What is the general solution for sin(3x+pi/4)=(sqrt(3))/2 ?

The general solution for sin(3x+pi/4)=(sqrt(3))/2 is x=(2pin)/3+pi/(36),x=(2pin)/3+(5pi)/(36)

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sin(3x+pi/4)=(sqrt(3))/2

Solution

sin (3 x + \frac{π}{4}) = \frac{3}{2}

Solution

x = \frac{2 πn}{3} + \frac{π}{36}, x = \frac{2 πn}{3} + \frac{5 π}{36}

Degrees

x = 5^{\circ} + 12 0^{\circ} n, x = 2 5^{\circ} + 12 0^{\circ} n

Solution steps

sin (3 x + \frac{π}{4}) = \frac{3}{2}

General solutions for

sin (3 x + \frac{π}{4}) = \frac{3}{2}

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}

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

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

\frac{π}{3} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{3} - \frac{π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 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}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 4}{12} - \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - π 3}{12}

Add similar elements:

4 π - 3 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}{4} = \frac{2 π}{3} + 2 πn : x = \frac{2 πn}{3} + \frac{5 π}{36}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

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

\frac{2 π}{3} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{2 π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

4

3

= 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}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= - \frac{π 3}{12} + \frac{8 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 8 π}{12}

Add similar elements:

- 3 π + 8 π = 5 π

= 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{5 π}{12}

Divide both sides by

3

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

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{5 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{5 π}{12}}{3} : \frac{2 πn}{3} + \frac{5 π}{36}

\frac{2 πn}{3} + \frac{\frac{5 π}{12}}{3}

\frac{\frac{5 π}{12}}{3} = \frac{5 π}{36}

\frac{\frac{5 π}{12}}{3}

Apply the fraction rule:

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

= \frac{5 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{5 π}{36}

= \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{π}{36}, x = \frac{2 πn}{3} + \frac{5 π}{36}

Popular Examples

(cos(x)-sin(x))/(cos(x))=tan(x)-1

\frac{cos ( x ) - sin ( x )}{cos ( x )} = tan (x) - 1

cos(φ)= 1/4

cos (φ) = \frac{1}{4}

-2cos(x)=sin(x)

- 2 cos (x) = sin (x)

sin(x)=-0.9781

sin (x) = - 0.9781

tan(x)=3.73

tan (x) = 3.73

Frequently Asked Questions (FAQ)

What is the general solution for sin(3x+pi/4)=(sqrt(3))/2 ?
The general solution for sin(3x+pi/4)=(sqrt(3))/2 is x=(2pin)/3+pi/(36),x=(2pin)/3+(5pi)/(36)