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Popular >

cos(x-pi/4)<1

Solution

cos (x - \frac{π}{4}) < 1

Solution

\frac{π}{4} + 2 πn < x < \frac{9 π}{4} + 2 πn

Interval Notation

(\frac{π}{4} + 2 πn, \frac{9 π}{4} + 2 πn)

Decimal Notation

0.78539 \dots + 2 πn < x < 7.06858 \dots + 2 πn

Solution steps

cos (x - \frac{π}{4}) < 1

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (1) + 2 πn < (x - \frac{π}{4}) < 2 π - arccos (1) + 2 πn

a < u < b

then

a < u and u < b

arccos (1) + 2 πn < x - \frac{π}{4} and x - \frac{π}{4} < 2 π - arccos (1) + 2 πn

arccos (1) + 2 πn < x - \frac{π}{4} : x > 2 πn + \frac{π}{4}

arccos (1) + 2 πn < x - \frac{π}{4}

Switch sides

x - \frac{π}{4} > arccos (1) + 2 πn

Simplify

arccos (1) + 2 πn : 2 πn

arccos (1) + 2 πn

Use the following trivial identity:

arccos (1) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

x - \frac{π}{4} > 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} > 2 πn

Add

\frac{π}{4}

to both sides

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

Simplify

x > 2 πn + \frac{π}{4}

x > 2 πn + \frac{π}{4}

x - \frac{π}{4} < 2 π - arccos (1) + 2 πn : x < \frac{9 π}{4} + 2 πn

x - \frac{π}{4} < 2 π - arccos (1) + 2 πn

Simplify

2 π - arccos (1) + 2 πn : 2 π + 2 πn

2 π - arccos (1) + 2 πn

Use the following trivial identity:

arccos (1) = 0

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 - 0 + 2 πn

2 π - 0 + 2 πn = 2 π + 2 πn

= 2 π + 2 πn

x - \frac{π}{4} < 2 π + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} < 2 π + 2 πn

Add

\frac{π}{4}

to both sides

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

Simplify

x < 2 π + 2 πn + \frac{π}{4}

x < 2 π + 2 πn + \frac{π}{4}

Simplify

2 π + \frac{π}{4} : \frac{9 π}{4}

2 π + \frac{π}{4}

Convert element to fraction:

2 π = \frac{2 π 4}{4}

= \frac{2 π 4}{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{2 π 4 + π}{4}

2 π 4 + π = 9 π

2 π 4 + π

Multiply the numbers:

2 \cdot 4 = 8

= 8 π + π

Add similar elements:

8 π + π = 9 π

= 9 π

= \frac{9 π}{4}

x < \frac{9 π}{4} + 2 πn

Combine the intervals

x > 2 πn + \frac{π}{4} and x < \frac{9 π}{4} + 2 πn

Merge Overlapping Intervals

\frac{π}{4} + 2 πn < x < \frac{9 π}{4} + 2 πn

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