What is the general solution for sinh(x)= 15/8 ?

The general solution for sinh(x)= 15/8 is x=2ln(2)

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Popular Trigonometry >

sinh(x)= 15/8

Solution

sinh (x) = \frac{15}{8}

Solution

x = 2 ln (2)

Degrees

x = 79.42881 \dots^{\circ}

Solution steps

sinh (x) = \frac{15}{8}

Rewrite using trig identities

sinh (x) = \frac{15}{8}

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8} : x = 2 ln (2)

\frac{e ^{x} - e ^{- x}}{2} = \frac{15}{8}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 8 = 2 \cdot 15

Simplify

(e^{x} - e^{- x}) \cdot 8 = 30

Apply exponent rules

(e^{x} - e^{- x}) \cdot 8 = 30

Apply exponent rule:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 8 = 30

(e^{x} - (e^{x})^{- 1}) \cdot 8 = 30

Rewrite the equation with

e^{x} = u

(u - (u)^{- 1}) \cdot 8 = 30

Solve

(u - u^{- 1}) \cdot 8 = 30 : u = 4, u = - \frac{1}{4}

(u - u^{- 1}) \cdot 8 = 30

Refine

(u - \frac{1}{u}) \cdot 8 = 30

Simplify

(u - \frac{1}{u}) \cdot 8 : 8 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 8

Apply the commutative law:

(u - \frac{1}{u}) \cdot 8 = 8 (u - \frac{1}{u})

8 (u - \frac{1}{u})

8 (u - \frac{1}{u}) = 30

Expand

8 (u - \frac{1}{u}) : 8 u - \frac{8}{u}

8 (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

a = 8, b = u, c = \frac{1}{u}

= 8 u - 8 \cdot \frac{1}{u}

8 \cdot \frac{1}{u} = \frac{8}{u}

8 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 8}{u}

Multiply the numbers:

1 \cdot 8 = 8

= \frac{8}{u}

= 8 u - \frac{8}{u}

8 u - \frac{8}{u} = 30

Multiply both sides by

u

8 u - \frac{8}{u} = 30

Multiply both sides by

u

8 uu - \frac{8}{u} u = 30 u

Simplify

8 uu - \frac{8}{u} u = 30 u

Simplify

8 uu : 8 u^{2}

8 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 8 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 8 u^{2}

Simplify

- \frac{8}{u} u : - 8

- \frac{8}{u} u

Multiply fractions:

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

= - \frac{8 u}{u}

Cancel the common factor:

u

= - 8

8 u^{2} - 8 = 30 u

8 u^{2} - 8 = 30 u

8 u^{2} - 8 = 30 u

Solve

8 u^{2} - 8 = 30 u : u = 4, u = - \frac{1}{4}

8 u^{2} - 8 = 30 u

Move

30 u

to the left side

8 u^{2} - 8 = 30 u

Subtract

30 u

from both sides

8 u^{2} - 8 - 30 u = 30 u - 30 u

Simplify

8 u^{2} - 8 - 30 u = 0

8 u^{2} - 8 - 30 u = 0

Write in the standard form

a x^{2} + b x + c = 0

8 u^{2} - 30 u - 8 = 0

Solve with the quadratic formula

8 u^{2} - 30 u - 8 = 0

Quadratic Equation Formula:

For

a = 8, b = - 30, c = - 8

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 8 ( - 8 )}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 8 ( - 8 )}{2 \cdot 8}

(- 30)^{2} - 4 \cdot 8 (- 8) = 34

(- 30)^{2} - 4 \cdot 8 (- 8)

Apply rule

- (- a) = a

= (- 30)^{2} + 4 \cdot 8 \cdot 8

Apply exponent rule:

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

n

is even

(- 30)^{2} = 3 0^{2}

= 3 0^{2} + 4 \cdot 8 \cdot 8

Multiply the numbers:

4 \cdot 8 \cdot 8 = 256

= 3 0^{2} + 256

3 0^{2} = 900

= 900 + 256

Add the numbers:

900 + 256 = 1156

= 1156

Factor the number:

1156 = 3 4^{2}

= 3 4^{2}

Apply radical rule:

3 4^{2} = 34

= 34

u_{1, 2} = \frac{- ( - 30 ) \pm 34}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( - 30 ) + 34}{2 \cdot 8}, u_{2} = \frac{- ( - 30 ) - 34}{2 \cdot 8}

u = \frac{- ( - 30 ) + 34}{2 \cdot 8} : 4

\frac{- ( - 30 ) + 34}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{30 + 34}{2 \cdot 8}

Add the numbers:

30 + 34 = 64

= \frac{64}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{64}{16}

Divide the numbers:

\frac{64}{16} = 4

= 4

u = \frac{- ( - 30 ) - 34}{2 \cdot 8} : - \frac{1}{4}

\frac{- ( - 30 ) - 34}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{30 - 34}{2 \cdot 8}

Subtract the numbers:

30 - 34 = - 4

= \frac{- 4}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 4}{16}

Apply the fraction rule:

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

= - \frac{4}{16}

Cancel the common factor:

4

= - \frac{1}{4}

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u - u^{- 1}) 8

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

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

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 4 : x = 2 ln (2)

e^{x} = 4

Apply exponent rules

e^{x} = 4

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (4)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (4)

Simplify

ln (4) : 2 ln (2)

ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (2^{2}) = 2 ln (2)

= 2 ln (2)

x = 2 ln (2)

x = 2 ln (2)

Solve

e^{x} = - \frac{1}{4} :

No Solution for

x \in R

e^{x} = - \frac{1}{4}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = 2 ln (2)

x = 2 ln (2)

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)= 15/8 ?
The general solution for sinh(x)= 15/8 is x=2ln(2)