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Popular Trigonometry >

sin(x-45)> 1/2 sqrt(3)

Solution

sin (x - 4 5^{\circ}) > \frac{1}{2} 3

Solution

\frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn < x < \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

Interval Notation

(\frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn, \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn)

Decimal

1.83259 \dots + 2 πn < x < 2.87979 \dots + 2 πn

Solution steps

sin (x - 4 5^{\circ}) > \frac{1}{2} 3

Simplify

\frac{1}{2} 3 : \frac{3}{2}

\frac{1}{2} 3

Multiply fractions:

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

= \frac{1 \cdot 3}{2}

Multiply:

1 \cdot 3 = 3

= \frac{3}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{3}{2}) + 2 πn < (x - 4 5^{\circ}) < π - arcsin (\frac{3}{2}) + 2 πn

a < u < b

then

a < u and u < b

arcsin (\frac{3}{2}) + 2 πn < x - 4 5^{\circ} and x - 4 5^{\circ} < π - arcsin (\frac{3}{2}) + 2 πn

arcsin (\frac{3}{2}) + 2 πn < x - 4 5^{\circ} : x > \frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

arcsin (\frac{3}{2}) + 2 πn < x - 4 5^{\circ}

Switch sides

x - 4 5^{\circ} > arcsin (\frac{3}{2}) + 2 πn

Simplify

arcsin (\frac{3}{2}) + 2 πn : \frac{π}{3} + 2 πn

arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3} + 2 πn

x - 4 5^{\circ} > \frac{π}{3} + 2 πn

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} > \frac{π}{3} + 2 πn

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} > \frac{π}{3} + 2 πn + 4 5^{\circ}

Simplify

x > \frac{π}{3} + 2 πn + 4 5^{\circ}

x > \frac{π}{3} + 2 πn + 4 5^{\circ}

Simplify

\frac{π}{3} + 4 5^{\circ} : \frac{π + 3 \cdot 4 5 ^{\circ}}{3}

\frac{π}{3} + 4 5^{\circ}

Convert element to fraction:

4 5^{\circ} = \frac{4 5 ^{\circ} \cdot 3}{3}

= \frac{π}{3} + \frac{4 5 ^{\circ} \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 4 5 ^{\circ} \cdot 3}{3}

x > \frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

x - 4 5^{\circ} < π - arcsin (\frac{3}{2}) + 2 πn : x < \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

x - 4 5^{\circ} < π - arcsin (\frac{3}{2}) + 2 πn

Simplify

π - arcsin (\frac{3}{2}) + 2 πn : π - \frac{π}{3} + 2 πn

π - arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3} + 2 πn

x - 4 5^{\circ} < π - \frac{π}{3} + 2 πn

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} < π - \frac{π}{3} + 2 πn

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} < π - \frac{π}{3} + 2 πn + 4 5^{\circ}

Simplify

x < π - \frac{π}{3} + 2 πn + 4 5^{\circ}

x < π - \frac{π}{3} + 2 πn + 4 5^{\circ}

Simplify

π - \frac{π}{3} + 4 5^{\circ} : \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3}

π - \frac{π}{3} + 4 5^{\circ}

Convert element to fraction:

π = \frac{π 3}{3}, 4 5^{\circ} = \frac{4 5 ^{\circ} \cdot 3}{3}

= \frac{π 3}{3} - \frac{π}{3} + \frac{4 5 ^{\circ} \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π + 4 5 ^{\circ} \cdot 3}{3}

Add similar elements:

3 π - π = 2 π

= \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3}

x < \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

Combine the intervals

x > \frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn and x < \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

Merge Overlapping Intervals

\frac{π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn < x < \frac{2 π + 3 \cdot 4 5 ^{\circ}}{3} + 2 πn

Popular Examples

cot(x)>=-1

sec(x)>= 0

sin(x)<= cos(x)

sin(2x)<=-(sqrt(3))/2

sin(2x)> 1/2