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

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

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arccos(2x)-arccos(x)= pi/3

Solution

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

Solution

x = - \frac{1}{2}

Solution steps

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

Rewrite using trig identities

arccos (2 x) - arccos (x)

Use the Sum to Product identity:

arccos (s) - arccos (t) = arccos (s t + (1 - s^{2}) (1 - t^{2}))

= arccos (2 xx + (1 - (2 x)^{2}) (1 - x^{2}))

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

Apply trig inverse properties

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

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

2 xx + (1 - (2 x)^{2}) (1 - x^{2}) = 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 xx + (1 - (2 x)^{2}) (1 - x^{2}) = \frac{1}{2}

2 xx + (1 - (2 x)^{2}) (1 - x^{2}) = \frac{1}{2}

Solve

2 xx + (1 - (2 x)^{2}) (1 - x^{2}) = \frac{1}{2} : x = \frac{1}{2}, x = - \frac{1}{2}

2 xx + (1 - (2 x)^{2}) (1 - x^{2}) = \frac{1}{2}

Multiply both sides by

2

2 xx \cdot 2 + (1 - (2 x)^{2}) (1 - x^{2}) \cdot 2 = \frac{1}{2} \cdot 2

Simplify

4 x^{2} + 2 (1 - (2 x)^{2}) (1 - x^{2}) = 1

Remove square roots

4 x^{2} + 2 (1 - (2 x)^{2}) (1 - x^{2}) = 1

Subtract

4 x^{2}

from both sides

4 x^{2} + 2 (1 - (2 x)^{2}) (1 - x^{2}) - 4 x^{2} = 1 - 4 x^{2}

Simplify

2 (1 - (2 x)^{2}) (1 - x^{2}) = 1 - 4 x^{2}

Square both sides:

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4 x^{2} + 2 (1 - (2 x)^{2}) (1 - x^{2}) = 1

(2 (1 - (2 x)^{2}) (1 - x^{2}))^{2} = (1 - 4 x^{2})^{2}

Expand

(2 (1 - (2 x)^{2}) (1 - x^{2}))^{2} : 4 - 20 x^{2} + 16 x^{4}

(2 (1 - (2 x)^{2}) (1 - x^{2}))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} ((1 - (2 x)^{2}) (1 - x^{2}))^{2}

((1 - (2 x)^{2}) (1 - x^{2}))^{2} : (1 - (2 x)^{2}) (1 - x^{2})

Apply radical rule:

a = a^{\frac{1}{2}}

= (((1 - (2 x)^{2}) (1 - x^{2}))^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= ((1 - (2 x)^{2}) (1 - x^{2}))^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= (1 - (2 x)^{2}) (1 - x^{2})

= 2^{2} (1 - (2 x)^{2}) (1 - x^{2})

2^{2} = 4

= 4 (1 - (2 x)^{2}) (1 - x^{2})

Expand

4 (1 - (2 x)^{2}) (1 - x^{2}) : 4 - 20 x^{2} + 16 x^{4}

4 (1 - (2 x)^{2}) (1 - x^{2})

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4 (- 2^{2} x^{2} + 1) (- x^{2} + 1)

2^{2} = 4

= 4 (- 4 x^{2} + 1) (- x^{2} + 1)

Expand

(1 - 4 x^{2}) (1 - x^{2}) : 1 - 5 x^{2} + 4 x^{4}

(1 - 4 x^{2}) (1 - x^{2})

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = - 4 x^{2}, c = 1, d = - x^{2}

= 1 \cdot 1 + 1 \cdot (- x^{2}) + (- 4 x^{2}) \cdot 1 + (- 4 x^{2}) (- x^{2})

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1 \cdot 1 - 1 \cdot x^{2} - 4 \cdot 1 \cdot x^{2} + 4 x^{2} x^{2}

Simplify

1 \cdot 1 - 1 \cdot x^{2} - 4 \cdot 1 \cdot x^{2} + 4 x^{2} x^{2} : 1 - 5 x^{2} + 4 x^{4}

1 \cdot 1 - 1 \cdot x^{2} - 4 \cdot 1 \cdot x^{2} + 4 x^{2} x^{2}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

1 \cdot x^{2} = x^{2}

1 \cdot x^{2}

Multiply:

1 \cdot x^{2} = x^{2}

= x^{2}

4 \cdot 1 \cdot x^{2} = 4 x^{2}

4 \cdot 1 \cdot x^{2}

Multiply the numbers:

4 \cdot 1 = 4

= 4 x^{2}

4 x^{2} x^{2} = 4 x^{4}

4 x^{2} x^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

x^{2} x^{2} = x^{2 + 2}

= 4 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 4 x^{4}

= 1 - x^{2} - 4 x^{2} + 4 x^{4}

Add similar elements:

