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Popular >

2(arcsin(x))^2+(2-pi)arcsin(x)-pi=0

Solution

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Solution

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Solution steps

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Solve by substitution

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Let:

arcsin (x) = u

2 u^{2} + (2 - π) u - π = 0

2 u^{2} + (2 - π) u - π = 0 : u = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, u = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

2 u^{2} + (2 - π) u - π = 0

Solve with the quadratic formula

2 u^{2} + (2 - π) u - π = 0

Quadratic Equation Formula:

For

a = 2, b = 2 - π, c = - π

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

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

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

(2 - π)^{2} - 4 \cdot 2 (- π)

Apply rule

- (- a) = a

= (2 - π)^{2} + 4 \cdot 2 π

Multiply the numbers:

4 \cdot 2 = 8

= (- π + 2)^{2} + 8 π

Expand

(2 - π)^{2} + 8 π : π^{2} + 4 π + 4

(2 - π)^{2} + 8 π

(2 - π)^{2} : 4 - 4 π + π^{2}

Apply Perfect Square Formula:

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

a = 2, b = π

= 2^{2} - 2 \cdot 2 π + π^{2}

Simplify

2^{2} - 2 \cdot 2 π + π^{2} : 4 - 4 π + π^{2}

2^{2} - 2 \cdot 2 π + π^{2}

2^{2} = 4

= 4 - 2 \cdot 2 π + π^{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 - 4 π + π^{2}

= 4 - 4 π + π^{2}

= 4 - 4 π + π^{2} + 8 π

Simplify

4 - 4 π + π^{2} + 8 π : π^{2} + 4 π + 4

4 - 4 π + π^{2} + 8 π

Group like terms

= π^{2} - 4 π + 8 π + 4

Add similar elements:

- 4 π + 8 π = 4 π

= π^{2} + 4 π + 4

= π^{2} + 4 π + 4

= π^{2} + 4 π + 4

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

Separate the solutions

u_{1} = \frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2}, u_{2} = \frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2}

u = \frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2} : \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

\frac{- ( 2 - π ) + π ^{2} + 4 π + 4}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( - π + 2 ) + π ^{2} + 4 π + 4}{4}

- (2 - π) : - 2 + π

- (2 - π)

Distribute parentheses

= - (2) - (- π)

Apply minus-plus rules

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

= - 2 + π

= \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

u = \frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2} : \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

\frac{- ( 2 - π ) - π ^{2} + 4 π + 4}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( - π + 2 ) - π ^{2} + 4 π + 4}{4}

- (2 - π) : - 2 + π

- (2 - π)

Distribute parentheses

= - (2) - (- π)

Apply minus-plus rules

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

= - 2 + π

= \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

The solutions to the quadratic equation are:

u = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, u = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

Substitute back

u = arcsin (x)

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

Apply trig inverse properties

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}

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

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

Apply trig inverse properties

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

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

x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Combine all the solutions

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

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