What is the general solution for 8cos(2x-5)=3 ?

The general solution for 8cos(2x-5)=3 is x=pin+5/2+(1.18639…)/2 ,x=pi+pin+5/2-(1.18639…)/2

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8cos(2x-5)=3

Solution

8 cos (2 x - 5) = 3

Solution

x = πn + \frac{5}{2} + \frac{1.18639 \dots}{2}, x = π + πn + \frac{5}{2} - \frac{1.18639 \dots}{2}

Degrees

x = 177.22729 \dots^{\circ} + 18 0^{\circ} n, x = 289.25160 \dots^{\circ} + 18 0^{\circ} n

Solution steps

8 cos (2 x - 5) = 3

Divide both sides by

8

8 cos (2 x - 5) = 3

Divide both sides by

8

\frac{8 cos ( 2 x - 5 )}{8} = \frac{3}{8}

Simplify

cos (2 x - 5) = \frac{3}{8}

cos (2 x - 5) = \frac{3}{8}

Apply trig inverse properties

cos (2 x - 5) = \frac{3}{8}

General solutions for

cos (2 x - 5) = \frac{3}{8}

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

2 x - 5 = arccos (\frac{3}{8}) + 2 πn, 2 x - 5 = 2 π - arccos (\frac{3}{8}) + 2 πn

2 x - 5 = arccos (\frac{3}{8}) + 2 πn, 2 x - 5 = 2 π - arccos (\frac{3}{8}) + 2 πn

Solve

2 x - 5 = arccos (\frac{3}{8}) + 2 πn : x = πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

2 x - 5 = arccos (\frac{3}{8}) + 2 πn

Move

5

to the right side

2 x - 5 = arccos (\frac{3}{8}) + 2 πn

Add

5

to both sides

2 x - 5 + 5 = arccos (\frac{3}{8}) + 2 πn + 5

Simplify

2 x = arccos (\frac{3}{8}) + 2 πn + 5

2 x = arccos (\frac{3}{8}) + 2 πn + 5

Divide both sides by

2

2 x = arccos (\frac{3}{8}) + 2 πn + 5

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2}

Simplify

\frac{2 x}{2} = \frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2} : πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

\frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2}

Group like terms

= \frac{5}{2} + \frac{2 πn}{2} + \frac{arccos ( \frac{3}{8} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{5}{2} + πn + \frac{arccos ( \frac{3}{8} )}{2}

Group like terms

= πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

x = πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

x = πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

x = πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}

Solve

2 x - 5 = 2 π - arccos (\frac{3}{8}) + 2 πn : x = π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

2 x - 5 = 2 π - arccos (\frac{3}{8}) + 2 πn

Move

5

to the right side

2 x - 5 = 2 π - arccos (\frac{3}{8}) + 2 πn

Add

5

to both sides

2 x - 5 + 5 = 2 π - arccos (\frac{3}{8}) + 2 πn + 5

Simplify

2 x = 2 π - arccos (\frac{3}{8}) + 2 πn + 5

2 x = 2 π - arccos (\frac{3}{8}) + 2 πn + 5

Divide both sides by

2

2 x = 2 π - arccos (\frac{3}{8}) + 2 πn + 5

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{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{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2} : π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

\frac{2 π}{2} - \frac{arccos ( \frac{3}{8} )}{2} + \frac{2 πn}{2} + \frac{5}{2}

Group like terms

= \frac{5}{2} + \frac{2 π}{2} + \frac{2 πn}{2} - \frac{arccos ( \frac{3}{8} )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{5}{2} + π + πn - \frac{arccos ( \frac{3}{8} )}{2}

Group like terms

= π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

x = π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

x = π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

x = π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

x = πn + \frac{5}{2} + \frac{arccos ( \frac{3}{8} )}{2}, x = π + πn + \frac{5}{2} - \frac{arccos ( \frac{3}{8} )}{2}

Show solutions in decimal form

x = πn + \frac{5}{2} + \frac{1.18639 \dots}{2}, x = π + πn + \frac{5}{2} - \frac{1.18639 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for 8cos(2x-5)=3 ?
The general solution for 8cos(2x-5)=3 is x=pin+5/2+(1.18639…)/2 ,x=pi+pin+5/2-(1.18639…)/2