What is the general solution for 3csc(x)=2csc(x)+sqrt(2) ?

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

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3csc(x)=2csc(x)+sqrt(2)

3 csc (x) = 2 csc (x) + 2

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

3 csc (x) = 2 csc (x) + 2

Solve by substitution

3 csc (x) = 2 csc (x) + 2

Let:

csc (x) = u

3 u = 2 u + 2

3 u = 2 u + 2 : u = 2

3 u = 2 u + 2

Move

2 u

to the left side

3 u = 2 u + 2

Subtract

2 u

from both sides

3 u - 2 u = 2 u + 2 - 2 u

Simplify

u = 2

u = 2

Substitute back

u = csc (x)

csc (x) = 2

csc (x) = 2

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

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{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

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