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Popular Trigonometry >

3sqrt(3)cos(x)-13/2 <-2

Solution

33 cos (x) - \frac{13}{2} < - 2

Solution

\frac{π}{6} + 2 πn < x < \frac{11 π}{6} + 2 πn

Interval Notation

(\frac{π}{6} + 2 πn, \frac{11 π}{6} + 2 πn)

Decimal

0.52359 \dots + 2 πn < x < 5.75958 \dots + 2 πn

Solution steps

33 cos (x) - \frac{13}{2} < - 2

Move

\frac{13}{2}

to the right side

33 cos (x) - \frac{13}{2} < - 2

Add

\frac{13}{2}

to both sides

33 cos (x) - \frac{13}{2} + \frac{13}{2} < - 2 + \frac{13}{2}

Simplify

33 cos (x) - \frac{13}{2} + \frac{13}{2} < - 2 + \frac{13}{2}

Simplify

33 cos (x) - \frac{13}{2} + \frac{13}{2} : 33 cos (x)

33 cos (x) - \frac{13}{2} + \frac{13}{2}

Add similar elements:

- \frac{13}{2} + \frac{13}{2} < 0

= 33 cos (x)

Simplify

- 2 + \frac{13}{2} : \frac{9}{2}

- 2 + \frac{13}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= - \frac{2 \cdot 2}{2} + \frac{13}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 \cdot 2 + 13}{2}

- 2 \cdot 2 + 13 = 9

- 2 \cdot 2 + 13

Multiply the numbers:

2 \cdot 2 = 4

= - 4 + 13

Add/Subtract the numbers:

- 4 + 13 = 9

= 9

= \frac{9}{2}

33 cos (x) < \frac{9}{2}

33 cos (x) < \frac{9}{2}

33 cos (x) < \frac{9}{2}

Divide both sides by

33

33 cos (x) < \frac{9}{2}

Divide both sides by

33

\frac{3 3 cos ( x )}{3 3} < \frac{\frac{9}{2}}{3 3}

Simplify

\frac{3 3 cos ( x )}{3 3} < \frac{\frac{9}{2}}{3 3}

Simplify

\frac{3 3 cos ( x )}{3 3} : cos (x)

\frac{3 3 cos ( x )}{3 3}

Divide the numbers:

\frac{3}{3} = 1

= \frac{3 cos ( x )}{3}

Cancel the common factor:

3

= cos (x)

Simplify

\frac{\frac{9}{2}}{3 3} : \frac{3}{2}

\frac{\frac{9}{2}}{3 3}

Apply the fraction rule:

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

= \frac{9}{2 \cdot 3 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{9}{6 3}

Cancel the common factor:

3

= \frac{3}{2 3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{2 \cdot 3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= \frac{3 ^{1 - \frac{1}{2}}}{2}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{3 ^{\frac{1}{2}}}{2}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{3}{2}

cos (x) < \frac{3}{2}

cos (x) < \frac{3}{2}

cos (x) < \frac{3}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

arccos (a) + 2 πn < x < 2 π - arccos (a) + 2 πn

arccos (\frac{3}{2}) + 2 πn < x < 2 π - arccos (\frac{3}{2}) + 2 πn

Simplify

arccos (\frac{3}{2}) : \frac{π}{6}

arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{6}

Simplify

2 π - arccos (\frac{3}{2}) : \frac{11 π}{6}

2 π - arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{6}

Simplify

2 π - \frac{π}{6}

Convert element to fraction:

2 π = \frac{2 π 6}{6}

= \frac{2 π 6}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 6 - π}{6}

2 π 6 - π = 11 π

2 π 6 - π

Multiply the numbers:

2 \cdot 6 = 12

= 12 π - π

Add similar elements:

12 π - π = 11 π

= 11 π

= \frac{11 π}{6}

= \frac{11 π}{6}

\frac{π}{6} + 2 πn < x < \frac{11 π}{6} + 2 πn

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