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

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

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arccos(x)-arcsin(x)=arccos((sqrt(3))/2)

Solution

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

Solution

x = \frac{1}{2}

Solution steps

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

a = b \Rightarrow cos (a) = cos (b)

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

Use the following identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

cos (arccos (x)) cos (arcsin (x)) + sin (arccos (x)) sin (arcsin (x)) = cos (arccos (\frac{3}{2}))

Use the following identity:

cos (arccos (x)) = x

Use the following identity:

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

Use the following identity:

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

Use the following identity:

sin (arcsin (x)) = x

x 1 - x^{2} + 1 - x^{2} x = \frac{3}{2}

Solve

x 1 - x^{2} + 1 - x^{2} x = \frac{3}{2} : x = \frac{1}{2}, x = \frac{3}{2}

x 1 - x^{2} + 1 - x^{2} x = \frac{3}{2}

Multiply both sides by

2

x 1 - x^{2} \cdot 2 + 1 - x^{2} x \cdot 2 = \frac{3}{2} \cdot 2

Simplify

4 1 - x^{2} x = 3

Square both sides:

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

4 1 - x^{2} x = 3

(4 1 - x^{2} x)^{2} = (3)^{2}

Expand

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

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

Apply exponent rule:

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (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 - x^{2}

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

4^{2} = 16

= 16 (1 - x^{2}) x^{2}

Expand

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

16 (1 - x^{2}) x^{2}

= 16 x^{2} (1 - x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 16 x^{2}, b = 1, c = x^{2}

= 16 x^{2} \cdot 1 - 16 x^{2} x^{2}

= 16 \cdot 1 \cdot x^{2} - 16 x^{2} x^{2}

Simplify

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

16 \cdot 1 \cdot x^{2} - 16 x^{2} x^{2}

16 \cdot 1 \cdot x^{2} = 16 x^{2}

16 \cdot 1 \cdot x^{2}

Multiply the numbers:

16 \cdot 1 = 16

= 16 x^{2}

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

16 x^{2} x^{2}

Apply exponent rule:

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

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

= 16 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 16 x^{4}

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

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

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

Expand

(3)^{2} : 3

(3)^{2}

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

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

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

Solve

16 x^{2} - 16 x^{4} = 3 : x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2}, x = - \frac{3}{2}

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

Move

3

to the left side

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

Subtract

3

from both sides

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

Simplify

16 x^{2} - 16 x^{4} - 3 = 0

16 x^{2} - 16 x^{4} - 3 = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 16 x^{4} + 16 x^{2} - 3 = 0

Rewrite the equation with

u = x^{2}

and

u^{2} = x^{4}

- 16 u^{2} + 16 u - 3 = 0

Solve

- 16 u^{2} + 16 u - 3 = 0 : u = \frac{1}{4}, u = \frac{3}{4}

- 16 u^{2} + 16 u - 3 = 0

Solve with the quadratic formula

- 16 u^{2} + 16 u - 3 = 0

Quadratic Equation Formula:

For

a = - 16, b = 16, c = - 3

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 ( - 16 ) ( - 3 )}{2 ( - 16 )}

u_{1, 2} = \frac{- 16 \pm 1 6 ^{2} - 4 ( - 16 ) ( - 3 )}{2 ( - 16 )}

1 6^{2} - 4 (- 16) (- 3) = 8

1 6^{2} - 4 (- 16) (- 3)

Apply rule

- (- a) = a

= 1 6^{2} - 4 \cdot 16 \cdot 3

Multiply the numbers:

4 \cdot 16 \cdot 3 = 192

= 1 6^{2} - 192

1 6^{2} = 256

= 256 - 192

Subtract the numbers:

256 - 192 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- 16 \pm 8}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- 16 + 8}{2 ( - 16 )}, u_{2} = \frac{- 16 - 8}{2 ( - 16 )}

u = \frac{- 16 + 8}{2 ( - 16 )} : \frac{1}{4}

\frac{- 16 + 8}{2 ( - 16 )}

Remove parentheses:

(- a) = - a

= \frac{- 16 + 8}{- 2 \cdot 16}

Add/Subtract the numbers:

- 16 + 8 = - 8

= \frac{- 8}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{- 8}{- 32}

Apply the fraction rule:

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

= \frac{8}{32}

Cancel the common factor:

8

= \frac{1}{4}

u = \frac{- 16 - 8}{2 ( - 16 )} : \frac{3}{4}

\frac{- 16 - 8}{2 ( - 16 )}

Remove parentheses:

(- a) = - a

= \frac{- 16 - 8}{- 2 \cdot 16}

Subtract the numbers:

- 16 - 8 = - 24

= \frac{- 24}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{- 24}{- 32}

Apply the fraction rule:

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

= \frac{24}{32}

Cancel the common factor:

8

= \frac{3}{4}

The solutions to the quadratic equation are:

u = \frac{1}{4}, u = \frac{3}{4}

u = \frac{1}{4}, u = \frac{3}{4}

Substitute back

u = x^{2},

solve for

x

Solve

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

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}

Solve

x^{2} = \frac{3}{4} : x = \frac{3}{2}, x = - \frac{3}{2}

x^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

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

x = \frac{3}{4}, x = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

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

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

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Apply radical rule:

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

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

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

The solutions are

x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2}, x = - \frac{3}{2}

x = \frac{1}{2}, x = - \frac{1}{2}, x = \frac{3}{2}, x = - \frac{3}{2}

Verify Solutions:

x = \frac{1}{2}

True

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

False

, x = \frac{3}{2}

True

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

False

Check the solutions by plugging them into

x 1 - x^{2} + 1 - x^{2} x = \frac{3}{2}

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

Plug in

x = \frac{1}{2} :

