What is the general solution for 3arccos(2x)=2pi ?

The general solution for 3arccos(2x)=2pi is x=-1/4

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3arccos(2x)=2pi

3 arccos (2 x) = 2 π

x = - \frac{1}{4}

Solution steps

3 arccos (2 x) = 2 π

Divide both sides by

3

3 arccos (2 x) = 2 π

Divide both sides by

3

\frac{3 arccos ( 2 x )}{3} = \frac{2 π}{3}

Simplify

arccos (2 x) = \frac{2 π}{3}

arccos (2 x) = \frac{2 π}{3}

Apply trig inverse properties

arccos (2 x) = \frac{2 π}{3}

arccos (x) = a \Rightarrow x = cos (a)

2 x = cos (\frac{2 π}{3})

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

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{1}{2}

= - \frac{1}{2}

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

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

Solve

2 x = - \frac{1}{2} : x = - \frac{1}{4}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{- \frac{1}{2}}{2} : - \frac{1}{4}

\frac{- \frac{1}{2}}{2}

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{1}{4}

x = - \frac{1}{4}

x = - \frac{1}{4}

x = - \frac{1}{4}

x = - \frac{1}{4}

3 cos (x) = 1

cos (a) = \frac{4}{5}

5 - 10 sin^{2} (x) = 0

sec^{2} (θ) - 2 = 0

2 - sin^{2} (x) = 2 cos^{2} (\frac{x}{2})

What is the general solution for 3arccos(2x)=2pi ?
The general solution for 3arccos(2x)=2pi is x=-1/4