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Popular >

3cos(t/2-pi/4)>0

Solution

3 cos (\frac{t}{2} - \frac{π}{4}) > 0

Solution

- \frac{π}{2} + 4 πn < t < \frac{3 π}{2} + 4 πn

Interval Notation

(- \frac{π}{2} + 4 πn, \frac{3 π}{2} + 4 πn)

Decimal Notation

- 1.57079 \dots + 4 πn < t < 4.71238 \dots + 4 πn

Solution steps

3 cos (\frac{t}{2} - \frac{π}{4}) > 0

Divide both sides by

3

3 cos (\frac{t}{2} - \frac{π}{4}) > 0

Divide both sides by

3

\frac{3 cos ( \frac{t}{2} - \frac{π}{4} )}{3} > \frac{0}{3}

Simplify

cos (\frac{t}{2} - \frac{π}{4}) > 0

cos (\frac{t}{2} - \frac{π}{4}) > 0

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

\frac{t}{2} - \frac{π}{4} > - 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{t}{2} - \frac{π}{4} > - \frac{π}{2} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{t}{2} - \frac{π}{4} > - \frac{π}{2} + 2 πn

Add

\frac{π}{4}

to both sides

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} > - \frac{π}{2} + 2 πn + \frac{π}{4}

Simplify

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} > - \frac{π}{2} + 2 πn + \frac{π}{4}

Simplify

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} : \frac{t}{2}

\frac{t}{2} - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} > 0

= \frac{t}{2}

Simplify

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

- \frac{π}{2} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{2} + \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= - \frac{π 2}{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{- π 2 + π}{4}

Add similar elements:

- 2 π + π = - π

= \frac{- π}{4}

Apply the fraction rule:

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

= 2 πn - \frac{π}{4}

\frac{t}{2} > 2 πn - \frac{π}{4}

\frac{t}{2} > 2 πn - \frac{π}{4}

\frac{t}{2} > 2 πn - \frac{π}{4}

Multiply both sides by

2

\frac{t}{2} > 2 πn - \frac{π}{4}

Multiply both sides by

2

\frac{2 t}{2} > 2 \cdot 2 πn - 2 \cdot \frac{π}{4}

Simplify

\frac{2 t}{2} > 2 \cdot 2 πn - 2 \cdot \frac{π}{4}

Simplify

\frac{2 t}{2} : t

\frac{2 t}{2}

Divide the numbers:

\frac{2}{2} = 1

= t

Simplify

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

2 \cdot 2 πn - 2 \cdot \frac{π}{4}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 4 πn - \frac{π}{2}

t > 4 πn - \frac{π}{2}

t > 4 πn - \frac{π}{2}

t > 4 πn - \frac{π}{2}

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

\frac{t}{2} - \frac{π}{4} < 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{t}{2} - \frac{π}{4} < \frac{π}{2} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{t}{2} - \frac{π}{4} < \frac{π}{2} + 2 πn

Add

\frac{π}{4}

to both sides

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} < \frac{π}{2} + 2 πn + \frac{π}{4}

Simplify

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} < \frac{π}{2} + 2 πn + \frac{π}{4}

Simplify

\frac{t}{2} - \frac{π}{4} + \frac{π}{4} : \frac{t}{2}

\frac{t}{2} - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} < 0

= \frac{t}{2}

Simplify

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

\frac{π}{2} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= \frac{π 2}{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{π 2 + π}{4}

Add similar elements:

2 π + π = 3 π

= 2 πn + \frac{3 π}{4}

\frac{t}{2} < 2 πn + \frac{3 π}{4}

\frac{t}{2} < 2 πn + \frac{3 π}{4}

\frac{t}{2} < 2 πn + \frac{3 π}{4}

Multiply both sides by

2

\frac{t}{2} < 2 πn + \frac{3 π}{4}

Multiply both sides by

2

\frac{2 t}{2} < 2 \cdot 2 πn + 2 \cdot \frac{3 π}{4}

Simplify

\frac{2 t}{2} < 2 \cdot 2 πn + 2 \cdot \frac{3 π}{4}

Simplify

\frac{2 t}{2} : t

\frac{2 t}{2}

Divide the numbers:

\frac{2}{2} = 1

= t

Simplify

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

2 \cdot 2 πn + 2 \cdot \frac{3 π}{4}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 4 πn + \frac{3 π}{2}

t < 4 πn + \frac{3 π}{2}

t < 4 πn + \frac{3 π}{2}

t < 4 πn + \frac{3 π}{2}

Combine the intervals

t > 4 πn - \frac{π}{2} and t < 4 πn + \frac{3 π}{2}

Merge Overlapping Intervals

- \frac{π}{2} + 4 πn < t < \frac{3 π}{2} + 4 πn

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