- x^{2} - 4 x^{2} = - 5 x^{2}

= 1 - 5 x^{2} + 4 x^{4}

= 1 - 5 x^{2} + 4 x^{4}

= 4 (1 - 5 x^{2} + 4 x^{4})

Expand

4 (1 - 5 x^{2} + 4 x^{4}) : 4 - 20 x^{2} + 16 x^{4}

4 (1 - 5 x^{2} + 4 x^{4})

Distribute parentheses

= 4 \cdot 1 + 4 (- 5 x^{2}) + 4 \cdot 4 x^{4}

Apply minus-plus rules

+ (- a) = - a

= 4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

Simplify

4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4} : 4 - 20 x^{2} + 16 x^{4}

4 \cdot 1 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 \cdot 5 x^{2} + 4 \cdot 4 x^{4}

Multiply the numbers:

4 \cdot 5 = 20

= 4 - 20 x^{2} + 4 \cdot 4 x^{4}

Multiply the numbers:

4 \cdot 4 = 16

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

= 4 - 20 x^{2} + 16 x^{4}

Expand

(1 - 4 x^{2})^{2} : 1 - 8 x^{2} + 16 x^{4}

(1 - 4 x^{2})^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = 4 x^{2}

= 1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

Simplify

1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2} : 1 - 8 x^{2} + 16 x^{4}

1^{2} - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 2 \cdot 1 \cdot 4 x^{2} + (4 x^{2})^{2}

2 \cdot 1 \cdot 4 x^{2} = 8 x^{2}

2 \cdot 1 \cdot 4 x^{2}

Multiply the numbers:

2 \cdot 1 \cdot 4 = 8

= 8 x^{2}

(4 x^{2})^{2} = 16 x^{4}

(4 x^{2})^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (x^{2})^{2}

(x^{2})^{2} : x^{4}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= x^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= x^{4}

= 4^{2} x^{4}

4^{2} = 16

= 16 x^{4}

= 1 - 8 x^{2} + 16 x^{4}

= 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

Solve

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4} : x = \frac{1}{2}, x = - \frac{1}{2}

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

Move

4

to the right side

4 - 20 x^{2} + 16 x^{4} = 1 - 8 x^{2} + 16 x^{4}

Subtract

4

from both sides

4 - 20 x^{2} + 16 x^{4} - 4 = 1 - 8 x^{2} + 16 x^{4} - 4

Simplify

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

Move

8 x^{2}

to the left side

- 20 x^{2} + 16 x^{4} = 16 x^{4} - 8 x^{2} - 3

Add

8 x^{2}

to both sides

- 20 x^{2} + 16 x^{4} + 8 x^{2} = 16 x^{4} - 8 x^{2} - 3 + 8 x^{2}

Simplify

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

Move

16 x^{4}

to the left side

16 x^{4} - 12 x^{2} = 16 x^{4} - 3

Subtract

16 x^{4}

from both sides

16 x^{4} - 12 x^{2} - 16 x^{4} = 16 x^{4} - 3 - 16 x^{4}

Simplify

- 12 x^{2} = - 3

- 12 x^{2} = - 3

Divide both sides by

- 12

- 12 x^{2} = - 3

Divide both sides by

- 12

\frac{- 12 x ^{2}}{- 12} = \frac{- 3}{- 12}

Simplify

\frac{- 12 x ^{2}}{- 12} = \frac{- 3}{- 12}

Simplify

\frac{- 12 x ^{2}}{- 12} : x^{2}

\frac{- 12 x ^{2}}{- 12}

Apply the fraction rule:

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

= \frac{12 x ^{2}}{12}

Divide the numbers:

\frac{12}{12} = 1

= x^{2}

Simplify

\frac{- 3}{- 12} : \frac{1}{4}

\frac{- 3}{- 12}

Apply the fraction rule:

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

= \frac{3}{12}

Cancel the common factor:

3

= \frac{1}{4}

x^{2} = \frac{1}{4}

x^{2} = \frac{1}{4}

x^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{1}{4}, x = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{4}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= \frac{1}{2}

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

- \frac{1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{4}

Apply radical rule:

1 = 1

1 = 1

= - \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= - \frac{1}{2}

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

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

Verify Solutions:

x = \frac{1}{2}

True

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

True

Check the solutions by plugging them into

2 xx + (1 - (2 x)^{2}) (1 - x^{2}) = \frac{1}{2}

Remove the ones that don't agree with the equation.

