What is the general solution for 3csc(x)=4+csc(x) ?

The general solution for 3csc(x)=4+csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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3csc(x)=4+csc(x)

3 csc (x) = 4 + csc (x)

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

3 csc (x) = 4 + csc (x)

Solve by substitution

3 csc (x) = 4 + csc (x)

Let:

csc (x) = u

3 u = 4 + u

3 u = 4 + u : u = 2

3 u = 4 + u

Move

u

to the left side

3 u = 4 + u

Subtract

u

from both sides

3 u - u = 4 + u - u

Simplify

2 u = 4

2 u = 4

Divide both sides by

2

2 u = 4

Divide both sides by

2

\frac{2 u}{2} = \frac{4}{2}

Simplify

u = 2

u = 2

Substitute back

u = csc (x)

csc (x) = 2

csc (x) = 2

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan (θ) = - \frac{8}{15}

4 cos (t) = 0

7 sec^{3} (x) + 16 sec^{2} (x) + 11 sec (x) + 2 = 0

1 + sin (5 x) = 0

0 = \frac{π}{2} - 2 arctan (\frac{\frac{π}{2} - 2 - c}{\frac{π}{2} - c})

What is the general solution for 3csc(x)=4+csc(x) ?
The general solution for 3csc(x)=4+csc(x) is x= pi/6+2pin,x=(5pi)/6+2pin