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Popular >

sin(pix)>= 0

Solution

sin (π x) \geq 0

Solution

2 n \leq x \leq 1 + 2 n

+1

Interval Notation

[2 n, 1 + 2 n]

Solution steps

sin (π x) \geq 0

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (0) + 2 πn \leq π x \leq π - arcsin (0) + 2 πn

If

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (0) + 2 πn \leq π x and π x \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq π x : x \geq 2 n

arcsin (0) + 2 πn \leq π x

Switch sides

π x \geq arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

π x \geq 2 πn

Divide both sides by

π

π x \geq 2 πn

Divide both sides by

π

\frac{π x}{π} \geq \frac{2 πn}{π}

Simplify

x \geq 2 n

x \geq 2 n

π x \leq π - arcsin (0) + 2 πn : x \leq 1 + 2 n

π x \leq π - arcsin (0) + 2 πn

Simplify

π - arcsin (0) + 2 πn : π + 2 πn

π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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 - 0 + 2 πn

π - 0 + 2 πn = π + 2 πn

= π + 2 πn

π x \leq π + 2 πn

Divide both sides by

π

π x \leq π + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{π} + \frac{2 πn}{π} : 1 + 2 n

\frac{π}{π} + \frac{2 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{2 πn}{π}

Cancel

\frac{2 πn}{π} : 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 1 + 2 n

x \leq 1 + 2 n

x \leq 1 + 2 n

x \leq 1 + 2 n

Combine the intervals

x \geq 2 n and x \leq 1 + 2 n

Merge Overlapping Intervals

2 n \leq x \leq 1 + 2 n

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