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Popular >

0<sin(2x)<2sqrt(2)

Solution

0 < sin (2 x) < 22

Solution

πn < x < \frac{π}{2} + πn

Interval Notation

(πn, \frac{π}{2} + πn)

Decimal Notation

πn < x < 1.57079 \dots + πn

Solution steps

0 < sin (2 x) < 22

a < u < b

then

a < u and u < b

0 < sin (2 x) and sin (2 x) < 22

0 < sin (2 x) : πn < x < \frac{π}{2} + πn

0 < sin (2 x)

Switch sides

sin (2 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 < 2 x < π - arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

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

arcsin (0) + 2 πn < 2 x : x > πn

arcsin (0) + 2 πn < 2 x

Switch sides

2 x > arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

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 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

2 x > 2 πn

Divide both sides by

2

2 x > 2 πn

Divide both sides by

2

\frac{2 x}{2} > \frac{2 πn}{2}

Simplify

x > πn

x > πn

2 x < π - arcsin (0) + 2 πn : x < \frac{π}{2} + πn

2 x < π - arcsin (0) + 2 πn

Simplify

π - arcsin (0) + 2 πn : π + 2 πn

π - arcsin (0) + 2 πn

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 + 2 πn

π - 0 + 2 πn = π + 2 πn

= π + 2 πn

2 x < π + 2 πn

Divide both sides by

2

2 x < π + 2 πn

Divide both sides by

2

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

Simplify

x < \frac{π}{2} + πn

x < \frac{π}{2} + πn

Combine the intervals

x > πn and x < \frac{π}{2} + πn

Merge Overlapping Intervals

πn < x < \frac{π}{2} + πn

sin (2 x) < 22 :

True for all

x \in R

sin (2 x) < 22

Range of

sin (2 x) : - 1 \leq sin (2 x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (2 x) \leq 1

- 1 \leq sin (2 x) \leq 1

sin (2 x) < 22 and - 1 \leq sin (2 x) \leq 1 : - 1 \leq sin (2 x) \leq 1

Let

y = sin (2 x)

Combine the intervals

y < 22 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < 22 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y < 22

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

πn < x < \frac{π}{2} + πn and True for all x \in R

Merge Overlapping Intervals

πn < x < \frac{π}{2} + πn

Popular Examples

sin(t)<0\land cos(t)>0

sin (t) < 0 and cos (t) > 0

0<sin(x)<= 1

0 < sin (x) \leq 1

cos^2(θ)0<θ<360

cos^{2} (θ) 0^{\circ} < θ < 36 0^{\circ}

derivative of (2sin(x-x)0)<= x<= 2pi

\frac{d}{d x} ((2 sin (x) - x) 0) \leq x \leq 2 π

sin(θ)cos(θ)=0.222\land tan(θ)<0

sin (θ) cos (θ) = 0.222 and tan (θ) < 0