What is the general solution for 4+2csc(θ/2)=8 ?

The general solution for 4+2csc(θ/2)=8 is θ= pi/3+4pin,θ=(5pi)/3+4pin

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4+2csc(θ/2)=8

Solution

4 + 2 csc (\frac{θ}{2}) = 8

Solution

θ = \frac{π}{3} + 4 πn, θ = \frac{5 π}{3} + 4 πn

Degrees

θ = 6 0^{\circ} + 72 0^{\circ} n, θ = 30 0^{\circ} + 72 0^{\circ} n

Solution steps

4 + 2 csc (\frac{θ}{2}) = 8

Move

4

to the right side

4 + 2 csc (\frac{θ}{2}) = 8

Subtract

4

from both sides

4 + 2 csc (\frac{θ}{2}) - 4 = 8 - 4

Simplify

2 csc (\frac{θ}{2}) = 4

2 csc (\frac{θ}{2}) = 4

Divide both sides by

2

2 csc (\frac{θ}{2}) = 4

Divide both sides by

2

\frac{2 csc ( \frac{θ}{2} )}{2} = \frac{4}{2}

Simplify

csc (\frac{θ}{2}) = 2

csc (\frac{θ}{2}) = 2

General solutions for

csc (\frac{θ}{2}) = 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

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

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

Solve

\frac{θ}{2} = \frac{π}{6} + 2 πn : θ = \frac{π}{3} + 4 πn

\frac{θ}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

θ = \frac{π}{3} + 4 πn

θ = \frac{π}{3} + 4 πn

θ = \frac{π}{3} + 4 πn

Solve

\frac{θ}{2} = \frac{5 π}{6} + 2 πn : θ = \frac{5 π}{3} + 4 πn

\frac{θ}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

θ = \frac{π}{3} + 4 πn, θ = \frac{5 π}{3} + 4 πn

Popular Examples

2cos(3x)= 1/2 ,x=(17pi)/4

2 cos (3 x) = \frac{1}{2}, x = \frac{17 π}{4}

cos(θ)=sin(36)

cos (θ) = sin (3 6^{\circ})

2sec^2(x)=tan^2(x)+2sec(x)+4

2 sec^{2} (x) = tan^{2} (x) + 2 sec (x) + 4

cos^2(θ)+sin(θ)+1=0

cos^{2} (θ) + sin (θ) + 1 = 0

cos(x)=(22.5)/(34)

cos (x) = \frac{22.5}{34}

Frequently Asked Questions (FAQ)

What is the general solution for 4+2csc(θ/2)=8 ?
The general solution for 4+2csc(θ/2)=8 is θ= pi/3+4pin,θ=(5pi)/3+4pin