What is the general solution for cos((3x-7)/2)=0 ?

The general solution for cos((3x-7)/2)=0 is x= pi/3+(4pin)/3+7/3 ,x= 7/3+pi+(4pin)/3

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cos((3x-7)/2)=0

Solution

cos (\frac{3 x - 7}{2}) = 0

Solution

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

Degrees

x = 193.69015 \dots^{\circ} + 24 0^{\circ} n, x = 313.69015 \dots^{\circ} + 24 0^{\circ} n

Solution steps

cos (\frac{3 x - 7}{2}) = 0

General solutions for

cos (\frac{3 x - 7}{2}) = 0

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}

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 3 x - 7 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 3 x - 7 )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 3 x - 7 )}{2} : 3 x - 7

\frac{2 ( 3 x - 7 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 x - 7

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x - 7 = π + 4 πn

3 x - 7 = π + 4 πn

3 x - 7 = π + 4 πn

Move

7

to the right side

3 x - 7 = π + 4 πn

Add

7

to both sides

3 x - 7 + 7 = π + 4 πn + 7

Simplify

3 x = π + 4 πn + 7

3 x = π + 4 πn + 7

Divide both sides by

3

3 x = π + 4 πn + 7

Divide both sides by

3

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

Simplify

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 ( 3 x - 7 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 3 x - 7 )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 3 x - 7 )}{2} : 3 x - 7

\frac{2 ( 3 x - 7 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 x - 7

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x - 7 = 3 π + 4 πn

3 x - 7 = 3 π + 4 πn

3 x - 7 = 3 π + 4 πn

Move

7

to the right side

3 x - 7 = 3 π + 4 πn

Add

7

to both sides

3 x - 7 + 7 = 3 π + 4 πn + 7

Simplify

3 x = 3 π + 4 πn + 7

3 x = 3 π + 4 πn + 7

Divide both sides by

3

3 x = 3 π + 4 πn + 7

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{3 π}{3} + \frac{4 πn}{3} + \frac{7}{3} : \frac{7}{3} + π + \frac{4 πn}{3}

\frac{3 π}{3} + \frac{4 πn}{3} + \frac{7}{3}

Group like terms

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

Divide the numbers:

\frac{3}{3} = 1

= \frac{7}{3} + π + \frac{4 πn}{3}

x = \frac{7}{3} + π + \frac{4 πn}{3}

x = \frac{7}{3} + π + \frac{4 πn}{3}

x = \frac{7}{3} + π + \frac{4 πn}{3}

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos((3x-7)/2)=0 ?
The general solution for cos((3x-7)/2)=0 is x= pi/3+(4pin)/3+7/3 ,x= 7/3+pi+(4pin)/3