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Popular Trigonometry >

sin(4x)>0.5

Solution

sin (4 x) > 0.5

Solution

\frac{π}{24} + \frac{π}{2} n < x < \frac{5 π}{24} + \frac{π}{2} n

Interval Notation

(\frac{π}{24} + \frac{π}{2} n, \frac{5 π}{24} + \frac{π}{2} n)

Decimal

0.13089 \dots + \frac{π}{2} n < x < 0.65449 \dots + \frac{π}{2} n

Solution steps

sin (4 x) > 0.5

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (0.5) + 2 πn < 4 x < π - arcsin (0.5) + 2 πn

a < u < b

then

a < u and u < b

arcsin (0.5) + 2 πn < 4 x and 4 x < π - arcsin (0.5) + 2 πn

arcsin (0.5) + 2 πn < 4 x : x > \frac{π}{24} + \frac{πn}{2}

arcsin (0.5) + 2 πn < 4 x

Switch sides

4 x > arcsin (0.5) + 2 πn

Simplify

arcsin (0.5) + 2 πn : \frac{π}{6} + 2 πn

arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{6}

arcsin (0.5)

= arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}{6}

= \frac{π}{6}

= \frac{π}{6} + 2 πn

4 x > \frac{π}{6} + 2 πn

Divide both sides by

4

4 x > \frac{π}{6} + 2 πn

Divide both sides by

4

\frac{4 x}{4} > \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} > \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{π}{6}}{4} + \frac{2 πn}{4} : \frac{π}{24} + \frac{πn}{2}

\frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{6}}{4} = \frac{π}{24}

\frac{\frac{π}{6}}{4}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{π}{24}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{24} + \frac{πn}{2}

x > \frac{π}{24} + \frac{πn}{2}

x > \frac{π}{24} + \frac{πn}{2}

x > \frac{π}{24} + \frac{πn}{2}

4 x < π - arcsin (0.5) + 2 πn : x < \frac{5 π}{24} + \frac{π}{2} n

4 x < π - arcsin (0.5) + 2 πn

Simplify

π - arcsin (0.5) + 2 πn : π - \frac{π}{6} + 2 πn

π - arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{6}

arcsin (0.5)

= arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}{6}

= \frac{π}{6}

= π - \frac{π}{6} + 2 πn

4 x < π - \frac{π}{6} + 2 πn

Divide both sides by

4

4 x < π - \frac{π}{6} + 2 πn

Divide both sides by

4

\frac{4 x}{4} < \frac{π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} < \frac{π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} : \frac{π}{4} - \frac{π}{24} + \frac{πn}{2}

\frac{π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{6}}{4} = \frac{π}{24}

\frac{\frac{π}{6}}{4}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{π}{24}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{4} - \frac{π}{24} + \frac{πn}{2}

x < \frac{π}{4} - \frac{π}{24} + \frac{πn}{2}

x < \frac{π}{4} - \frac{π}{24} + \frac{πn}{2}

Simplify

\frac{π}{4} - \frac{π}{24} : \frac{5 π}{24}

\frac{π}{4} - \frac{π}{24}

Least Common Multiplier of

4, 24 : 24

4, 24

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

24 : 2 \cdot 2 \cdot 2 \cdot 3

24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

24

= 2 \cdot 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 3 = 24

= 24

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

24

For

\frac{π}{4} :

multiply the denominator and numerator by

6

\frac{π}{4} = \frac{π 6}{4 \cdot 6} = \frac{π 6}{24}

= \frac{π 6}{24} - \frac{π}{24}

Since the denominators are equal, combine the fractions:

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

= \frac{π 6 - π}{24}

Add similar elements:

6 π - π = 5 π

= \frac{5 π}{24}

x < \frac{5 π}{24} + \frac{π}{2} n

x < \frac{5 π}{24} + \frac{π}{2} n

Combine the intervals

x > \frac{π}{24} + \frac{πn}{2} and x < \frac{5 π}{24} + \frac{π}{2} n

Merge Overlapping Intervals

\frac{π}{24} + \frac{π}{2} n < x < \frac{5 π}{24} + \frac{π}{2} n

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