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Popular Trigonometry >

sin(x/2)>= 0

Solution

sin (\frac{x}{2}) \geq 0

Solution

4 πn \leq x \leq 2 π + 4 πn

+2

Interval Notation

[4 πn, 2 π + 4 πn]

Decimal

4 πn \leq x \leq 6.28318 \dots + 4 πn

Solution steps

sin (\frac{x}{2}) \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 \frac{x}{2} \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 \frac{x}{2} and \frac{x}{2} \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq \frac{x}{2} : x \geq 4 πn

arcsin (0) + 2 πn \leq \frac{x}{2}

Switch sides

\frac{x}{2} \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

\frac{x}{2} \geq 2 πn

Multiply both sides by

2

\frac{x}{2} \geq 2 πn

Multiply both sides by

2

\frac{2 x}{2} \geq 2 \cdot 2 πn

Simplify

x \geq 4 πn

x \geq 4 πn

\frac{x}{2} \leq π - arcsin (0) + 2 πn : x \leq 2 π + 4 πn

\frac{x}{2} \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

\frac{x}{2} \leq π + 2 πn

Multiply both sides by

2

\frac{x}{2} \leq π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} \leq 2 π + 2 \cdot 2 πn

Simplify

x \leq 2 π + 4 πn

x \leq 2 π + 4 πn

Combine the intervals

x \geq 4 πn and x \leq 2 π + 4 πn

Merge Overlapping Intervals

4 πn \leq x \leq 2 π + 4 πn

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