What is the general solution for sin((3pi)/4+1/12 x)=1 ?

The general solution for sin((3pi)/4+1/12 x)=1 is x=24pin-3pi

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

sin((3pi)/4+1/12 x)=1

Solution

sin (\frac{3 π}{4} + \frac{1}{12} x) = 1

Solution

x = 24 πn - 3 π

Degrees

x = - 54 0^{\circ} + 432 0^{\circ} n

Solution steps

sin (\frac{3 π}{4} + \frac{1}{12} x) = 1

General solutions for

sin (\frac{3 π}{4} + \frac{1}{12} x) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 π}{4} + \frac{1}{12} x = \frac{π}{2} + 2 πn

\frac{3 π}{4} + \frac{1}{12} x = \frac{π}{2} + 2 πn

Solve

\frac{3 π}{4} + \frac{1}{12} x = \frac{π}{2} + 2 πn : x = 24 πn - 3 π

\frac{3 π}{4} + \frac{1}{12} x = \frac{π}{2} + 2 πn

Move

\frac{3 π}{4}

to the right side

\frac{3 π}{4} + \frac{1}{12} x = \frac{π}{2} + 2 πn

Subtract

\frac{3 π}{4}

from both sides

\frac{3 π}{4} + \frac{1}{12} x - \frac{3 π}{4} = \frac{π}{2} + 2 πn - \frac{3 π}{4}

Simplify

\frac{3 π}{4} + \frac{1}{12} x - \frac{3 π}{4} = \frac{π}{2} + 2 πn - \frac{3 π}{4}

Simplify

\frac{3 π}{4} + \frac{1}{12} x - \frac{3 π}{4} : \frac{1}{12} x

\frac{3 π}{4} + \frac{1}{12} x - \frac{3 π}{4}

Add similar elements:

\frac{3 π}{4} - \frac{3 π}{4} = 0

= \frac{1}{12} x

Simplify

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

\frac{π}{2} + 2 πn - \frac{3 π}{4}

Group like terms

= 2 πn + \frac{π}{2} - \frac{3 π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{4} - \frac{3 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 3 π}{4}

Add similar elements:

2 π - 3 π = - π

= \frac{- π}{4}

Apply the fraction rule:

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

= 2 πn - \frac{π}{4}

\frac{1}{12} x = 2 πn - \frac{π}{4}

\frac{1}{12} x = 2 πn - \frac{π}{4}

\frac{1}{12} x = 2 πn - \frac{π}{4}

Multiply both sides by

12

\frac{1}{12} x = 2 πn - \frac{π}{4}

Multiply both sides by

12

12 \cdot \frac{1}{12} x = 12 \cdot 2 πn - 12 \cdot \frac{π}{4}

Simplify

12 \cdot \frac{1}{12} x = 12 \cdot 2 πn - 12 \cdot \frac{π}{4}

Simplify

12 \cdot \frac{1}{12} x : x

12 \cdot \frac{1}{12} x

Multiply fractions:

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

= \frac{1 \cdot 12}{12} x

Cancel the common factor:

12

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

12 \cdot 2 πn - 12 \cdot \frac{π}{4} : 24 πn - 3 π

12 \cdot 2 πn - 12 \cdot \frac{π}{4}

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

12 \cdot \frac{π}{4} = 3 π

12 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 π

= 24 πn - 3 π

x = 24 πn - 3 π

x = 24 πn - 3 π

x = 24 πn - 3 π

x = 24 πn - 3 π

Graph

Popular Examples

Frequently Asked Questions (FAQ)

What is the general solution for sin((3pi)/4+1/12 x)=1 ?
The general solution for sin((3pi)/4+1/12 x)=1 is x=24pin-3pi