What is the general solution for csc(pi/(42)x)=1 ?

The general solution for csc(pi/(42)x)=1 is x=21+84n

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csc(pi/(42)x)=1

Solution

csc (\frac{π}{42} x) = 1

Solution

x = 21 + 84 n

Degrees

x = 1203.21136 \dots^{\circ} + 4812.84547 \dots^{\circ} n

Solution steps

csc (\frac{π}{42} x) = 1

General solutions for

csc (\frac{π}{42} x) = 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{π}{42} x = \frac{π}{2} + 2 πn

\frac{π}{42} x = \frac{π}{2} + 2 πn

Solve

\frac{π}{42} x = \frac{π}{2} + 2 πn : x = 21 + 84 n

\frac{π}{42} x = \frac{π}{2} + 2 πn

Multiply both sides by

42

\frac{π}{42} x = \frac{π}{2} + 2 πn

Multiply both sides by

42

42 \cdot \frac{π}{42} x = 42 \cdot \frac{π}{2} + 42 \cdot 2 πn

Simplify

42 \cdot \frac{π}{42} x = 42 \cdot \frac{π}{2} + 42 \cdot 2 πn

Simplify

42 \cdot \frac{π}{42} x : π x

42 \cdot \frac{π}{42} x

Multiply fractions:

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

= \frac{42 π}{42} x

Cancel the common factor:

42

= x π

Simplify

42 \cdot \frac{π}{2} + 42 \cdot 2 πn : 21 π + 84 πn

42 \cdot \frac{π}{2} + 42 \cdot 2 πn

42 \cdot \frac{π}{2} = 21 π

42 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 42}{2}

Divide the numbers:

\frac{42}{2} = 21

= 21 π

42 \cdot 2 πn = 84 πn

42 \cdot 2 πn

Multiply the numbers:

42 \cdot 2 = 84

= 84 πn

= 21 π + 84 πn

π x = 21 π + 84 πn

π x = 21 π + 84 πn

π x = 21 π + 84 πn

Divide both sides by

π

π x = 21 π + 84 πn

Divide both sides by

π

\frac{π x}{π} = \frac{21 π}{π} + \frac{84 πn}{π}

Simplify

\frac{π x}{π} = \frac{21 π}{π} + \frac{84 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{21 π}{π} + \frac{84 πn}{π} : 21 + 84 n

\frac{21 π}{π} + \frac{84 πn}{π}

Cancel

\frac{21 π}{π} : 21

\frac{21 π}{π}

Cancel the common factor:

π

= 21

= 21 + \frac{84 πn}{π}

Cancel

\frac{84 πn}{π} : 84 n

\frac{84 πn}{π}

Cancel the common factor:

π

= 84 n

= 21 + 84 n

x = 21 + 84 n

x = 21 + 84 n

x = 21 + 84 n

x = 21 + 84 n

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Frequently Asked Questions (FAQ)

What is the general solution for csc(pi/(42)x)=1 ?
The general solution for csc(pi/(42)x)=1 is x=21+84n