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Popular Trigonometry >

6-3cos((pit)/2)>6

Solution

6 - 3 cos (\frac{π t}{2}) > 6

Solution

1 + 4 n < t < 3 + 4 n

Interval Notation

(1 + 4 n, 3 + 4 n)

Solution steps

6 - 3 cos (\frac{π t}{2}) > 6

Subtract

6

from both sides

6 - 3 cos (\frac{π t}{2}) - 6 > 6 - 6

Simplify

- 3 cos (\frac{π t}{2}) > 0

Multiply both sides by

- 1

- 3 cos (\frac{π t}{2}) > 0

Multiply both sides by -1 (reverse the inequality)

(- 3 cos (\frac{π t}{2})) (- 1) < 0 \cdot (- 1)

Simplify

3 cos (\frac{π t}{2}) < 0

3 cos (\frac{π t}{2}) < 0

Divide both sides by

3

3 cos (\frac{π t}{2}) < 0

Divide both sides by

3

\frac{3 cos ( \frac{π t}{2} )}{3} < \frac{0}{3}

Simplify

cos (\frac{π t}{2}) < 0

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

a < u < b

then

a < u and u < b

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

arccos (0) + 2 πn < \frac{π t}{2} : t > 1 + 4 n

arccos (0) + 2 πn < \frac{π t}{2}

Switch sides

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

Multiply both sides by

2

\frac{π t}{2} > \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π t}{2} : π t

\frac{2 π t}{2}

Divide the numbers:

\frac{2}{2} = 1

= π t

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

π t > π + 4 πn

π t > π + 4 πn

π t > π + 4 πn

Divide both sides by

π

π t > π + 4 πn

Divide both sides by

π

\frac{π t}{π} > \frac{π}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} > \frac{π}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{π}{π} + \frac{4 πn}{π} : 1 + 4 n

\frac{π}{π} + \frac{4 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 1 + 4 n

t > 1 + 4 n

t > 1 + 4 n

t > 1 + 4 n

\frac{π t}{2} < 2 π - arccos (0) + 2 πn : t < 3 + 4 n

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π t}{2} : π t

\frac{2 π t}{2}

Divide the numbers:

\frac{2}{2} = 1

= π t

Simplify

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

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

2 \cdot 2 π = 4 π

2 \cdot 2 π

Multiply the numbers:

2 \cdot 2 = 4

= 4 π

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 4 π - π + 4 πn

Add similar elements:

4 π - π = 3 π

= 3 π + 4 πn

π t < 3 π + 4 πn

π t < 3 π + 4 πn

π t < 3 π + 4 πn

Divide both sides by

π

π t < 3 π + 4 πn

Divide both sides by

π

\frac{π t}{π} < \frac{3 π}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} < \frac{3 π}{π} + \frac{4 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{3 π}{π} + \frac{4 πn}{π} : 3 + 4 n

\frac{3 π}{π} + \frac{4 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 3 + 4 n

t < 3 + 4 n

t < 3 + 4 n

t < 3 + 4 n

Combine the intervals

t > 1 + 4 n and t < 3 + 4 n

Merge Overlapping Intervals

1 + 4 n < t < 3 + 4 n

Popular Examples

(10pi)/9 <= arctan(θ)