What is the general solution for 2csc(x)+17=15+csc(x) ?

The general solution for 2csc(x)+17=15+csc(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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2csc(x)+17=15+csc(x)

2 csc (x) + 17 = 15 + csc (x)

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

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

2 csc (x) + 17 = 15 + csc (x)

Solve by substitution

2 csc (x) + 17 = 15 + csc (x)

Let:

csc (x) = u

2 u + 17 = 15 + u

2 u + 17 = 15 + u : u = - 2

2 u + 17 = 15 + u

Move

17

to the right side

2 u + 17 = 15 + u

Subtract

17

from both sides

2 u + 17 - 17 = 15 + u - 17

Simplify

2 u = u - 2

2 u = u - 2

Move

u

to the left side

2 u = u - 2

Subtract

u

from both sides

2 u - u = u - 2 - u

Simplify

u = - 2

u = - 2

Substitute back

u = csc (x)

csc (x) = - 2

csc (x) = - 2

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

Combine all the solutions

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

cos (x) - sin (x) = 2

cot (θ) = (5.8473)

6 = 2 + 5 arctan (\frac{x}{2})

tan^{2} (θ) - 2 sec (θ) = 2

- sin (B) - 4 = 3 sin (B) - 5

What is the general solution for 2csc(x)+17=15+csc(x) ?
The general solution for 2csc(x)+17=15+csc(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin