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

The general solution for arcsin(x)+arcsin(sqrt(3)x)= pi/2 is x= 1/2

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arcsin(x)+arcsin(sqrt(3)x)= pi/2

Solution

arcsin (x) + arcsin (3 x) = \frac{π}{2}

Solution

x = \frac{1}{2}

Solution steps

arcsin (x) + arcsin (3 x) = \frac{π}{2}

Rewrite using trig identities

arcsin (x) + arcsin (3 x)

Use the Sum to Product identity:

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

= arcsin (x 1 - (3 x)^{2} + 3 x 1 - x^{2})

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

Apply trig inverse properties

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

arcsin (x) = a \Rightarrow x = sin (a)

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

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1

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

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

Solve

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

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

Remove square roots

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

Subtract

3 x 1 - x^{2}

from both sides

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

Simplify

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

Square both sides:

x^{2} - 3 x^{4} = 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

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

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

Expand

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

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

Apply exponent rule:

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

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

(1 - (3 x)^{2})^{2} : 1 - (3 x)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Expand

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

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

(3 x)^{2} = 3 x^{2}

(3 x)^{2}

Apply exponent rule:

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

= (3)^{2} x^{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

= 3 x^{2}

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = x^{2}, b = 1, c = 3 x^{2}

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

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

Simplify

1 \cdot x^{2} - 3 x^{2} x^{2} : x^{2} - 3 x^{4}

1 \cdot x^{2} - 3 x^{2} x^{2}

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

1 \cdot x^{2}

Multiply:

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

= x^{2}

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

3 x^{2} x^{2}

Apply exponent rule:

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

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

= 3 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 3 x^{4}

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

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

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

Expand

(1 - 3 x 1 - x^{2})^{2} : 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

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

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

Apply Perfect Square Formula:

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

a = 1, b = 3 x 1 - x^{2}

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

Simplify

1^{2} - 2 \cdot 1 \cdot 3 x 1 - x^{2} + (3 x 1 - x^{2})^{2} : 1 - 23 1 - x^{2} x + 31 - x^{2} x^{2}

1^{2} - 2 \cdot 1 \cdot 3 x 1 - x^{2} + (3 x 1 - x^{2})^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

2 \cdot 1 \cdot 3 x 1 - x^{2} = 23 1 - x^{2} x

2 \cdot 1 \cdot 3 x 1 - x^{2}

Multiply the numbers:

2 \cdot 1 = 2

= 23 1 - x^{2} x

(3 x 1 - x^{2})^{2} = 31 - x^{2} x^{2}

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

Apply exponent rule:

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

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

= 3 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}

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

= 1 - 23 1 - x^{2} x + 3 (1 - x^{2}) x^{2}

= 1 - 23 1 - x^{2} x + 3 (1 - x^{2}) x^{2}

Expand

1 - 23 1 - x^{2} x + 3 (1 - x^{2}) x^{2} : 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

1 - 23 1 - x^{2} x + 3 (1 - x^{2}) x^{2}

= 1 - 23 x 1 - x^{2} + 3 x^{2} (1 - x^{2})

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3 x^{2}, b = 1, c = x^{2}

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

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

Simplify

3 \cdot 1 \cdot x^{2} - 3 x^{2} x^{2} : 3 x^{2} - 3 x^{4}

3 \cdot 1 \cdot x^{2} - 3 x^{2} x^{2}

3 \cdot 1 \cdot x^{2} = 3 x^{2}

3 \cdot 1 \cdot x^{2}

Multiply the numbers:

3 \cdot 1 = 3

= 3 x^{2}

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

3 x^{2} x^{2}

Apply exponent rule:

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

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

= 3 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 3 x^{4}

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

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

= 1 - 23 1 - x^{2} x + 3 x^{2} - 3 x^{4}

= 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

= 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

x^{2} - 3 x^{4} = 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

x^{2} - 3 x^{4} = 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

Subtract

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

from both sides

x^{2} - 3 x^{4} - (3 x^{2} - 3 x^{4}) = 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4} - (3 x^{2} - 3 x^{4})

Simplify

- 2 x^{2} = - 23 1 - x^{2} x + 1

Subtract

1

from both sides

- 2 x^{2} - 1 = - 23 1 - x^{2} x + 1 - 1

Simplify

- 2 x^{2} - 1 = - 23 1 - x^{2} x

Square both sides:

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

x^{2} - 3 x^{4} = 1 - 23 x 1 - x^{2} + 3 x^{2} - 3 x^{4}

(- 2 x^{2} - 1)^{2} = (- 23 1 - x^{2} x)^{2}

Expand

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

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

Apply Perfect Square Formula:

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

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

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Apply rule

- (- a) = a

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

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

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

= 2^{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}

= 2^{2} x^{4}

2^{2} = 4

= 4 x^{4}

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

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

Multiply the numbers:

2 \cdot 2 \cdot 1 = 4

= 4 x^{2}

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

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

Expand

(- 23 1 - x^{2} x)^{2} : 12 x^{2} - 12 x^{4}

(- 23 1 - x^{2} x)^{2}

Apply exponent rule:

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

n

is even

(- 23 1 - x^{2} x)^{2} = (23 1 - x^{2} x)^{2}

= (23 1 - x^{2} x)^{2}

Apply exponent rule:

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

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

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

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

Refine

= 12 (1 - x^{2}) x^{2}

Expand

12 (1 - x^{2}) x^{2} : 12 x^{2} - 12 x^{4}

12 (1 - x^{2}) x^{2}

= 12 x^{2} (1 - x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 12 x^{2}, b = 1, c = x^{2}

