What is the general solution for cos(2x-1)=0.4 ?

The general solution for cos(2x-1)=0.4 is x=pin+1/2+(1.15927…)/2 ,x=pi+pin+1/2-(1.15927…)/2

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cos(2x-1)=0.4

Solution

cos (2 x - 1) = 0.4

Solution

x = πn + \frac{1}{2} + \frac{1.15927 \dots}{2}, x = π + πn + \frac{1}{2} - \frac{1.15927 \dots}{2}

Degrees

x = 61.85880 \dots^{\circ} + 18 0^{\circ} n, x = 175.43697 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cos (2 x - 1) = 0.4

Apply trig inverse properties

cos (2 x - 1) = 0.4

General solutions for

cos (2 x - 1) = 0.4

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

2 x - 1 = arccos (0.4) + 2 πn, 2 x - 1 = 2 π - arccos (0.4) + 2 πn

2 x - 1 = arccos (0.4) + 2 πn, 2 x - 1 = 2 π - arccos (0.4) + 2 πn

Solve

2 x - 1 = arccos (0.4) + 2 πn : x = πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

2 x - 1 = arccos (0.4) + 2 πn

Move

1

to the right side

2 x - 1 = arccos (0.4) + 2 πn

Add

1

to both sides

2 x - 1 + 1 = arccos (0.4) + 2 πn + 1

Simplify

2 x = arccos (0.4) + 2 πn + 1

2 x = arccos (0.4) + 2 πn + 1

Divide both sides by

2

2 x = arccos (0.4) + 2 πn + 1

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2} : πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

\frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Group like terms

= \frac{1}{2} + \frac{2 πn}{2} + \frac{arccos ( 0.4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{2} + πn + \frac{arccos ( 0.4 )}{2}

Group like terms

= πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

x = πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

x = πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

x = πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}

Solve

2 x - 1 = 2 π - arccos (0.4) + 2 πn : x = π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

2 x - 1 = 2 π - arccos (0.4) + 2 πn

Move

1

to the right side

2 x - 1 = 2 π - arccos (0.4) + 2 πn

Add

1

to both sides

2 x - 1 + 1 = 2 π - arccos (0.4) + 2 πn + 1

Simplify

2 x = 2 π - arccos (0.4) + 2 πn + 1

2 x = 2 π - arccos (0.4) + 2 πn + 1

Divide both sides by

2

2 x = 2 π - arccos (0.4) + 2 πn + 1

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} - \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2} : π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

\frac{2 π}{2} - \frac{arccos ( 0.4 )}{2} + \frac{2 πn}{2} + \frac{1}{2}

Group like terms

= \frac{1}{2} + \frac{2 π}{2} + \frac{2 πn}{2} - \frac{arccos ( 0.4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{2} + π + πn - \frac{arccos ( 0.4 )}{2}

Group like terms

= π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

x = π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

x = π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

x = π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

x = πn + \frac{1}{2} + \frac{arccos ( 0.4 )}{2}, x = π + πn + \frac{1}{2} - \frac{arccos ( 0.4 )}{2}

Show solutions in decimal form

x = πn + \frac{1}{2} + \frac{1.15927 \dots}{2}, x = π + πn + \frac{1}{2} - \frac{1.15927 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x-1)=0.4 ?
The general solution for cos(2x-1)=0.4 is x=pin+1/2+(1.15927…)/2 ,x=pi+pin+1/2-(1.15927…)/2