What is the general solution for 2=4cos(3x+1) ?

The general solution for 2=4cos(3x+1) is x=-1/3+pi/9+(2pin)/3 ,x=-1/3+(5pi)/9+(2pin)/3

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2=4cos(3x+1)

Solution

2 = 4 cos (3 x + 1)

Solution

x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}, x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

Degrees

x = 0.90140 \dots^{\circ} + 12 0^{\circ} n, x = 80.90140 \dots^{\circ} + 12 0^{\circ} n

Solution steps

2 = 4 cos (3 x + 1)

Switch sides

4 cos (3 x + 1) = 2

Divide both sides by

4

4 cos (3 x + 1) = 2

Divide both sides by

4

\frac{4 cos ( 3 x + 1 )}{4} = \frac{2}{4}

Simplify

cos (3 x + 1) = \frac{1}{2}

cos (3 x + 1) = \frac{1}{2}

General solutions for

cos (3 x + 1) = \frac{1}{2}

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

3 x + 1 = \frac{π}{3} + 2 πn, 3 x + 1 = \frac{5 π}{3} + 2 πn

3 x + 1 = \frac{π}{3} + 2 πn, 3 x + 1 = \frac{5 π}{3} + 2 πn

Solve

3 x + 1 = \frac{π}{3} + 2 πn : x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

3 x + 1 = \frac{π}{3} + 2 πn

Move

1

to the right side

3 x + 1 = \frac{π}{3} + 2 πn

Subtract

1

from both sides

3 x + 1 - 1 = \frac{π}{3} + 2 πn - 1

Simplify

3 x = \frac{π}{3} + 2 πn - 1

3 x = \frac{π}{3} + 2 πn - 1

Divide both sides by

3

3 x = \frac{π}{3} + 2 πn - 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3} : - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3}

Group like terms

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= - \frac{1}{3} + \frac{2 πn}{3} + \frac{π}{9}

Group like terms

= - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}

Solve

3 x + 1 = \frac{5 π}{3} + 2 πn : x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

3 x + 1 = \frac{5 π}{3} + 2 πn

Move

1

to the right side

3 x + 1 = \frac{5 π}{3} + 2 πn

Subtract

1

from both sides

3 x + 1 - 1 = \frac{5 π}{3} + 2 πn - 1

Simplify

3 x = \frac{5 π}{3} + 2 πn - 1

3 x = \frac{5 π}{3} + 2 πn - 1

Divide both sides by

3

3 x = \frac{5 π}{3} + 2 πn - 1

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3} : - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} - \frac{1}{3}

Group like terms

= - \frac{1}{3} + \frac{2 πn}{3} + \frac{\frac{5 π}{3}}{3}

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= - \frac{1}{3} + \frac{2 πn}{3} + \frac{5 π}{9}

Group like terms

= - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

x = - \frac{1}{3} + \frac{π}{9} + \frac{2 πn}{3}, x = - \frac{1}{3} + \frac{5 π}{9} + \frac{2 πn}{3}

Popular Examples

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tan (x) = \frac{13}{6}

3sinh(2x)=5

3 sinh (2 x) = 5

sin(x/(25.4))=35

sin (\frac{x}{25.4}) = 35

2/3 =(sin(x))/(sin(135))

\frac{2}{3} = \frac{sin ( x )}{sin ( 13 5 ^{\circ} )}

sin(x)+2(cos(x))^2=1

sin (x) + 2 (cos (x))^{2} = 1

Frequently Asked Questions (FAQ)

What is the general solution for 2=4cos(3x+1) ?
The general solution for 2=4cos(3x+1) is x=-1/3+pi/9+(2pin)/3 ,x=-1/3+(5pi)/9+(2pin)/3