What is the general solution for 2sin(x)csc(x)-csc(x)=4sin(x)-2 ?

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

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2sin(x)csc(x)-csc(x)=4sin(x)-2

Solution

2 sin (x) csc (x) - csc (x) = 4 sin (x) - 2

Solution

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

2 sin (x) csc (x) - csc (x) = 4 sin (x) - 2

Subtract

4 sin (x) - 2

from both sides

2 sin (x) csc (x) - csc (x) - 4 sin (x) + 2 = 0

Factor

2 sin (x) csc (x) - csc (x) - 4 sin (x) + 2 : (2 sin (x) - 1) (csc (x) - 2)

2 sin (x) csc (x) - csc (x) - 4 sin (x) + 2

= (2 sin (x) csc (x) - csc (x)) + (- 4 sin (x) + 2)

Factor out

- 2

from

- 4 sin (x) + 2 : - 2 (2 sin (x) - 1)

- 4 sin (x) + 2

Rewrite

- 4

2 \cdot 2

= 2 \cdot 2 sin (x) + 2

Factor out common term

- 2

= - 2 (2 sin (x) - 1)

Factor out

csc (x)

from

2 sin (x) csc (x) - csc (x) : csc (x) (2 sin (x) - 1)

2 sin (x) csc (x) - csc (x)

Factor out common term

csc (x)

= csc (x) (2 sin (x) - 1)

= - 2 (2 sin (x) - 1) + csc (x) (2 sin (x) - 1)

Factor out common term

2 sin (x) - 1

= (2 sin (x) - 1) (csc (x) - 2)

(2 sin (x) - 1) (csc (x) - 2) = 0

Solving each part separately

2 sin (x) - 1 = 0 or csc (x) - 2 = 0

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

2 sin (x) - 1 + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

csc (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) - 2 = 0

Move

2

to the right side

csc (x) - 2 = 0

Add

2

to 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{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

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Frequently Asked Questions (FAQ)

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