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Popular Trigonometry >

arcsin(3pix+2)>= 0

Solution

arcsin (3 π x + 2) \geq 0

Solution

- \frac{2}{3 π} \leq x \leq - \frac{1}{3 π}

Interval Notation

[- \frac{2}{3 π}, - \frac{1}{3 π}]

Decimal

- 0.21220 \dots \leq x \leq - 0.10610 \dots

Solution steps

arcsin (3 π x + 2) \geq 0

arcsin (x) \geq a

then

x \geq sin (a)

3 π x + 2 \geq sin (0)

sin (0) = 0

sin (0)

Use the following trivial identity:

sin (0) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

3 π x + 2 \geq 0

3 π x + 2 \geq 0 : x \geq - \frac{2}{3 π}

3 π x + 2 \geq 0

Move

2

to the right side

3 π x + 2 \geq 0

Subtract

2

from both sides

3 π x + 2 - 2 \geq 0 - 2

Simplify

3 π x \geq - 2

3 π x \geq - 2

Divide both sides by

3 π

3 π x \geq - 2

Divide both sides by

3 π

\frac{3 π x}{3 π} \geq \frac{- 2}{3 π}

Simplify

x \geq - \frac{2}{3 π}

x \geq - \frac{2}{3 π}

x \geq - \frac{2}{3 π}

Domain of

arcsin (3 π x + 2) : - \frac{1}{π} \leq x \leq - \frac{1}{3 π}

Domain definition

Find known functions domain restrictions:

- \frac{1}{π} \leq x \leq - \frac{1}{3 π}

arcsin (f (x)) \Rightarrow - 1 \leq f (x) \leq 1

Solve

- 1 \leq (3 π x + 2) \leq 1 : - \frac{1}{π} \leq x \leq - \frac{1}{3 π}

- 1 \leq (3 π x + 2) \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- 1 \leq (3 π x + 2) and (3 π x + 2) \leq 1

- 1 \leq 3 π x + 2 : x \geq - \frac{1}{π}

- 1 \leq 3 π x + 2

Switch sides

3 π x + 2 \geq - 1

Move

2

to the right side

3 π x + 2 \geq - 1

Subtract

2

from both sides

3 π x + 2 - 2 \geq - 1 - 2

Simplify

3 π x \geq - 3

3 π x \geq - 3

Divide both sides by

3 π

3 π x \geq - 3

Divide both sides by

3 π

\frac{3 π x}{3 π} \geq \frac{- 3}{3 π}

Simplify

\frac{3 π x}{3 π} \geq \frac{- 3}{3 π}

Simplify

\frac{3 π x}{3 π} : x

\frac{3 π x}{3 π}

Divide the numbers:

\frac{3}{3} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{- 3}{3 π} : - \frac{1}{π}

\frac{- 3}{3 π}

Apply the fraction rule:

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

= - \frac{3}{3 π}

Divide the numbers:

\frac{3}{3} = 1

= - \frac{1}{π}

x \geq - \frac{1}{π}

x \geq - \frac{1}{π}

x \geq - \frac{1}{π}

3 π x + 2 \leq 1 : x \leq - \frac{1}{3 π}

3 π x + 2 \leq 1

Move

2

to the right side

3 π x + 2 \leq 1

Subtract

2

from both sides

3 π x + 2 - 2 \leq 1 - 2

Simplify

3 π x \leq - 1

3 π x \leq - 1

Divide both sides by

3 π

3 π x \leq - 1

Divide both sides by

3 π

\frac{3 π x}{3 π} \leq \frac{- 1}{3 π}

Simplify

x \leq - \frac{1}{3 π}

x \leq - \frac{1}{3 π}

Combine the intervals

x \geq - \frac{1}{π} and x \leq - \frac{1}{3 π}

Merge Overlapping Intervals

x \geq - \frac{1}{π} and x \leq - \frac{1}{3 π}

The intersection of two intervals is the set of numbers which are in both intervals

x \geq - \frac{1}{π}

and

x \leq - \frac{1}{3 π}

- \frac{1}{π} \leq x \leq - \frac{1}{3 π}

- \frac{1}{π} \leq x \leq - \frac{1}{3 π}

The function domain

- \frac{1}{π} \leq x \leq - \frac{1}{3 π}

Combine the intervals

x \geq - \frac{2}{3 π} and - \frac{1}{π} \leq x \leq - \frac{1}{3 π}

Merge Overlapping Intervals

x \geq - \frac{2}{3 π} and - \frac{1}{π} \leq x \leq - \frac{1}{3 π}

The intersection of two intervals is the set of numbers which are in both intervals

x \geq - \frac{2}{3 π}

and

- \frac{1}{π} \leq x \leq - \frac{1}{3 π}

- \frac{2}{3 π} \leq x \leq - \frac{1}{3 π}

- \frac{2}{3 π} \leq x \leq - \frac{1}{3 π}

Graph

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