What is the general solution for csc(x)(2sin(x)-sqrt(2))=0 ?

The general solution for csc(x)(2sin(x)-sqrt(2))=0 is x= pi/4+2pin,x=(3pi)/4+2pin

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csc(x)(2sin(x)-sqrt(2))=0

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

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Solving each part separately

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

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

2 sin (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

2 sin (x) - 2 = 0

Move

2

to the right side

2 sin (x) - 2 = 0

Add

2

to both sides

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

Simplify

2 sin (x) = 2

2 sin (x) = 2

Divide both sides by

2

2 sin (x) = 2

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{2}{2}

Simplify

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

General solutions for

sin (x) = \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

\frac{700}{sin ( 9 0 ^{\circ} )} = \frac{236.3277}{sin ( θ )}

cos (x) + cos (2 x) = - 0.75

x, 2 y = sin (2 x)

- csc (θ) + 5 = cot (θ) + 6

sin (θ) - 0.2 \cdot cos (θ) = 0.11836

What is the general solution for csc(x)(2sin(x)-sqrt(2))=0 ?
The general solution for csc(x)(2sin(x)-sqrt(2))=0 is x= pi/4+2pin,x=(3pi)/4+2pin