x) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (0) + 2 πn < x < arcsin (0) + 2 πn

Simplify

- π - arcsin (0) : - π

- π - arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - 0

- π - 0 = - π

= - π

Simplify

arcsin (0) : 0

arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

- π + 2 πn < x < 0 + 2 πn

Simplify

- π + 2 πn < x < 2 πn

sin (x) > 0 : 2 πn < x < π + 2 πn

sin (x) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (0) + 2 πn < x < π - arcsin (0) + 2 πn

Simplify

arcsin (0) : 0

arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

Simplify

π - arcsin (0) : π

π - arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - 0

π - 0 = π

= π

0 + 2 πn < x < π + 2 πn

Simplify

2 πn < x < π + 2 πn

Combine the intervals

- π + 2 πn < x < 2 πn or 2 πn < x < π + 2 πn

Merge Overlapping Intervals

2 πn < x < π + 2 πn or - π + 2 πn < x < 2 πn

Popular Examples

solvefor n, pi/2-arctan(n)<= 0.001 solve for

n, \frac{π}{2} - arctan (n) \leq 0.001

sin(x)+cos^2(x)<1

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

-2cos(x+pi/6)>1

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tan(x)*sin(x)> 1/(2cos(x))

tan (x) \cdot sin (x) > \frac{1}{2 cos ( x )}

sec^2(θ)tan(θ)>= 2tan(θ)

sec^{2} (θ) tan (θ) \geq 2 tan (θ)