What is the general solution for cosh(x)=10 ?

The general solution for cosh(x)=10 is x=ln(10+3sqrt(11)),x=ln(10-3sqrt(11))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cosh(x)=10

Solution

cosh (x) = 10

Solution

x = ln (10 + 311), x = ln (10 - 311)

Degrees

x = 171.49903 \dots^{\circ}, x = - 171.49903 \dots^{\circ}

Solution steps

cosh (x) = 10

Rewrite using trig identities

cosh (x) = 10

Use the Hyperbolic identity:

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

\frac{e ^{x} + e ^{- x}}{2} = 10

\frac{e ^{x} + e ^{- x}}{2} = 10

\frac{e ^{x} + e ^{- x}}{2} = 10 : x = ln (10 + 311), x = ln (10 - 311)

\frac{e ^{x} + e ^{- x}}{2} = 10

Multiply both sides by

2

\frac{e ^{x} + e ^{- x}}{2} \cdot 2 = 10 \cdot 2

Simplify

e^{x} + e^{- x} = 20

Apply exponent rules

e^{x} + e^{- x} = 20

Apply exponent rule:

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

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

e^{x} + (e^{x})^{- 1} = 20

e^{x} + (e^{x})^{- 1} = 20

Rewrite the equation with

e^{x} = u

u + (u)^{- 1} = 20

Solve

u + u^{- 1} = 20 : u = 10 + 311, u = 10 - 311

u + u^{- 1} = 20

Refine

u + \frac{1}{u} = 20

Multiply both sides by

u

u + \frac{1}{u} = 20

Multiply both sides by

u

uu + \frac{1}{u} u = 20 u

Simplify

uu + \frac{1}{u} u = 20 u

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= 1

u^{2} + 1 = 20 u

u^{2} + 1 = 20 u

u^{2} + 1 = 20 u

Solve

u^{2} + 1 = 20 u : u = 10 + 311, u = 10 - 311

u^{2} + 1 = 20 u

Move

20 u

to the left side

u^{2} + 1 = 20 u

Subtract

20 u

from both sides

u^{2} + 1 - 20 u = 20 u - 20 u

Simplify

u^{2} + 1 - 20 u = 0

u^{2} + 1 - 20 u = 0

Write in the standard form

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

u^{2} - 20 u + 1 = 0

Solve with the quadratic formula

u^{2} - 20 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 20, c = 1

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 20)^{2} - 4 \cdot 1 \cdot 1 = 611

(- 20)^{2} - 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

(- 20)^{2} = 2 0^{2}

= 2 0^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2 0^{2} - 4

2 0^{2} = 400

= 400 - 4

Subtract the numbers:

400 - 4 = 396

= 396

Prime factorization of

396 : 2^{2} \cdot 3^{2} \cdot 11

396

396

divides by

2396 = 198 \cdot 2

= 2 \cdot 198

198

divides by

2198 = 99 \cdot 2

= 2 \cdot 2 \cdot 99

99

divides by

399 = 33 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 11

= 2^{2} \cdot 3^{2} \cdot 11

= 2^{2} \cdot 3^{2} \cdot 11

Apply radical rule:

= 11 2^{2} 3^{2}

Apply radical rule:

2^{2} = 2

= 211 3^{2}

Apply radical rule:

3^{2} = 3

= 2 \cdot 311

Refine

= 611

u_{1, 2} = \frac{- ( - 20 ) \pm 6 11}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 20 ) + 6 11}{2 \cdot 1}, u_{2} = \frac{- ( - 20 ) - 6 11}{2 \cdot 1}

u = \frac{- ( - 20 ) + 6 11}{2 \cdot 1} : 10 + 311

\frac{- ( - 20 ) + 6 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{20 + 6 11}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{20 + 6 11}{2}

Factor

20 + 611 : 2 (10 + 311)

20 + 611

Rewrite as

= 2 \cdot 10 + 2 \cdot 311

Factor out common term

2

= 2 (10 + 311)

= \frac{2 ( 10 + 3 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 + 311

u = \frac{- ( - 20 ) - 6 11}{2 \cdot 1} : 10 - 311

\frac{- ( - 20 ) - 6 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{20 - 6 11}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{20 - 6 11}{2}

Factor

20 - 611 : 2 (10 - 311)

20 - 611

Rewrite as

= 2 \cdot 10 - 2 \cdot 311

Factor out common term

2

= 2 (10 - 311)

= \frac{2 ( 10 - 3 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 - 311

The solutions to the quadratic equation are:

u = 10 + 311, u = 10 - 311

u = 10 + 311, u = 10 - 311

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u + u^{- 1}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 10 + 311, u = 10 - 311

u = 10 + 311, u = 10 - 311

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 10 + 311 : x = ln (10 + 311)

e^{x} = 10 + 311

Apply exponent rules

e^{x} = 10 + 311

f (x) = g (x)

, then

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

ln (e^{x}) = ln (10 + 311)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (10 + 311)

x = ln (10 + 311)

Solve

e^{x} = 10 - 311 : x = ln (10 - 311)

e^{x} = 10 - 311

Apply exponent rules

e^{x} = 10 - 311

f (x) = g (x)

, then

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

ln (e^{x}) = ln (10 - 311)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (10 - 311)

x = ln (10 - 311)

x = ln (10 + 311), x = ln (10 - 311)

x = ln (10 + 311), x = ln (10 - 311)

Popular Examples

solvefor x,2sin^2(x)-sin(x)=1 solve for

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

arctan(x)=180

arctan (x) = 180

cos(θ)=(2sqrt(2))/3

cos (θ) = \frac{2 2}{3}

sqrt(2)cos(x)-sqrt(2)sin(x)=-sqrt(3)

2 cos (x) - 2 sin (x) = - 3

cos^4(x)=1

cos^{4} (x) = 1

Frequently Asked Questions (FAQ)

What is the general solution for cosh(x)=10 ?
The general solution for cosh(x)=10 is x=ln(10+3sqrt(11)),x=ln(10-3sqrt(11))