Plug in

x = \frac{1}{2} :

True

2 (\frac{1}{2}) (\frac{1}{2}) + (1 - (2 (\frac{1}{2}))^{2}) (1 - (\frac{1}{2})^{2}) = \frac{1}{2}

2 (\frac{1}{2}) (\frac{1}{2}) + (1 - (2 (\frac{1}{2}))^{2}) (1 - (\frac{1}{2})^{2}) = \frac{1}{2}

2 (\frac{1}{2}) (\frac{1}{2}) + (1 - (2 (\frac{1}{2}))^{2}) (1 - (\frac{1}{2})^{2})

Remove parentheses:

(a) = a

= 2 \cdot \frac{1}{2} \cdot \frac{1}{2} + (1 - (2 \cdot \frac{1}{2})^{2}) (1 - (\frac{1}{2})^{2})

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{2}

(1 - (2 \cdot \frac{1}{2})^{2}) (1 - (\frac{1}{2})^{2}) = 0

(1 - (2 \cdot \frac{1}{2})^{2}) (1 - (\frac{1}{2})^{2})

(2 \cdot \frac{1}{2})^{2} = 1

(2 \cdot \frac{1}{2})^{2}

Multiply

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 1^{2}

Apply rule

1^{a} = 1

= 1

= (1 - 1) (- (\frac{1}{2})^{2} + 1)

(\frac{1}{2})^{2} = \frac{1}{4}

(\frac{1}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

2^{2} = 4

= \frac{1}{4}

= (1 - 1) (- \frac{1}{4} + 1)

Subtract the numbers:

1 - 1 = 0

= 0 \cdot (- \frac{1}{4} + 1)

Join

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{4}

= 0 \cdot \frac{3}{4}

Apply rule

0 \cdot a = 0

= 0

Apply rule

0 = 0

= 0

= \frac{1}{2} + 0

\frac{1}{2} + 0 = \frac{1}{2}

= \frac{1}{2}

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

True

Plug in

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

True

2 (- \frac{1}{2}) (- \frac{1}{2}) + (1 - (2 (- \frac{1}{2}))^{2}) (1 - (- \frac{1}{2})^{2}) = \frac{1}{2}

2 (- \frac{1}{2}) (- \frac{1}{2}) + (1 - (2 (- \frac{1}{2}))^{2}) (1 - (- \frac{1}{2})^{2}) = \frac{1}{2}

2 (- \frac{1}{2}) (- \frac{1}{2}) + (1 - (2 (- \frac{1}{2}))^{2}) (1 - (- \frac{1}{2})^{2})

Remove parentheses:

(- a) = - a, - (- a) = a

= 2 \cdot \frac{1}{2} \cdot \frac{1}{2} + (1 - (- 2 \cdot \frac{1}{2})^{2}) (1 - (- \frac{1}{2})^{2})

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{2}

(1 - (- 2 \cdot \frac{1}{2})^{2}) (1 - (- \frac{1}{2})^{2}) = 0

(1 - (- 2 \cdot \frac{1}{2})^{2}) (1 - (- \frac{1}{2})^{2})

(- 2 \cdot \frac{1}{2})^{2} = 1

(- 2 \cdot \frac{1}{2})^{2}

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 1

= (- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

= (1 - 1) (- (- \frac{1}{2})^{2} + 1)

(- \frac{1}{2})^{2} = \frac{1}{4}

(- \frac{1}{2})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

2^{2} = 4

= \frac{1}{4}

= (1 - 1) (- \frac{1}{4} + 1)

Subtract the numbers:

1 - 1 = 0

= 0 \cdot (- \frac{1}{4} + 1)

Join

1 - \frac{1}{4} : \frac{3}{4}

1 - \frac{1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{4}

= 0 \cdot \frac{3}{4}

Apply rule

0 \cdot a = 0

= 0

Apply rule

0 = 0

= 0

= \frac{1}{2} + 0

\frac{1}{2} + 0 = \frac{1}{2}

= \frac{1}{2}

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

True

The solutions are

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

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

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

Remove the ones that don't agree with the equation.

Check the solution

\frac{1}{2} :

False

\frac{1}{2}

Plug in

n = 1

\frac{1}{2}

For

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

plug in

x = \frac{1}{2}

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

Refine

- 1.04719 \dots = 1.04719 \dots

\Rightarrow False

Check the solution

- \frac{1}{2} :

True

- \frac{1}{2}

Plug in

n = 1

- \frac{1}{2}

For

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

plug in

x = - \frac{1}{2}

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

Refine

1.04719 \dots = 1.04719 \dots

\Rightarrow True

x = - \frac{1}{2}

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Frequently Asked Questions (FAQ)

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