What is the general solution for cos(2/3 x-pi/2)-1=0 ?

The general solution for cos(2/3 x-pi/2)-1=0 is x=3pin+(3pi)/4

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cos(2/3 x-pi/2)-1=0

Solution

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

Solution

x = 3 πn + \frac{3 π}{4}

Degrees

x = 13 5^{\circ} + 54 0^{\circ} n

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

cos (\frac{2}{3} x - \frac{π}{2}) = 1

cos (\frac{2}{3} x - \frac{π}{2}) = 1

General solutions for

cos (\frac{2}{3} x - \frac{π}{2}) = 1

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{2}{3} x - \frac{π}{2} = 0 + 2 πn

\frac{2}{3} x - \frac{π}{2} = 0 + 2 πn

Solve

\frac{2}{3} x - \frac{π}{2} = 0 + 2 πn : x = 3 πn + \frac{3 π}{4}

\frac{2}{3} x - \frac{π}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{2}{3} x - \frac{π}{2} = 2 πn

Move

\frac{π}{2}

to the right side

\frac{2}{3} x - \frac{π}{2} = 2 πn

Add

\frac{π}{2}

to both sides

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

Simplify

\frac{2}{3} x = 2 πn + \frac{π}{2}

\frac{2}{3} x = 2 πn + \frac{π}{2}

Multiply both sides by

3

\frac{2}{3} x = 2 πn + \frac{π}{2}

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{2}{3} x : 2 x

3 \cdot \frac{2}{3} x

Multiply fractions:

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

= \frac{2 \cdot 3}{3} x

Cancel the common factor:

3

= x \cdot 2

Simplify

3 \cdot 2 πn + 3 \cdot \frac{π}{2} : 6 πn + \frac{3 π}{2}

3 \cdot 2 πn + 3 \cdot \frac{π}{2}

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn + 3 \cdot \frac{π}{2}

Multiply

3 \cdot \frac{π}{2} : \frac{3 π}{2}

3 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 3}{2}

= 6 πn + \frac{3 π}{2}

2 x = 6 πn + \frac{3 π}{2}

2 x = 6 πn + \frac{3 π}{2}

2 x = 6 πn + \frac{3 π}{2}

Divide both sides by

2

2 x = 6 πn + \frac{3 π}{2}

Divide both sides by

2

\frac{2 x}{2} = \frac{6 πn}{2} + \frac{\frac{3 π}{2}}{2}

Simplify

\frac{2 x}{2} = \frac{6 πn}{2} + \frac{\frac{3 π}{2}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{6 πn}{2} + \frac{\frac{3 π}{2}}{2} : 3 πn + \frac{3 π}{4}

\frac{6 πn}{2} + \frac{\frac{3 π}{2}}{2}

\frac{6 πn}{2} = 3 πn

\frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 πn

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= 3 πn + \frac{3 π}{4}

x = 3 πn + \frac{3 π}{4}

x = 3 πn + \frac{3 π}{4}

x = 3 πn + \frac{3 π}{4}

x = 3 πn + \frac{3 π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2/3 x-pi/2)-1=0 ?
The general solution for cos(2/3 x-pi/2)-1=0 is x=3pin+(3pi)/4