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Popular >

cot(pi-x)<-1

Solution

cot (π - x) < - 1

Solution

π + πn < x < \frac{5 π}{4} + πn

Interval Notation

(π + πn, \frac{5 π}{4} + πn)

Decimal Notation

3.14159 \dots + πn < x < 3.92699 \dots + πn

Solution steps

cot (π - x) < - 1

Factor out

- 1

from

π - x : - (- π + x)

cot (- (- π + x)) < - 1

Use the following identity:

cot (- x) = - cot (x)

- cot (x - π) < - 1

Multiply both sides by

- 1

- cot (x - π) < - 1

Multiply both sides by -1 (reverse the inequality)

(- cot (x - π)) (- 1) > (- 1) (- 1)

Simplify

cot (x - π) > 1

cot (x - π) > 1

cot (x) > a

then

πn < x < arccot (a) + πn

πn < (x - π) < arccot (1) + πn

a < u < b

then

a < u and u < b

πn < x - π and x - π < arccot (1) + πn

πn < x - π : x > πn + π

πn < x - π

Switch sides

x - π > πn

Move

π

to the right side

x - π > πn

Add

π

to both sides

x - π + π > πn + π

Simplify

x > πn + π

x > πn + π

x - π < arccot (1) + πn : x < \frac{5 π}{4} + πn

x - π < arccot (1) + πn

Simplify

arccot (1) + πn : \frac{π}{4} + πn

arccot (1) + πn

Use the following trivial identity:

arccot (1) = \frac{π}{4}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{π}{4} + πn

x - π < \frac{π}{4} + πn

Move

π

to the right side

x - π < \frac{π}{4} + πn

Add

π

to both sides

x - π + π < \frac{π}{4} + πn + π

Simplify

x < \frac{π}{4} + πn + π

x < \frac{π}{4} + πn + π

Simplify

\frac{π}{4} + π : \frac{5 π}{4}

\frac{π}{4} + π

Convert element to fraction:

π = \frac{π 4}{4}

= \frac{π}{4} + \frac{π 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π + π 4}{4}

Add similar elements:

π + 4 π = 5 π

= \frac{5 π}{4}

x < \frac{5 π}{4} + πn

Combine the intervals

x > πn + π and x < \frac{5 π}{4} + πn

Merge Overlapping Intervals

π + πn < x < \frac{5 π}{4} + πn

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