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Popular >

sqrt(2)sin(θ)-1<0

Solution

2 sin (θ) - 1 < 0

Solution

- \frac{5 π}{4} + 2 πn < θ < \frac{π}{4} + 2 πn

Interval Notation

(- \frac{5 π}{4} + 2 πn, \frac{π}{4} + 2 πn)

Decimal Notation

- 3.92699 \dots + 2 πn < θ < 0.78539 \dots + 2 πn

Solution steps

2 sin (θ) - 1 < 0

Move

1

to the right side

2 sin (θ) - 1 < 0

Add

1

to both sides

2 sin (θ) - 1 + 1 < 0 + 1

Simplify

2 sin (θ) < 1

2 sin (θ) < 1

Divide both sides by

2

2 sin (θ) < 1

Divide both sides by

2

\frac{2 sin ( θ )}{2} < \frac{1}{2}

Simplify

\frac{2 sin ( θ )}{2} < \frac{1}{2}

Simplify

\frac{2 sin ( θ )}{2} : sin (θ)

\frac{2 sin ( θ )}{2}

Cancel the common factor:

2

= sin (θ)

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 (θ) < \frac{2}{2}

sin (θ) < \frac{2}{2}

sin (θ) < \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 < θ < arcsin (\frac{2}{2}) + 2 πn

Simplify

- π - arcsin (\frac{2}{2}) : - \frac{5 π}{4}

- π - arcsin (\frac{2}{2})

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}

Simplify

- π - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 - π}{4}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

Apply the fraction rule:

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

= - \frac{5 π}{4}

= - \frac{5 π}{4}

Simplify

arcsin (\frac{2}{2}) : \frac{π}{4}

arcsin (\frac{2}{2})

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}

- \frac{5 π}{4} + 2 πn < θ < \frac{π}{4} + 2 πn

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