What is the general solution for cos(2+3*x)-cos(-0.5)=0 ?

The general solution for cos(2+3*x)-cos(-0.5)=0 is x=(2pin)/3-1/2 ,x=-5/6+(2pi)/3+(2pin)/3

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cos(2+3*x)-cos(-0.5)=0

Solution

cos (2 + 3 \cdot x) - cos (- 0.5) = 0

Solution

x = \frac{2 πn}{3} - \frac{1}{2}, x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

Degrees

x = - 28.64788 \dots^{\circ} + 12 0^{\circ} n, x = 72.25351 \dots^{\circ} + 12 0^{\circ} n

Solution steps

cos (2 + 3 x) - cos (- 0.5) = 0

cos (- 0.5) = cos (\frac{1}{2})

cos (- 0.5)

= cos (- \frac{1}{2})

Use the following property:

cos (- x) = cos (x)

cos (- \frac{1}{2}) = cos (\frac{1}{2})

= cos (\frac{1}{2})

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

Move

cos (\frac{1}{2})

to the right side

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

Add

cos (\frac{1}{2})

to both sides

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

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

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

Simplify

2 + 3 x - 2 : 3 x

2 + 3 x - 2

Add similar elements:

2 - 2 = 0

= 3 x

Simplify

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

arccos (cos (\frac{1}{2})) + 2 πn - 2

arccos (cos (\frac{1}{2})) = \frac{1}{2}

arccos (cos (\frac{1}{2}))

For

0 \leq x \leq π, a rccos (cos (x)) = x

0 \leq \frac{1}{2} \leq π

= \frac{1}{2}

= \frac{1}{2} + 2 πn - 2

Combine the fractions

- 2 + \frac{1}{2} : - \frac{3}{2}

- 2 + \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= - \frac{2 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 \cdot 2 + 1}{2}

- 2 \cdot 2 + 1 = - 3

- 2 \cdot 2 + 1

Multiply the numbers:

2 \cdot 2 = 4

= - 4 + 1

Add/Subtract the numbers:

- 4 + 1 = - 3

= - 3

= \frac{- 3}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{3}{2}

= 2 πn - \frac{3}{2}

3 x = 2 πn - \frac{3}{2}

3 x = 2 πn - \frac{3}{2}

3 x = 2 πn - \frac{3}{2}

Divide both sides by

3

3 x = 2 πn - \frac{3}{2}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} - \frac{\frac{3}{2}}{3} : \frac{2 πn}{3} - \frac{1}{2}

\frac{2 πn}{3} - \frac{\frac{3}{2}}{3}

\frac{\frac{3}{2}}{3} = \frac{1}{2}

\frac{\frac{3}{2}}{3}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

= \frac{2 πn}{3} - \frac{1}{2}

x = \frac{2 πn}{3} - \frac{1}{2}

x = \frac{2 πn}{3} - \frac{1}{2}

x = \frac{2 πn}{3} - \frac{1}{2}

Solve

2 + 3 x = 2 π - arccos (cos (\frac{1}{2})) + 2 πn : x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

Simplify

2 + 3 x - 2 : 3 x

2 + 3 x - 2

Add similar elements:

2 - 2 = 0

= 3 x

Simplify

2 π - arccos (cos (\frac{1}{2})) + 2 πn - 2 : 2 π + 2 πn - \frac{5}{2}

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

arccos (cos (\frac{1}{2})) = \frac{1}{2}

arccos (cos (\frac{1}{2}))

For

0 \leq x \leq π, a rccos (cos (x)) = x

0 \leq \frac{1}{2} \leq π

= \frac{1}{2}

= 2 π - \frac{1}{2} + 2 πn - 2

Combine the fractions

- 2 - \frac{1}{2} : - \frac{5}{2}

- 2 - \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= - \frac{2 \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 \cdot 2 - 1}{2}

- 2 \cdot 2 - 1 = - 5

- 2 \cdot 2 - 1

Multiply the numbers:

2 \cdot 2 = 4

= - 4 - 1

Subtract the numbers:

- 4 - 1 = - 5

= - 5

= \frac{- 5}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{5}{2}

= 2 π + 2 πn - \frac{5}{2}

3 x = 2 π + 2 πn - \frac{5}{2}

3 x = 2 π + 2 πn - \frac{5}{2}

3 x = 2 π + 2 πn - \frac{5}{2}

Divide both sides by

3

3 x = 2 π + 2 πn - \frac{5}{2}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{2 πn}{3} - \frac{\frac{5}{2}}{3}

Simplify

\frac{3 x}{3} = \frac{2 π}{3} + \frac{2 πn}{3} - \frac{\frac{5}{2}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 π}{3} + \frac{2 πn}{3} - \frac{\frac{5}{2}}{3} : - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

\frac{2 π}{3} + \frac{2 πn}{3} - \frac{\frac{5}{2}}{3}

\frac{\frac{5}{2}}{3} = \frac{5}{6}

\frac{\frac{5}{2}}{3}

Apply the fraction rule:

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

= \frac{5}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5}{6}

= \frac{2 π}{3} + \frac{2 πn}{3} - \frac{5}{6}

Group like terms

= - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x = \frac{2 πn}{3} - \frac{1}{2}, x = - \frac{5}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2+3*x)-cos(-0.5)=0 ?
The general solution for cos(2+3*x)-cos(-0.5)=0 is x=(2pin)/3-1/2 ,x=-5/6+(2pi)/3+(2pin)/3