True

(\frac{1}{2}) 1 - (\frac{1}{2})^{2} + 1 - (\frac{1}{2})^{2} (\frac{1}{2}) = \frac{3}{2}

(\frac{1}{2}) 1 - (\frac{1}{2})^{2} + 1 - (\frac{1}{2})^{2} (\frac{1}{2}) = \frac{3}{2}

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

Remove parentheses:

(a) = a

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

Factor out common term

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

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

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

\frac{1}{2} + \frac{1}{2}

Apply rule

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

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

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

(\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}}

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

2^{2} = 4

= 1 - \frac{1}{4}

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}

= \frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

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

True

Plug in

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

False

(- \frac{1}{2}) 1 - (- \frac{1}{2})^{2} + 1 - (- \frac{1}{2})^{2} (- \frac{1}{2}) = \frac{3}{2}

(- \frac{1}{2}) 1 - (- \frac{1}{2})^{2} + 1 - (- \frac{1}{2})^{2} (- \frac{1}{2}) = - \frac{3}{2}

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

Remove parentheses:

(- a) = - a

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

Factor out common term

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

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

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

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

Apply rule

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

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

Subtract the numbers:

- 1 - 1 = - 2

= \frac{- 2}{2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= - 1

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

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

(- \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}}

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

2^{2} = 4

= - - \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}

= - \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

False

Plug in

x = \frac{3}{2} :

True

(\frac{3}{2}) 1 - (\frac{3}{2})^{2} + 1 - (\frac{3}{2})^{2} (\frac{3}{2}) = \frac{3}{2}

(\frac{3}{2}) 1 - (\frac{3}{2})^{2} + 1 - (\frac{3}{2})^{2} (\frac{3}{2}) = \frac{3}{2}

(\frac{3}{2}) 1 - (\frac{3}{2})^{2} + 1 - (\frac{3}{2})^{2} (\frac{3}{2})

Remove parentheses:

(a) = a

= \frac{3}{2} 1 - (\frac{3}{2})^{2} + 1 - (\frac{3}{2})^{2} \frac{3}{2}

Factor out common term

1 - (\frac{3}{2})^{2}

= 1 - (\frac{3}{2})^{2} (\frac{3}{2} + \frac{3}{2})

\frac{3}{2} + \frac{3}{2} = 3

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

Apply rule

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

= \frac{3 + 3}{2}

Factor

3 + 3 : 23

3 + 3

Factor out common term

3

= 3 (1 + 1)

Refine

= 23

= \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3

= 3 - (\frac{3}{2})^{2} + 1

1 - (\frac{3}{2})^{2} = \frac{1}{2}

1 - (\frac{3}{2})^{2}

(\frac{3}{2})^{2} = \frac{3}{4}

(\frac{3}{2})^{2}

Apply exponent rule:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

Join

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

1 - \frac{3}{4}

Convert element to fraction:

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

= \frac{1 \cdot 4}{4} - \frac{3}{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 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

= 3 \frac{1}{2}

Multiply fractions:

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

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

Multiply:

1 \cdot 3 = 3

= \frac{3}{2}

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

True

Plug in

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

False

(- \frac{3}{2}) 1 - (- \frac{3}{2})^{2} + 1 - (- \frac{3}{2})^{2} (- \frac{3}{2}) = \frac{3}{2}

(- \frac{3}{2}) 1 - (- \frac{3}{2})^{2} + 1 - (- \frac{3}{2})^{2} (- \frac{3}{2}) = - \frac{3}{2}

(- \frac{3}{2}) 1 - (- \frac{3}{2})^{2} + 1 - (- \frac{3}{2})^{2} (- \frac{3}{2})

Remove parentheses:

(- a) = - a

= - \frac{3}{2} 1 - (- \frac{3}{2})^{2} - 1 - (- \frac{3}{2})^{2} \frac{3}{2}

Factor out common term

1 - (- \frac{3}{2})^{2}

= 1 - (- \frac{3}{2})^{2} (- \frac{3}{2} - \frac{3}{2})

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

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

Apply rule

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

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

Factor

- 3 - 3 : - 23

- 3 - 3

Factor out common term

3

= - 3 (1 + 1)

Refine

= - 23

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

= - 3 - (- \frac{3}{2})^{2} + 1

1 - (- \frac{3}{2})^{2} = \frac{1}{2}

1 - (- \frac{3}{2})^{2}

(- \frac{3}{2})^{2} = \frac{3}{4}

(- \frac{3}{2})^{2}

Apply exponent rule:

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

n

is even

(- \frac{3}{2})^{2} = (\frac{3}{2})^{2}

= (\frac{3}{2})^{2}

Apply exponent rule:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

Join

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

1 - \frac{3}{4}

Convert element to fraction:

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

= \frac{1 \cdot 4}{4} - \frac{3}{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 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

= - 3 \frac{1}{2}

Multiply fractions:

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

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

Multiply:

1 \cdot 3 = 3

= - \frac{3}{2}

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

False

The solutions are

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

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

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

\frac{1}{2} :

True

\frac{1}{2}

Plug in

n = 1

\frac{1}{2}

For

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

plug in

x = \frac{1}{2}

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

Refine

0.52359 \dots = 0.52359 \dots

\Rightarrow True

Check the solution

\frac{3}{2} :

False

\frac{3}{2}

Plug in

n = 1

\frac{3}{2}

For

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

plug in

x = \frac{3}{2}

arccos (\frac{3}{2}) - arcsin (\frac{3}{2}) = arccos (\frac{3}{2})

Refine

- 0.52359 \dots = 0.52359 \dots

\Rightarrow False

x = \frac{1}{2}

Graph

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Frequently Asked Questions (FAQ)

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