What is the general solution for cos(3x-pi/8)=(sqrt(2))/2 ?

The general solution for cos(3x-pi/8)=(sqrt(2))/2 is x=(2pin)/3+pi/8 ,x=(2pin)/3+(5pi)/8

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cos(3x-pi/8)=(sqrt(2))/2

Solution

cos (3 x - \frac{π}{8}) = \frac{2}{2}

Solution

x = \frac{2 πn}{3} + \frac{π}{8}, x = \frac{2 πn}{3} + \frac{5 π}{8}

Degrees

x = 22. 5^{\circ} + 12 0^{\circ} n, x = 112. 5^{\circ} + 12 0^{\circ} n

Solution steps

cos (3 x - \frac{π}{8}) = \frac{2}{2}

General solutions for

cos (3 x - \frac{π}{8}) = \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}{8} = \frac{π}{4} + 2 πn, 3 x - \frac{π}{8} = \frac{7 π}{4} + 2 πn

3 x - \frac{π}{8} = \frac{π}{4} + 2 πn, 3 x - \frac{π}{8} = \frac{7 π}{4} + 2 πn

Solve

3 x - \frac{π}{8} = \frac{π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{π}{8}

3 x - \frac{π}{8} = \frac{π}{4} + 2 πn

Move

\frac{π}{8}

to the right side

3 x - \frac{π}{8} = \frac{π}{4} + 2 πn

Add

\frac{π}{8}

to both sides

3 x - \frac{π}{8} + \frac{π}{8} = \frac{π}{4} + 2 πn + \frac{π}{8}

Simplify

3 x - \frac{π}{8} + \frac{π}{8} = \frac{π}{4} + 2 πn + \frac{π}{8}

Simplify

3 x - \frac{π}{8} + \frac{π}{8} : 3 x

3 x - \frac{π}{8} + \frac{π}{8}

Add similar elements:

- \frac{π}{8} + \frac{π}{8} = 0

= 3 x

Simplify

\frac{π}{4} + 2 πn + \frac{π}{8} : 2 πn + \frac{3 π}{8}

\frac{π}{4} + 2 πn + \frac{π}{8}

Group like terms

= 2 πn + \frac{π}{4} + \frac{π}{8}

Least Common Multiplier of

4, 8 : 8

4, 8

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

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

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

4

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

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

8

For

\frac{π}{4} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{8} + \frac{π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + π}{8}

Add similar elements:

2 π + π = 3 π

= 2 πn + \frac{3 π}{8}

3 x = 2 πn + \frac{3 π}{8}

3 x = 2 πn + \frac{3 π}{8}

3 x = 2 πn + \frac{3 π}{8}

Divide both sides by

3

3 x = 2 πn + \frac{3 π}{8}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{3 π}{8}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{3 π}{8}}{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}{8}}{3} : \frac{2 πn}{3} + \frac{π}{8}

\frac{2 πn}{3} + \frac{\frac{3 π}{8}}{3}

\frac{\frac{3 π}{8}}{3} = \frac{π}{8}

\frac{\frac{3 π}{8}}{3}

Apply the fraction rule:

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

= \frac{3 π}{8 \cdot 3}

Multiply the numbers:

8 \cdot 3 = 24

= \frac{3 π}{24}

Cancel the common factor:

3

= \frac{π}{8}

= \frac{2 πn}{3} + \frac{π}{8}

x = \frac{2 πn}{3} + \frac{π}{8}

x = \frac{2 πn}{3} + \frac{π}{8}

x = \frac{2 πn}{3} + \frac{π}{8}

Solve

3 x - \frac{π}{8} = \frac{7 π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{5 π}{8}

3 x - \frac{π}{8} = \frac{7 π}{4} + 2 πn

Move

\frac{π}{8}

to the right side

3 x - \frac{π}{8} = \frac{7 π}{4} + 2 πn

Add

\frac{π}{8}

to both sides

3 x - \frac{π}{8} + \frac{π}{8} = \frac{7 π}{4} + 2 πn + \frac{π}{8}

Simplify

3 x - \frac{π}{8} + \frac{π}{8} = \frac{7 π}{4} + 2 πn + \frac{π}{8}

Simplify

3 x - \frac{π}{8} + \frac{π}{8} : 3 x

3 x - \frac{π}{8} + \frac{π}{8}

Add similar elements:

- \frac{π}{8} + \frac{π}{8} = 0

= 3 x

Simplify

\frac{7 π}{4} + 2 πn + \frac{π}{8} : 2 πn + \frac{15 π}{8}

\frac{7 π}{4} + 2 πn + \frac{π}{8}

Group like terms

= 2 πn + \frac{π}{8} + \frac{7 π}{4}

Least Common Multiplier of

8, 4 : 8

8, 4

Least Common Multiplier (LCM)

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 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

8

4

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

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

8

For

\frac{7 π}{4} :

multiply the denominator and numerator by

2

\frac{7 π}{4} = \frac{7 π 2}{4 \cdot 2} = \frac{14 π}{8}

= \frac{π}{8} + \frac{14 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 14 π}{8}

Add similar elements:

π + 14 π = 15 π

= 2 πn + \frac{15 π}{8}

3 x = 2 πn + \frac{15 π}{8}

3 x = 2 πn + \frac{15 π}{8}

3 x = 2 πn + \frac{15 π}{8}

Divide both sides by

3

3 x = 2 πn + \frac{15 π}{8}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{15 π}{8}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{15 π}{8}}{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{15 π}{8}}{3} : \frac{2 πn}{3} + \frac{5 π}{8}

\frac{2 πn}{3} + \frac{\frac{15 π}{8}}{3}

\frac{\frac{15 π}{8}}{3} = \frac{5 π}{8}

\frac{\frac{15 π}{8}}{3}

Apply the fraction rule:

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

= \frac{15 π}{8 \cdot 3}

Multiply the numbers:

8 \cdot 3 = 24

= \frac{15 π}{24}

Cancel the common factor:

3

= \frac{5 π}{8}

= \frac{2 πn}{3} + \frac{5 π}{8}

x = \frac{2 πn}{3} + \frac{5 π}{8}

x = \frac{2 πn}{3} + \frac{5 π}{8}

x = \frac{2 πn}{3} + \frac{5 π}{8}

x = \frac{2 πn}{3} + \frac{π}{8}, x = \frac{2 πn}{3} + \frac{5 π}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(3x-pi/8)=(sqrt(2))/2 ?
The general solution for cos(3x-pi/8)=(sqrt(2))/2 is x=(2pin)/3+pi/8 ,x=(2pin)/3+(5pi)/8