What is the general solution for arccosh(x)=(2pi)/5 ?

The general solution for arccosh(x)=(2pi)/5 is No Solution for x\in\mathbb{R}

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Popular >

arccosh(x)=(2pi)/5

Solution

arccosh (x) = \frac{2 π}{5}

Solution

No Solution for x \in R

Solution steps

arccosh (x) = \frac{2 π}{5}

Subtract

\frac{2 π}{5}

from both sides

arccosh (x) - \frac{2 π}{5} = 0

Simplify

arccosh (x) - \frac{2 π}{5} : \frac{5 arccosh ( x ) - 2 π}{5}

arccosh (x) - \frac{2 π}{5}

Convert element to fraction:

arccosh (x) = \frac{arccosh ( x ) 5}{5}

= \frac{arccosh ( x ) \cdot 5}{5} - \frac{2 π}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{arccosh ( x ) \cdot 5 - 2 π}{5}

\frac{5 arccosh ( x ) - 2 π}{5} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

5 arccosh (x) - 2 π = 0

Solve by substitution

5 arccosh (x) - 2 π = 0

Let:

arccosh (x) = u

5 u - 2 π = 0

5 u - 2 π = 0 : u = \frac{2 π}{5}

5 u - 2 π = 0

Move

2 π

to the right side

5 u - 2 π = 0

Add

2 π

to both sides

5 u - 2 π + 2 π = 0 + 2 π

Simplify

5 u = 2 π

5 u = 2 π

Divide both sides by

5

5 u = 2 π

Divide both sides by

5

\frac{5 u}{5} = \frac{2 π}{5}

Simplify

u = \frac{2 π}{5}

u = \frac{2 π}{5}

Substitute back

u = arccosh (x)

arccosh (x) = \frac{2 π}{5}

arccosh (x) = \frac{2 π}{5}

Rewrite using trig identities

- 2 π + 5 arccosh (x)

Use the Hyperbolic identity:

arccosh (x) = ln (x + x^{2} - 1)

= - 2 π + 5 ln (x + x^{2} - 1)

- 2 π + 5 ln (- 1 + x^{2} + x) = 0

Move

2 π

to the right side

- 2 π + 5 ln (- 1 + x^{2} + x) = 0

Add

2 π

to both sides

- 2 π + 5 ln (- 1 + x^{2} + x) + 2 π = 0 + 2 π

Simplify

5 ln (- 1 + x^{2} + x) = 2 π

5 ln (- 1 + x^{2} + x) = 2 π

Divide both sides by

5

5 ln (- 1 + x^{2} + x) = 2 π

Divide both sides by

5

\frac{5 ln ( - 1 + x ^{2} + x )}{5} = \frac{2 π}{5}

Simplify

ln (- 1 + x^{2} + x) = \frac{2 π}{5}

ln (- 1 + x^{2} + x) = \frac{2 π}{5}

Apply trig inverse properties

ln (- 1 + x^{2} + x) = \frac{2 π}{5}

General solutions for

ln (- 1 + x^{2} + x) = \frac{2 π}{5}

Solve

No Solution for

x \in R

Remove square roots

Subtract

x

from both sides

Simplify

- 1 + x^{2} = Undefined

Square both sides:

- 1 + x^{2} =

Undefined

^{2}

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

Expand

(- 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}

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

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

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

Solve

- 1 + x^{2} =

Undefined

^{2} :

No Solution for

x \in R

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Therefore

No Solution for x \in R

No Solution for x \in R

Solve

No Solution for

x \in R

Remove square roots

Subtract

x

from both sides

Simplify

- 1 + x^{2} = Undefined

Square both sides:

- 1 + x^{2} =

Undefined

^{2}

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

Expand

(- 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}

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

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

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

Solve

- 1 + x^{2} =

Undefined

^{2} :

No Solution for

x \in R

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Therefore

No Solution for x \in R

No Solution for x \in R

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for arccosh(x)=(2pi)/5 ?
The general solution for arccosh(x)=(2pi)/5 is No Solution for x\in\mathbb{R}