= 12 x^{2} \cdot 1 - 12 x^{2} x^{2}

= 12 \cdot 1 \cdot x^{2} - 12 x^{2} x^{2}

Simplify

12 \cdot 1 \cdot x^{2} - 12 x^{2} x^{2} : 12 x^{2} - 12 x^{4}

12 \cdot 1 \cdot x^{2} - 12 x^{2} x^{2}

12 \cdot 1 \cdot x^{2} = 12 x^{2}

12 \cdot 1 \cdot x^{2}

Multiply the numbers:

12 \cdot 1 = 12

= 12 x^{2}

12 x^{2} x^{2} = 12 x^{4}

12 x^{2} x^{2}

Apply exponent rule:

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

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

= 12 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 12 x^{4}

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

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

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

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

Solve

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4} : x = \frac{1}{2}, x = - \frac{1}{2}

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

Move

12 x^{4}

to the left side

4 x^{4} + 4 x^{2} + 1 = 12 x^{2} - 12 x^{4}

Add

12 x^{4}

to both sides

4 x^{4} + 4 x^{2} + 1 + 12 x^{4} = 12 x^{2} - 12 x^{4} + 12 x^{4}

Simplify

16 x^{4} + 4 x^{2} + 1 = 12 x^{2}

16 x^{4} + 4 x^{2} + 1 = 12 x^{2}

Move

12 x^{2}

to the left side

16 x^{4} + 4 x^{2} + 1 = 12 x^{2}

Subtract

12 x^{2}

from both sides

16 x^{4} + 4 x^{2} + 1 - 12 x^{2} = 12 x^{2} - 12 x^{2}

Simplify

16 x^{4} - 8 x^{2} + 1 = 0

16 x^{4} - 8 x^{2} + 1 = 0

Rewrite the equation with

u = x^{2}

and

u^{2} = x^{4}

16 u^{2} - 8 u + 1 = 0

Solve

16 u^{2} - 8 u + 1 = 0 : u = \frac{1}{4}

16 u^{2} - 8 u + 1 = 0

Solve with the quadratic formula

16 u^{2} - 8 u + 1 = 0

Quadratic Equation Formula:

For

a = 16, b = - 8, c = 1

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 16 \cdot 1}{2 \cdot 16}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 16 \cdot 1}{2 \cdot 16}

(- 8)^{2} - 4 \cdot 16 \cdot 1 = 0

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

Apply exponent rule:

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

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 16 \cdot 1

Multiply the numbers:

4 \cdot 16 \cdot 1 = 64

= 8^{2} - 64

8^{2} = 64

= 64 - 64

Subtract the numbers:

64 - 64 = 0

= 0

u_{1, 2} = \frac{- ( - 8 ) \pm 0}{2 \cdot 16}

u = \frac{- ( - 8 )}{2 \cdot 16}

\frac{- ( - 8 )}{2 \cdot 16} = \frac{1}{4}

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8}{32}

Cancel the common factor:

8

= \frac{1}{4}

u = \frac{1}{4}

The solution to the quadratic equation is:

u = \frac{1}{4}

u = \frac{1}{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}

The solutions are

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}

False

Check the solutions by plugging them into

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

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

Plug in

x = \frac{1}{2} :

True

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

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

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

Remove parentheses:

(a) = a

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

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

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

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

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

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

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

Multiply

3 \frac{1}{2} : \frac{3}{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})^{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:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

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

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

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

(\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 - \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:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

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

Multiply fractions:

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

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

1 \cdot 33 = 3

1 \cdot 33

Apply radical rule:

a a = a

33 = 3

= 1 \cdot 3

Multiply the numbers:

1 \cdot 3 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

= \frac{1}{4} + \frac{3}{4}

Apply rule

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

= \frac{1 + 3}{4}

Add the numbers:

1 + 3 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

1 = 1

True

Plug in

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

False

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

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

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

Remove parentheses:

(- a) = - a

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

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

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

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

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

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

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

Multiply

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

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

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

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

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

(- \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 - \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:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

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

Multiply fractions:

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

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

1 \cdot 33 = 3

1 \cdot 33

Apply radical rule:

a a = a

33 = 3

= 1 \cdot 3

Multiply the numbers:

1 \cdot 3 = 3

= 3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

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

Apply rule

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

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

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

- 1 = 1

False

The solution is

x = \frac{1}{2}

x = \frac{1}{2}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arcsin (x) + arcsin (3 x) = \frac{π}{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

arcsin (x) + arcsin (3 x) = \frac{π}{2}

plug in

x = \frac{1}{2}

arcsin (\frac{1}{2}) + arcsin (3 \frac{1}{2}) = \frac{π}{2}

Refine

1.57079 \dots = 1.57079 \dots

\Rightarrow True

x = \frac{1}{2}

Popular Examples

cos(x)sin(x)= 1/2

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

(cos(x)-3)(cos(x)+1)=0

(cos (x) - 3) (cos (x) + 1) = 0

4tan^2(x)-16tan(x)+7=0

4 tan^{2} (x) - 16 tan (x) + 7 = 0

1-cos(x)=0.5

1 - cos (x) = 0.5

arctan(100x)-arctan(x)=45

arctan (100 x) - arctan (x) = 4 5^{\circ}

Frequently Asked Questions (FAQ)

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