What is the general solution for 4=sqrt(2)csc(3x) ?

The general solution for 4=sqrt(2)csc(3x) is x=(0.36136…)/3+(2pin)/3 ,x= pi/3-(0.36136…)/3+(2pin)/3

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Popular >

4=sqrt(2)csc(3x)

Solution

4 = 2 csc (3 x)

Solution

x = \frac{0.36136 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{0.36136 \dots}{3} + \frac{2 πn}{3}

Degrees

x = 6.90160 \dots^{\circ} + 12 0^{\circ} n, x = 53.09839 \dots^{\circ} + 12 0^{\circ} n

Solution steps

4 = 2 csc (3 x)

Switch sides

2 csc (3 x) = 4

Divide both sides by

2

2 csc (3 x) = 4

Divide both sides by

2

\frac{2 csc ( 3 x )}{2} = \frac{4}{2}

Simplify

\frac{2 csc ( 3 x )}{2} = \frac{4}{2}

Simplify

\frac{2 csc ( 3 x )}{2} : csc (3 x)

\frac{2 csc ( 3 x )}{2}

Cancel the common factor:

2

= csc (3 x)

Simplify

\frac{4}{2} : 22

\frac{4}{2}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 ^{2}}{2}

Cancel

\frac{2 ^{2}}{2} : 2^{\frac{3}{2}}

\frac{2 ^{2}}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{2}}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{2}}{2 ^{\frac{1}{2}}} = 2^{2 - \frac{1}{2}}

= 2^{2 - \frac{1}{2}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= 2^{\frac{3}{2}}

= 2^{\frac{3}{2}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= 22

csc (3 x) = 22

csc (3 x) = 22

csc (3 x) = 22

Apply trig inverse properties

csc (3 x) = 22

General solutions for

csc (3 x) = 22

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

3 x = arccsc (22) + 2 πn, 3 x = π - arccsc (22) + 2 πn

3 x = arccsc (22) + 2 πn, 3 x = π - arccsc (22) + 2 πn

Solve

3 x = arccsc (22) + 2 πn : x = \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

3 x = arccsc (22) + 2 πn

Divide both sides by

3

3 x = arccsc (22) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

x = \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

Solve

3 x = π - arccsc (22) + 2 πn : x = \frac{π}{3} - \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

3 x = π - arccsc (22) + 2 πn

Divide both sides by

3

3 x = π - arccsc (22) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} - \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

Simplify

x = \frac{π}{3} - \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

x = \frac{π}{3} - \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

x = \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{arccsc ( 2 2 )}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.36136 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{0.36136 \dots}{3} + \frac{2 πn}{3}

Popular Examples

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21 = 24 + 8 cos (\frac{π x}{6})

sin(x)=(16sin(31))/(12)

sin (x) = \frac{16 sin ( 3 1 ^{\circ} )}{12}

sinh(z)-cosh(z)=0

sinh (z) - cosh (z) = 0

cos(x)=-1.588908648

cos (x) = - 1.588908648

sin(-x)=1

sin (- x) = 1

Frequently Asked Questions (FAQ)

What is the general solution for 4=sqrt(2)csc(3x) ?
The general solution for 4=sqrt(2)csc(3x) is x=(0.36136…)/3+(2pin)/3 ,x= pi/3-(0.36136…)/3+(2pin)/3