What is the general solution for solvefor x,cos(3x+pi/4)= 1/(sqrt(2)) ?

The general solution for solvefor x,cos(3x+pi/4)= 1/(sqrt(2)) is x=(2pin)/3 ,x=(2pin)/3+pi/2

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solvefor x,cos(3x+pi/4)= 1/(sqrt(2))

Solution

solve for

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

Solution

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

Radians

x = 0 + \frac{2 π}{3} n, x = \frac{π}{2} + \frac{2 π}{3} n

Solution steps

cos (3 x + \frac{π}{4}) = \frac{1}{2}

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

cos (3 x + \frac{π}{4}) = \frac{2}{2}

General solutions for

cos (3 x + \frac{π}{4}) = \frac{2}{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 + \frac{π}{4} = \frac{π}{4} + 2 πn, 3 x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

3 x + \frac{π}{4} = \frac{π}{4} + 2 πn, 3 x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

Solve

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

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Solve

3 x + \frac{π}{4} = \frac{7 π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{π}{2}

3 x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{4} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{4} + 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{7 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{7 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{7 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{- π + 7 π}{4}

Add similar elements:

- π + 7 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

3 x = 2 πn + \frac{3 π}{2}

3 x = 2 πn + \frac{3 π}{2}

3 x = 2 πn + \frac{3 π}{2}

Divide both sides by

3

3 x = 2 πn + \frac{3 π}{2}

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{2 πn}{3} + \frac{π}{2}

x = \frac{2 πn}{3} + \frac{π}{2}

x = \frac{2 πn}{3} + \frac{π}{2}

x = \frac{2 πn}{3} + \frac{π}{2}

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

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Frequently Asked Questions (FAQ)

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