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

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

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arccos(2x)= pi/3

arccos (2 x) = \frac{π}{3}

x = \frac{1}{4}

Solution steps

arccos (2 x) = \frac{π}{3}

Apply trig inverse properties

arccos (2 x) = \frac{π}{3}

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

2 x = cos (\frac{π}{3})

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

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

= \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}

2 tan (x) = 2

43 cos (x) - 8 sin (x) cos (x) = 0

sin (x + \frac{π}{4}) = 2 cos (x)

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

cos (x) - 2 = - 8 + 5

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