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Popular Trigonometry >

4cos(2x-30)>0

Solution

4 cos (2 x - 30) > 0

Solution

\frac{60 - π}{4} + πn < x < \frac{60 + π}{4} + πn

Interval Notation

(\frac{60 - π}{4} + πn, \frac{60 + π}{4} + πn)

Decimal

14.21460 \dots + πn < x < 15.78539 \dots + πn

Solution steps

4 cos (2 x - 30) > 0

Divide both sides by

4

4 cos (2 x - 30) > 0

Divide both sides by

4

\frac{4 cos ( 2 x - 30 )}{4} > \frac{0}{4}

Simplify

cos (2 x - 30) > 0

cos (2 x - 30) > 0

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (0) + 2 πn < (2 x - 30) < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < 2 x - 30 and 2 x - 30 < arccos (0) + 2 πn

- arccos (0) + 2 πn < 2 x - 30 : x > \frac{60 - π}{4} + πn

- arccos (0) + 2 πn < 2 x - 30

Switch sides

2 x - 30 > - arccos (0) + 2 πn

Simplify

- arccos (0) + 2 πn : - \frac{π}{2} + 2 πn

- arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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}{2} + 2 πn

2 x - 30 > - \frac{π}{2} + 2 πn

Move

30

to the right side

2 x - 30 > - \frac{π}{2} + 2 πn

Add

30

to both sides

2 x - 30 + 30 > - \frac{π}{2} + 2 πn + 30

Simplify

2 x > - \frac{π}{2} + 2 πn + 30

2 x > - \frac{π}{2} + 2 πn + 30

Divide both sides by

2

2 x > - \frac{π}{2} + 2 πn + 30

Divide both sides by

2

\frac{2 x}{2} > - \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} > - \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2} : 15 + πn - \frac{π}{4}

- \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Group like terms

= \frac{30}{2} + \frac{2 πn}{2} - \frac{\frac{π}{2}}{2}

\frac{30}{2} = 15

\frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= 15

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= 15 + πn - \frac{π}{4}

x > 15 + πn - \frac{π}{4}

x > 15 + πn - \frac{π}{4}

Simplify

15 - \frac{π}{4} : \frac{60 - π}{4}

15 - \frac{π}{4}

Convert element to fraction:

15 = \frac{15 \cdot 4}{4}

= \frac{15 \cdot 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{15 \cdot 4 - π}{4}

Multiply the numbers:

15 \cdot 4 = 60

= \frac{60 - π}{4}

x > \frac{60 - π}{4} + πn

x > \frac{60 - π}{4} + πn

2 x - 30 < arccos (0) + 2 πn : x < \frac{60 + π}{4} + πn

2 x - 30 < arccos (0) + 2 πn

Simplify

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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}{2} + 2 πn

2 x - 30 < \frac{π}{2} + 2 πn

Move

30

to the right side

2 x - 30 < \frac{π}{2} + 2 πn

Add

30

to both sides

2 x - 30 + 30 < \frac{π}{2} + 2 πn + 30

Simplify

2 x < \frac{π}{2} + 2 πn + 30

2 x < \frac{π}{2} + 2 πn + 30

Divide both sides by

2

2 x < \frac{π}{2} + 2 πn + 30

Divide both sides by

2

\frac{2 x}{2} < \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} < \frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2} : 15 + πn + \frac{π}{4}

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} + \frac{30}{2}

Group like terms

= \frac{30}{2} + \frac{2 πn}{2} + \frac{\frac{π}{2}}{2}

\frac{30}{2} = 15

\frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= 15

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= 15 + πn + \frac{π}{4}

x < 15 + πn + \frac{π}{4}

x < 15 + πn + \frac{π}{4}

Simplify

15 + \frac{π}{4} : \frac{60 + π}{4}

15 + \frac{π}{4}

Convert element to fraction:

15 = \frac{15 \cdot 4}{4}

= \frac{15 \cdot 4}{4} + \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{15 \cdot 4 + π}{4}

Multiply the numbers:

15 \cdot 4 = 60

= \frac{60 + π}{4}

x < \frac{60 + π}{4} + πn

x < \frac{60 + π}{4} + πn

Combine the intervals

x > \frac{60 - π}{4} + πn and x < \frac{60 + π}{4} + πn

Merge Overlapping Intervals

\frac{60 - π}{4} + πn < x < \frac{60 + π}{4} + πn

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