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Popular Trigonometry >

2sin(5x)<= sqrt(2)

Solution

2 sin (5 x) \leq 2

Solution

- \frac{π}{4} + \frac{2 π}{5} n \leq x \leq \frac{π}{20} + \frac{2 π}{5} n

Interval Notation

[- \frac{π}{4} + \frac{2 π}{5} n, \frac{π}{20} + \frac{2 π}{5} n]

Decimal

- 0.78539 \dots + \frac{2 π}{5} n \leq x \leq 0.15707 \dots + \frac{2 π}{5} n

Solution steps

2 sin (5 x) \leq 2

Divide both sides by

2

2 sin (5 x) \leq 2

Divide both sides by

2

\frac{2 sin ( 5 x )}{2} \leq \frac{2}{2}

Simplify

sin (5 x) \leq \frac{2}{2}

sin (5 x) \leq \frac{2}{2}

For

sin (x) \leq a

, if

- 1 < a < 1

then

- π - arcsin (a) + 2 πn \leq x \leq arcsin (a) + 2 πn

- π - arcsin (\frac{2}{2}) + 2 πn \leq 5 x \leq arcsin (\frac{2}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

- π - arcsin (\frac{2}{2}) + 2 πn \leq 5 x and 5 x \leq arcsin (\frac{2}{2}) + 2 πn

- π - arcsin (\frac{2}{2}) + 2 πn \leq 5 x : x \geq - \frac{π}{4} + \frac{2 π}{5} n

- π - arcsin (\frac{2}{2}) + 2 πn \leq 5 x

Switch sides

5 x \geq - 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

5 x \geq - π - \frac{π}{4} + 2 πn

Divide both sides by

5

5 x \geq - π - \frac{π}{4} + 2 πn

Divide both sides by

5

\frac{5 x}{5} \geq - \frac{π}{5} - \frac{\frac{π}{4}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} \geq - \frac{π}{5} - \frac{\frac{π}{4}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

- \frac{π}{5} - \frac{\frac{π}{4}}{5} + \frac{2 πn}{5} : - \frac{π}{5} - \frac{π}{20} + \frac{2 πn}{5}

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

\frac{\frac{π}{4}}{5} = \frac{π}{20}

\frac{\frac{π}{4}}{5}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 5}

Multiply the numbers:

4 \cdot 5 = 20

= \frac{π}{20}

= - \frac{π}{5} - \frac{π}{20} + \frac{2 πn}{5}

x \geq - \frac{π}{5} - \frac{π}{20} + \frac{2 πn}{5}

x \geq - \frac{π}{5} - \frac{π}{20} + \frac{2 πn}{5}

Simplify

- \frac{π}{5} - \frac{π}{20} : - \frac{π}{4}

- \frac{π}{5} - \frac{π}{20}

Least Common Multiplier of

5, 20 : 20

5, 20

Least Common Multiplier (LCM)

Prime factorization of

5 : 5

5

5

is a prime number, therefore no factorization is possible

= 5

Prime factorization of

20 : 2 \cdot 2 \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

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

5

20

= 5 \cdot 2 \cdot 2

Multiply the numbers:

5 \cdot 2 \cdot 2 = 20

= 20

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

20

For

\frac{π}{5} :

multiply the denominator and numerator by

4

\frac{π}{5} = \frac{π 4}{5 \cdot 4} = \frac{π 4}{20}

= - \frac{π 4}{20} - \frac{π}{20}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 - π}{20}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{20}

Apply the fraction rule:

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

= - \frac{5 π}{20}

Cancel the common factor:

5

= - \frac{π}{4}

x \geq - \frac{π}{4} + \frac{2 π}{5} n

x \geq - \frac{π}{4} + \frac{2 π}{5} n

5 x \leq arcsin (\frac{2}{2}) + 2 πn : x \leq \frac{π}{20} + \frac{2 πn}{5}

5 x \leq 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

5 x \leq \frac{π}{4} + 2 πn

Divide both sides by

5

5 x \leq \frac{π}{4} + 2 πn

Divide both sides by

5

\frac{5 x}{5} \leq \frac{\frac{π}{4}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} \leq \frac{\frac{π}{4}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{π}{4}}{5} + \frac{2 πn}{5} : \frac{π}{20} + \frac{2 πn}{5}

\frac{\frac{π}{4}}{5} + \frac{2 πn}{5}

\frac{\frac{π}{4}}{5} = \frac{π}{20}

\frac{\frac{π}{4}}{5}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 5}

Multiply the numbers:

4 \cdot 5 = 20

= \frac{π}{20}

= \frac{π}{20} + \frac{2 πn}{5}

x \leq \frac{π}{20} + \frac{2 πn}{5}

x \leq \frac{π}{20} + \frac{2 πn}{5}

x \leq \frac{π}{20} + \frac{2 πn}{5}

Combine the intervals

x \geq - \frac{π}{4} + \frac{2 π}{5} n and x \leq \frac{π}{20} + \frac{2 πn}{5}

Merge Overlapping Intervals

- \frac{π}{4} + \frac{2 π}{5} n \leq x \leq \frac{π}{20} + \frac{2 π}{5} n

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