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Popular Trigonometry >

cos((x-60)/2)>0

Solution

cos (\frac{x - 60}{2}) > 0

Solution

- π + 60 + 4 πn < x < π + 60 + 4 πn

Interval Notation

(- π + 60 + 4 πn, π + 60 + 4 πn)

Decimal

56.85840 \dots + 4 πn < x < 63.14159 \dots + 4 πn

Solution steps

cos (\frac{x - 60}{2}) > 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{x - 60}{2} < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < \frac{x - 60}{2} and \frac{x - 60}{2} < arccos (0) + 2 πn

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

- arccos (0) + 2 πn < \frac{x - 60}{2}

Switch sides

\frac{x - 60}{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{x - 60}{2} > - \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x - 60}{2} > - \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( x - 60 )}{2} > - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( x - 60 )}{2} > - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( x - 60 )}{2} : x - 60

\frac{2 ( x - 60 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x - 60

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

x - 60 > - π + 4 πn

x - 60 > - π + 4 πn

x - 60 > - π + 4 πn

Move

60

to the right side

x - 60 > - π + 4 πn

Add

60

to both sides

x - 60 + 60 > - π + 4 πn + 60

Simplify

x > - π + 4 πn + 60

x > - π + 4 πn + 60

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

\frac{x - 60}{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{x - 60}{2} < \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x - 60}{2} < \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( x - 60 )}{2} < 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( x - 60 )}{2} < 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( x - 60 )}{2} : x - 60

\frac{2 ( x - 60 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x - 60

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

x - 60 < π + 4 πn

x - 60 < π + 4 πn

x - 60 < π + 4 πn

Move

60

to the right side

x - 60 < π + 4 πn

Add

60

to both sides

x - 60 + 60 < π + 4 πn + 60

Simplify

x < π + 4 πn + 60

x < π + 4 πn + 60

Combine the intervals

x > - π + 4 πn + 60 and x < π + 4 πn + 60

Merge Overlapping Intervals

- π + 60 + 4 πn < x < π + 60 + 4 πn

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