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Popular >

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

Solution

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

Solution

- 3 π + 4 πn < x < 4 πn

Interval Notation

(- 3 π + 4 πn, 4 πn)

Decimal Notation

- 9.42477 \dots + 4 πn < x < 4 πn

Solution steps

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

Use the following identity:

cos (x) + sin (x) = 2 sin (\frac{π}{4} + x)

2 sin (\frac{π}{4} + \frac{x}{2}) < 1

Divide both sides by

2

2 sin (\frac{π}{4} + \frac{x}{2}) < 1

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} + \frac{x}{2} )}{2} < \frac{1}{2}

Simplify

\frac{2 sin ( \frac{π}{4} + \frac{x}{2} )}{2} < \frac{1}{2}

Simplify

\frac{2 sin ( \frac{π}{4} + \frac{x}{2} )}{2} : sin (\frac{π}{4} + \frac{x}{2})

\frac{2 sin ( \frac{π}{4} + \frac{x}{2} )}{2}

Cancel the common factor:

2

= sin (\frac{π}{4} + \frac{x}{2})

Simplify

\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}

sin (\frac{π}{4} + \frac{x}{2}) < \frac{2}{2}

sin (\frac{π}{4} + \frac{x}{2}) < \frac{2}{2}

sin (\frac{π}{4} + \frac{x}{2}) < \frac{2}{2}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (\frac{2}{2}) + 2 πn < (\frac{π}{4} + \frac{x}{2}) < arcsin (\frac{2}{2}) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + \frac{x}{2} and \frac{π}{4} + \frac{x}{2} < arcsin (\frac{2}{2}) + 2 πn

- π - arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + \frac{x}{2} : x > - 3 π + 4 πn

- π - arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + \frac{x}{2}

Switch sides

\frac{π}{4} + \frac{x}{2} > - π - arcsin (\frac{2}{2}) + 2 πn

Simplify

- π - arcsin (\frac{2}{2}) + 2 πn : - π - \frac{π}{4} + 2 πn

- π - arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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 - \frac{π}{4} + 2 πn

\frac{π}{4} + \frac{x}{2} > - π - \frac{π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + \frac{x}{2} > - π - \frac{π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} > - π - \frac{π}{4} + 2 πn - \frac{π}{4}

Simplify

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} > - π - \frac{π}{4} + 2 πn - \frac{π}{4}

Simplify

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} : \frac{x}{2}

\frac{π}{4} + \frac{x}{2} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} > 0

= \frac{x}{2}

Simplify

- π - \frac{π}{4} + 2 πn - \frac{π}{4} : - \frac{π}{2} - π + 2 πn

- π - \frac{π}{4} + 2 πn - \frac{π}{4}

Group like terms

= - \frac{π}{4} - \frac{π}{4} - π + 2 πn

Combine the fractions

- \frac{π}{4} - \frac{π}{4} : - \frac{π}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π - π}{4}

Add similar elements:

- π - π = - 2 π

= \frac{- 2 π}{4}

Apply the fraction rule:

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

= - \frac{2 π}{4}

Cancel the common factor:

2

= - \frac{π}{2}

= - \frac{π}{2} - π + 2 πn

\frac{x}{2} > - \frac{π}{2} - π + 2 πn

\frac{x}{2} > - \frac{π}{2} - π + 2 πn

\frac{x}{2} > - \frac{π}{2} - π + 2 πn

Multiply both sides by

2

\frac{x}{2} > - \frac{π}{2} - π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} > - 2 \cdot \frac{π}{2} - 2 π + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} > - 2 \cdot \frac{π}{2} - 2 π + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- 2 \cdot \frac{π}{2} - 2 π + 2 \cdot 2 πn : - 3 π + 4 πn

- 2 \cdot \frac{π}{2} - 2 π + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= - π - 2 π + 4 πn

Add similar elements:

- π - 2 π = - 3 π

= - 3 π + 4 πn

x > - 3 π + 4 πn

x > - 3 π + 4 πn

x > - 3 π + 4 πn

\frac{π}{4} + \frac{x}{2} < arcsin (\frac{2}{2}) + 2 πn : x < 4 πn

\frac{π}{4} + \frac{x}{2} < arcsin (\frac{2}{2}) + 2 πn

Simplify

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}

= \frac{π}{4} + 2 πn

\frac{π}{4} + \frac{x}{2} < \frac{π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + \frac{x}{2} < \frac{π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} < \frac{π}{4} + 2 πn - \frac{π}{4}

Simplify

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} < \frac{π}{4} + 2 πn - \frac{π}{4}

Simplify

\frac{π}{4} + \frac{x}{2} - \frac{π}{4} : \frac{x}{2}

\frac{π}{4} + \frac{x}{2} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} < 0

= \frac{x}{2}

Simplify

\frac{π}{4} + 2 πn - \frac{π}{4} : 2 πn

\frac{π}{4} + 2 πn - \frac{π}{4}

Group like terms

= \frac{π}{4} - \frac{π}{4} + 2 πn

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 2 πn

\frac{x}{2} < 2 πn

\frac{x}{2} < 2 πn

\frac{x}{2} < 2 πn

Multiply both sides by

2

\frac{x}{2} < 2 πn

Multiply both sides by

2

\frac{2 x}{2} < 2 \cdot 2 πn

Simplify

x < 4 πn

x < 4 πn

Combine the intervals

x > - 3 π + 4 πn and x < 4 πn

Merge Overlapping Intervals

- 3 π + 4 πn < x < 4 πn

Popular Examples

2cos^2(x)+sin(2x)>= 1

2 cos^{2} (x) + sin (2 x) \geq 1

sin(t)>0

sin (t) > 0

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

sin (x) - cos (x) \geq 0

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

2 sin (x) - 2 < 0

0<2cos(2x)-sqrt(3)

0 < 2 cos (2 x) - 3