What is the general solution for csc(3x+pi/(12))=2 ?

The general solution for csc(3x+pi/(12))=2 is x=(2pin)/3+pi/(36),x=(2pin)/3+pi/4

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Popular Trigonometry >

csc(3x+pi/(12))=2

Solution

csc (3 x + \frac{π}{12}) = 2

Solution

x = \frac{2 πn}{3} + \frac{π}{36}, x = \frac{2 πn}{3} + \frac{π}{4}

Degrees

x = 5^{\circ} + 12 0^{\circ} n, x = 4 5^{\circ} + 12 0^{\circ} n

Solution steps

csc (3 x + \frac{π}{12}) = 2

General solutions for

csc (3 x + \frac{π}{12}) = 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

3 x + \frac{π}{12} = \frac{π}{6} + 2 πn, 3 x + \frac{π}{12} = \frac{5 π}{6} + 2 πn

3 x + \frac{π}{12} = \frac{π}{6} + 2 πn, 3 x + \frac{π}{12} = \frac{5 π}{6} + 2 πn

Solve

3 x + \frac{π}{12} = \frac{π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{π}{36}

3 x + \frac{π}{12} = \frac{π}{6} + 2 πn

Move

\frac{π}{12}

to the right side

3 x + \frac{π}{12} = \frac{π}{6} + 2 πn

Subtract

\frac{π}{12}

from both sides

3 x + \frac{π}{12} - \frac{π}{12} = \frac{π}{6} + 2 πn - \frac{π}{12}

Simplify

3 x + \frac{π}{12} - \frac{π}{12} = \frac{π}{6} + 2 πn - \frac{π}{12}

Simplify

3 x + \frac{π}{12} - \frac{π}{12} : 3 x

3 x + \frac{π}{12} - \frac{π}{12}

Add similar elements:

\frac{π}{12} - \frac{π}{12} = 0

= 3 x

Simplify

\frac{π}{6} + 2 πn - \frac{π}{12} : 2 πn + \frac{π}{12}

\frac{π}{6} + 2 πn - \frac{π}{12}

Group like terms

= 2 πn + \frac{π}{6} - \frac{π}{12}

Least Common Multiplier of

6, 12 : 12

6, 12

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

6

12

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= \frac{π 2}{12} - \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{12}

Add similar elements:

2 π - π = π

= 2 πn + \frac{π}{12}

3 x = 2 πn + \frac{π}{12}

3 x = 2 πn + \frac{π}{12}

3 x = 2 πn + \frac{π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{π}{12}}{3} : \frac{2 πn}{3} + \frac{π}{36}

\frac{2 πn}{3} + \frac{\frac{π}{12}}{3}

\frac{\frac{π}{12}}{3} = \frac{π}{36}

\frac{\frac{π}{12}}{3}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{π}{36}

= \frac{2 πn}{3} + \frac{π}{36}

x = \frac{2 πn}{3} + \frac{π}{36}

x = \frac{2 πn}{3} + \frac{π}{36}

x = \frac{2 πn}{3} + \frac{π}{36}

Solve

3 x + \frac{π}{12} = \frac{5 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{π}{4}

3 x + \frac{π}{12} = \frac{5 π}{6} + 2 πn

Move

\frac{π}{12}

to the right side

3 x + \frac{π}{12} = \frac{5 π}{6} + 2 πn

Subtract

\frac{π}{12}

from both sides

3 x + \frac{π}{12} - \frac{π}{12} = \frac{5 π}{6} + 2 πn - \frac{π}{12}

Simplify

3 x + \frac{π}{12} - \frac{π}{12} = \frac{5 π}{6} + 2 πn - \frac{π}{12}

Simplify

3 x + \frac{π}{12} - \frac{π}{12} : 3 x

3 x + \frac{π}{12} - \frac{π}{12}

Add similar elements:

\frac{π}{12} - \frac{π}{12} = 0

= 3 x

Simplify

\frac{5 π}{6} + 2 πn - \frac{π}{12} : 2 πn + \frac{3 π}{4}

\frac{5 π}{6} + 2 πn - \frac{π}{12}

Group like terms

= 2 πn + \frac{5 π}{6} - \frac{π}{12}

Least Common Multiplier of

6, 12 : 12

6, 12

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

6

12

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{5 π}{6} :

multiply the denominator and numerator by

2

\frac{5 π}{6} = \frac{5 π 2}{6 \cdot 2} = \frac{10 π}{12}

= \frac{10 π}{12} - \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{10 π - π}{12}

Add similar elements:

10 π - π = 9 π

= \frac{9 π}{12}

Cancel the common factor:

3

= 2 πn + \frac{3 π}{4}

3 x = 2 πn + \frac{3 π}{4}

3 x = 2 πn + \frac{3 π}{4}

3 x = 2 πn + \frac{3 π}{4}

Divide both sides by

3

3 x = 2 πn + \frac{3 π}{4}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{3 π}{4}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{3 π}{4}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

= \frac{2 πn}{3} + \frac{π}{4}

x = \frac{2 πn}{3} + \frac{π}{4}

x = \frac{2 πn}{3} + \frac{π}{4}

x = \frac{2 πn}{3} + \frac{π}{4}

x = \frac{2 πn}{3} + \frac{π}{36}, x = \frac{2 πn}{3} + \frac{π}{4}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for csc(3x+pi/(12))=2 ?
The general solution for csc(3x+pi/(12))=2 is x=(2pin)/3+pi/(36),x=(2pin)/3+pi/4