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Popular >

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

Solution

0 < sin (x) cos (x) < 2

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 (x) cos (x) < 2

a < u < b

then

a < u and u < b

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

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

0 < sin (x) cos (x)

Switch sides

sin (x) cos (x) > 0

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

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

\frac{sin ( 2 x )}{2} > 0

Multiply both sides by

2

\frac{sin ( 2 x )}{2} > 0

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} > 0 \cdot 2

Simplify

sin (2 x) > 0

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 (x) cos (x) < 2 :

True for all

x \in R

sin (x) cos (x) < 2

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

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

\frac{sin ( 2 x )}{2} < 2

Multiply both sides by

2

\frac{sin ( 2 x )}{2} < 2

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} < 22

Simplify

sin (2 x) < 22

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

csc(θ)<0\land (csc(θ))(cot(θ))>0

csc (θ) < 0 and (csc (θ)) (cot (θ)) > 0

1>arctan(x)>0

1 > arctan (x) > 0

cosh(θ)= 8/3 \land θ<0,sinh(θ)

cosh (θ) = \frac{8}{3} and θ < 0, sinh (θ)

cos(θ)=(sqrt(3))/2 \land csc(θ)<0

cos (θ) = \frac{3}{2} and csc (θ) < 0

0<= y<= sin(3.1416)

0 \leq y \leq sin (3.1416)