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

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

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arcsin(x)-arccos(x)=arcsin(1/2)

Solution

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

Solution

x = \frac{3}{2}

Solution steps

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

a = b \Rightarrow sin (a) = sin (b)

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

Use the following identity:

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

sin (arcsin (x)) cos (arccos (x)) - cos (arcsin (x)) sin (arccos (x)) = sin (arcsin (\frac{1}{2}))

Use the following identity:

sin (arcsin (x)) = x

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}

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

Solve

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

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

Multiply both sides by

2

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

Simplify

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

Expand

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

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

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

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

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

Expand

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

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

Expand

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

- 2 (1 - x^{2})

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= - 2 + 2 x^{2}

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

Simplify

2 x^{2} - 2 + 2 x^{2} : 4 x^{2} - 2

2 x^{2} - 2 + 2 x^{2}

Group like terms

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

Add similar elements:

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

= 4 x^{2} - 2

= 4 x^{2} - 2

= 4 x^{2} - 2

4 x^{2} - 2 = 1

Solve

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

4 x^{2} - 2 = 1

Move

2

to the right side

4 x^{2} - 2 = 1

Add

2

to both sides

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

Simplify

4 x^{2} = 3

4 x^{2} = 3

Divide both sides by

4

4 x^{2} = 3

Divide both sides by

4

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

Simplify

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

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}

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

Verify Solutions:

x = \frac{3}{2}

True

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

True

Check the solutions by plugging them into

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

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

Plug in

x = \frac{3}{2} :

True

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

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

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

Remove parentheses:

(a) = a

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

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

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

1 - (\frac{3}{2})^{2} 1 - (\frac{3}{2})^{2} = \frac{1}{4}

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

Apply radical rule:

a a = a

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

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

Apply rule

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

= \frac{3 - 1}{4}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

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

True

Plug in

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

True

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

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

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

Remove parentheses:

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

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

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

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

Multiply fractions:

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

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

1 - (- \frac{3}{2})^{2} 1 - (- \frac{3}{2})^{2} = \frac{1}{4}

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

Apply radical rule:

a a = a

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

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

Apply rule

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

= \frac{3 - 1}{4}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

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

True

The solutions are

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

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

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

\frac{3}{2} :

True

\frac{3}{2}

Plug in

n = 1

\frac{3}{2}

For

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

plug in

x = \frac{3}{2}

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

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

plug in

x = - \frac{3}{2}

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

Refine

- 3.66519 \dots = 0.52359 \dots

\Rightarrow False

x = \frac{3}{2}

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Frequently Asked Questions (FAQ)

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