What is the general solution for csc((2θ)/3)-1=0 ?

The general solution for csc((2θ)/3)-1=0 is θ=(3pi)/4+3pin

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Popular >

csc((2θ)/3)-1=0

Solution

csc (\frac{2 θ}{3}) - 1 = 0

Solution

θ = \frac{3 π}{4} + 3 πn

Degrees

θ = 13 5^{\circ} + 54 0^{\circ} n

Solution steps

csc (\frac{2 θ}{3}) - 1 = 0

Move

1

to the right side

csc (\frac{2 θ}{3}) - 1 = 0

Add

1

to both sides

csc (\frac{2 θ}{3}) - 1 + 1 = 0 + 1

Simplify

csc (\frac{2 θ}{3}) = 1

csc (\frac{2 θ}{3}) = 1

General solutions for

csc (\frac{2 θ}{3}) = 1

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{2 θ}{3} = \frac{π}{2} + 2 πn

\frac{2 θ}{3} = \frac{π}{2} + 2 πn

Solve

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

\frac{2 θ}{3} = \frac{π}{2} + 2 πn

Multiply both sides by

3

\frac{2 θ}{3} = \frac{π}{2} + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 θ}{3} = 3 \cdot \frac{π}{2} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 θ}{3} = 3 \cdot \frac{π}{2} + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 θ}{3} : 2 θ

\frac{3 \cdot 2 θ}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 θ}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 θ

Simplify

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

3 \cdot \frac{π}{2} + 3 \cdot 2 πn

Multiply

3 \cdot \frac{π}{2} : \frac{3 π}{2}

3 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 3}{2}

= \frac{3 π}{2} + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= \frac{3 π}{2} + 6 πn

2 θ = \frac{3 π}{2} + 6 πn

2 θ = \frac{3 π}{2} + 6 πn

2 θ = \frac{3 π}{2} + 6 πn

Divide both sides by

2

2 θ = \frac{3 π}{2} + 6 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

\frac{\frac{3 π}{2}}{2} + \frac{6 πn}{2}

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

\frac{6 πn}{2} = 3 πn

\frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 πn

= \frac{3 π}{4} + 3 πn

θ = \frac{3 π}{4} + 3 πn

θ = \frac{3 π}{4} + 3 πn

θ = \frac{3 π}{4} + 3 πn

θ = \frac{3 π}{4} + 3 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc((2θ)/3)-1=0 ?
The general solution for csc((2θ)/3)-1=0 is θ=(3pi)/4+3pin