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(csc(x)+2)(csc(x)-sqrt(2))=0,0<= x<= 2pi

Solution

(csc (x) + 2) (csc (x) - 2) = 0, 0 \leq x \leq 2 π

x = \frac{7 π}{6}, x = \frac{11 π}{6}, x = \frac{π}{4}, x = \frac{3 π}{4}

Degrees

x = 21 0^{\circ}, x = 33 0^{\circ}, x = 4 5^{\circ}, x = 13 5^{\circ}

Solution steps

(csc (x) + 2) (csc (x) - 2) = 0, 0 \leq x \leq 2 π

Solving each part separately

csc (x) + 2 = 0 or csc (x) - 2 = 0

csc (x) + 2 = 0, 0 \leq x \leq 2 π : x = \frac{7 π}{6}, x = \frac{11 π}{6}

csc (x) + 2 = 0, 0 \leq x \leq 2 π

Move

2

to the right side

csc (x) + 2 = 0

Subtract

2

from both sides

csc (x) + 2 - 2 = 0 - 2

Simplify

csc (x) = - 2

csc (x) = - 2

General solutions for

csc (x) = - 2

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{7 π}{6}, x = \frac{11 π}{6}

csc (x) - 2 = 0, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{3 π}{4}

csc (x) - 2 = 0, 0 \leq x \leq 2 π