en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

8+4sin(pi/6 (x))>= 8

Solution

8 + 4 sin (\frac{π}{6} (x)) \geq 8

Solution

12 n \leq x \leq 6 + 12 n

+1

Interval Notation

[12 n, 6 + 12 n]

Solution steps

8 + 4 sin (\frac{π}{6} x) \geq 8

Subtract

8

from both sides

8 + 4 sin (\frac{π}{6} x) - 8 \geq 8 - 8

Simplify

4 sin (\frac{π}{6} x) \geq 0

Divide both sides by

4

4 sin (\frac{π}{6} x) \geq 0

Divide both sides by

4

\frac{4 sin ( \frac{π}{6} x )}{4} \geq \frac{0}{4}

Simplify

sin (\frac{π}{6} x) \geq 0

sin (\frac{π}{6} 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 \frac{π}{6} 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 \frac{π}{6} x and \frac{π}{6} x \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq \frac{π}{6} x : x \geq 12 n

arcsin (0) + 2 πn \leq \frac{π}{6} x

Switch sides

\frac{π}{6} 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

\frac{π}{6} x \geq 2 πn

Multiply both sides by

6

\frac{π}{6} x \geq 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x \geq 6 \cdot 2 πn

Simplify

π x \geq 12 πn

π x \geq 12 πn

Divide both sides by

π

π x \geq 12 πn

Divide both sides by

π

\frac{π x}{π} \geq \frac{12 πn}{π}

Simplify

x \geq 12 n

x \geq 12 n

\frac{π}{6} x \leq π - arcsin (0) + 2 πn : x \leq 6 + 12 n

\frac{π}{6} 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

\frac{π}{6} x \leq π + 2 πn

Multiply both sides by

6

\frac{π}{6} x \leq π + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x \leq 6 π + 6 \cdot 2 πn

Simplify

π x \leq 6 π + 12 πn

π x \leq 6 π + 12 πn

Divide both sides by

π

π x \leq 6 π + 12 πn

Divide both sides by

π

\frac{π x}{π} \leq \frac{6 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} \leq \frac{6 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{6 π}{π} + \frac{12 πn}{π} : 6 + 12 n

\frac{6 π}{π} + \frac{12 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 6 + 12 n

x \leq 6 + 12 n

x \leq 6 + 12 n

x \leq 6 + 12 n

Combine the intervals

x \geq 12 n and x \leq 6 + 12 n

Merge Overlapping Intervals

12 n \leq x \leq 6 + 12 n

Popular Examples

cos(θ)>= (sqrt(3))/2 ,2pi>θ>= 0

sin^2(θ)<= cos^2(θ)

-2+3csc(2y-pi)>= 0

sin(x)<sin^2(x)