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Popular >

cos(pi/6 t)<0

Solution

cos (\frac{π}{6} t) < 0

Solution

3 + 12 n < t < 9 + 12 n

Interval Notation

(3 + 12 n, 9 + 12 n)

Solution steps

cos (\frac{π}{6} t) < 0

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (0) + 2 πn < \frac{π}{6} t < 2 π - arccos (0) + 2 πn

a < u < b

then

a < u and u < b

arccos (0) + 2 πn < \frac{π}{6} t and \frac{π}{6} t < 2 π - arccos (0) + 2 πn

arccos (0) + 2 πn < \frac{π}{6} t : t > 3 + 12 n

arccos (0) + 2 πn < \frac{π}{6} t

Switch sides

\frac{π}{6} t > 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

\frac{π}{6} t > \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{π}{6} t > \frac{π}{2} + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} t > 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t > 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiply fractions:

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

= \frac{6 π}{6} t

Cancel the common factor:

6

= t π

Simplify

6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 3 π + 12 πn

6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 3 π + 12 πn

π t > 3 π + 12 πn

π t > 3 π + 12 πn

π t > 3 π + 12 πn

Divide both sides by

π

π t > 3 π + 12 πn

Divide both sides by

π

\frac{π t}{π} > \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π t}{π} > \frac{3 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{3 π}{π} + \frac{12 πn}{π} : 3 + 12 n

\frac{3 π}{π} + \frac{12 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 3 + 12 n

t > 3 + 12 n

t > 3 + 12 n

t > 3 + 12 n

\frac{π}{6} t < 2 π - arccos (0) + 2 πn : t < 9 + 12 n

\frac{π}{6} t < 2 π - arccos (0) + 2 πn

Simplify

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

2 π - 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}

= 2 π - \frac{π}{2} + 2 πn

\frac{π}{6} t < 2 π - \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{π}{6} t < 2 π - \frac{π}{2} + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} t < 6 \cdot 2 π - 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t < 6 \cdot 2 π - 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiply fractions:

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

= \frac{6 π}{6} t

Cancel the common factor:

6

= t π

Simplify

6 \cdot 2 π - 6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 9 π + 12 πn

6 \cdot 2 π - 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot 2 π = 12 π

6 \cdot 2 π

Multiply the numbers:

6 \cdot 2 = 12

= 12 π

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 12 π - 3 π + 12 πn

Add similar elements:

12 π - 3 π = 9 π

= 9 π + 12 πn

π t < 9 π + 12 πn

π t < 9 π + 12 πn

π t < 9 π + 12 πn

Divide both sides by

π

π t < 9 π + 12 πn

Divide both sides by

π

\frac{π t}{π} < \frac{9 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π t}{π} < \frac{9 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{9 π}{π} + \frac{12 πn}{π} : 9 + 12 n

\frac{9 π}{π} + \frac{12 πn}{π}

Cancel

\frac{9 π}{π} : 9

\frac{9 π}{π}

Cancel the common factor:

π

= 9

= 9 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 9 + 12 n

t < 9 + 12 n

t < 9 + 12 n

t < 9 + 12 n

Combine the intervals

t > 3 + 12 n and t < 9 + 12 n

Merge Overlapping Intervals

3 + 12 n < t < 9 + 12 n

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