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Popular Trigonometry >

-pi/(12)sin^2(pi/(12)t)<0

Solution

- \frac{π}{12} sin^{2} (\frac{π}{12} t) < 0

Solution

24 n < t < 12 + 24 n or - 12 + 24 n < t < 24 n

Interval Notation

(24 n, 12 + 24 n) \cup (- 12 + 24 n, 24 n)

Solution steps

- \frac{π}{12} sin^{2} (\frac{π}{12} t) < 0

Multiply both sides by

- 1

- \frac{π}{12} sin^{2} (\frac{π}{12} t) < 0

Multiply both sides by -1 (reverse the inequality)

(- \frac{π}{12} sin^{2} (\frac{π}{12} t)) (- 1) > 0 \cdot (- 1)

Simplify

\frac{π}{12} sin^{2} (\frac{π}{12} t) > 0

\frac{π}{12} sin^{2} (\frac{π}{12} t) > 0

Multiply both sides by

12

\frac{π}{12} sin^{2} (\frac{π}{12} t) > 0

Multiply both sides by

12

12 \cdot \frac{π}{12} sin^{2} (\frac{π}{12} t) > 0 \cdot 12

Simplify

π sin^{2} (\frac{π}{12} t) > 0

π sin^{2} (\frac{π}{12} t) > 0

Divide both sides by

π

π sin^{2} (\frac{π}{12} t) > 0

Divide both sides by

π

\frac{π sin ^{2} ( \frac{π}{12} t )}{π} > \frac{0}{π}

Simplify

sin^{2} (\frac{π}{12} t) > 0

sin^{2} (\frac{π}{12} t) > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

sin (\frac{π}{12} t) < 0 or sin (\frac{π}{12} t) > 0

sin (\frac{π}{12} t) < 0 : - 12 + 24 n < t < 24 n

sin (\frac{π}{12} t) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (0) + 2 πn < \frac{π}{12} t < arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0) + 2 πn < \frac{π}{12} t and \frac{π}{12} t < arcsin (0) + 2 πn

- π - arcsin (0) + 2 πn < \frac{π}{12} t : t > 24 n - 12

- π - arcsin (0) + 2 πn < \frac{π}{12} t

Switch sides

\frac{π}{12} t > - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- 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

= 2 πn - π

\frac{π}{12} t > 2 πn - π

Multiply both sides by

12

\frac{π}{12} t > 2 πn - π

Multiply both sides by

12

12 \cdot \frac{π}{12} t > 12 \cdot 2 πn - 12 π

Simplify

π t > 24 πn - 12 π

π t > 24 πn - 12 π

Divide both sides by

π

π t > 24 πn - 12 π

Divide both sides by

π

\frac{π t}{π} > \frac{24 πn}{π} - \frac{12 π}{π}

Simplify

\frac{π t}{π} > \frac{24 πn}{π} - \frac{12 π}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{24 πn}{π} - \frac{12 π}{π} : 24 n - 12

\frac{24 πn}{π} - \frac{12 π}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 24 n - \frac{12 π}{π}

Cancel

\frac{12 π}{π} : 12

\frac{12 π}{π}

Cancel the common factor:

π

= 12

= 24 n - 12

t > 24 n - 12

t > 24 n - 12

t > 24 n - 12

\frac{π}{12} t < arcsin (0) + 2 πn : t < 24 n

\frac{π}{12} t < 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}{12} t < 2 πn

Multiply both sides by

12

\frac{π}{12} t < 2 πn

Multiply both sides by

12

12 \cdot \frac{π}{12} t < 12 \cdot 2 πn

Simplify

π t < 24 πn

π t < 24 πn

Divide both sides by

π

π t < 24 πn

Divide both sides by

π

\frac{π t}{π} < \frac{24 πn}{π}

Simplify

t < 24 n

t < 24 n

Combine the intervals

t > 24 n - 12 and t < 24 n

Merge Overlapping Intervals

- 12 + 24 n < t < 24 n

sin (\frac{π}{12} t) > 0 : 24 n < t < 12 + 24 n

sin (\frac{π}{12} t) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (0) + 2 πn < \frac{π}{12} t < π - arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

arcsin (0) + 2 πn < \frac{π}{12} t and \frac{π}{12} t < π - arcsin (0) + 2 πn

arcsin (0) + 2 πn < \frac{π}{12} t : t > 24 n

arcsin (0) + 2 πn < \frac{π}{12} t

Switch sides

\frac{π}{12} t > 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}{12} t > 2 πn

Multiply both sides by

12

\frac{π}{12} t > 2 πn

Multiply both sides by

12

12 \cdot \frac{π}{12} t > 12 \cdot 2 πn

Simplify

π t > 24 πn

π t > 24 πn

Divide both sides by

π

π t > 24 πn

Divide both sides by

π

\frac{π t}{π} > \frac{24 πn}{π}

Simplify

t > 24 n

t > 24 n

\frac{π}{12} t < π - arcsin (0) + 2 πn : t < 12 + 24 n

\frac{π}{12} t < 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}{12} t < π + 2 πn

Multiply both sides by

12

\frac{π}{12} t < π + 2 πn

Multiply both sides by

12

12 \cdot \frac{π}{12} t < 12 π + 12 \cdot 2 πn

Simplify

π t < 12 π + 24 πn

π t < 12 π + 24 πn

Divide both sides by

π

π t < 12 π + 24 πn

Divide both sides by

π

\frac{π t}{π} < \frac{12 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} < \frac{12 π}{π} + \frac{24 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{12 π}{π} + \frac{24 πn}{π} : 12 + 24 n

\frac{12 π}{π} + \frac{24 πn}{π}

Cancel

\frac{12 π}{π} : 12

\frac{12 π}{π}

Cancel the common factor:

π

= 12

= 12 + \frac{24 πn}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 12 + 24 n

t < 12 + 24 n

t < 12 + 24 n

t < 12 + 24 n

Combine the intervals

t > 24 n and t < 12 + 24 n

Merge Overlapping Intervals

24 n < t < 12 + 24 n

Combine the intervals

- 12 + 24 n < t < 24 n or 24 n < t < 12 + 24 n

Merge Overlapping Intervals

24 n < t < 12 + 24 n or - 12 + 24 n < t < 24 n

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