What is the general solution for 3cos(2x+1)=2 ?

The general solution for 3cos(2x+1)=2 is x=pin-1/2+(0.84106…)/2 ,x=pi+pin-1/2-(0.84106…)/2

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Popular >

3cos(2x+1)=2

Solution

3 cos (2 x + 1) = 2

Solution

x = πn - \frac{1}{2} + \frac{0.84106 \dots}{2}, x = π + πn - \frac{1}{2} - \frac{0.84106 \dots}{2}

Degrees

x = - 4.55304 \dots^{\circ} + 18 0^{\circ} n, x = 127.25726 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 cos (2 x + 1) = 2

Divide both sides by

3

3 cos (2 x + 1) = 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

2 x + 1 = arccos (\frac{2}{3}) + 2 πn, 2 x + 1 = 2 π - arccos (\frac{2}{3}) + 2 πn

2 x + 1 = arccos (\frac{2}{3}) + 2 πn, 2 x + 1 = 2 π - arccos (\frac{2}{3}) + 2 πn

Solve

2 x + 1 = arccos (\frac{2}{3}) + 2 πn : x = πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

2 x + 1 = arccos (\frac{2}{3}) + 2 πn

Move

1

to the right side

2 x + 1 = arccos (\frac{2}{3}) + 2 πn

Subtract

1

from both sides

2 x + 1 - 1 = arccos (\frac{2}{3}) + 2 πn - 1

Simplify

2 x = arccos (\frac{2}{3}) + 2 πn - 1

2 x = arccos (\frac{2}{3}) + 2 πn - 1

Divide both sides by

2

2 x = arccos (\frac{2}{3}) + 2 πn - 1

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{arccos ( \frac{2}{3} )}{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 ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2} : πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

\frac{arccos ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2}

Group like terms

= - \frac{1}{2} + \frac{2 πn}{2} + \frac{arccos ( \frac{2}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{1}{2} + πn + \frac{arccos ( \frac{2}{3} )}{2}

Group like terms

= πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

x = πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

x = πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

x = πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}

Solve

2 x + 1 = 2 π - arccos (\frac{2}{3}) + 2 πn : x = π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

2 x + 1 = 2 π - arccos (\frac{2}{3}) + 2 πn

Move

1

to the right side

2 x + 1 = 2 π - arccos (\frac{2}{3}) + 2 πn

Subtract

1

from both sides

2 x + 1 - 1 = 2 π - arccos (\frac{2}{3}) + 2 πn - 1

Simplify

2 x = 2 π - arccos (\frac{2}{3}) + 2 πn - 1

2 x = 2 π - arccos (\frac{2}{3}) + 2 πn - 1

Divide both sides by

2

2 x = 2 π - arccos (\frac{2}{3}) + 2 πn - 1

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{2}{3} )}{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 ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2} : π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

\frac{2 π}{2} - \frac{arccos ( \frac{2}{3} )}{2} + \frac{2 πn}{2} - \frac{1}{2}

Group like terms

= - \frac{1}{2} + \frac{2 π}{2} + \frac{2 πn}{2} - \frac{arccos ( \frac{2}{3} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{1}{2} + π + πn - \frac{arccos ( \frac{2}{3} )}{2}

Group like terms

= π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

x = π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

x = π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

x = π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

x = πn - \frac{1}{2} + \frac{arccos ( \frac{2}{3} )}{2}, x = π + πn - \frac{1}{2} - \frac{arccos ( \frac{2}{3} )}{2}

Show solutions in decimal form

x = πn - \frac{1}{2} + \frac{0.84106 \dots}{2}, x = π + πn - \frac{1}{2} - \frac{0.84106 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(2x+1)=2 ?
The general solution for 3cos(2x+1)=2 is x=pin-1/2+(0.84106…)/2 ,x=pi+pin-1/2-(0.84106…)/2