What is the general solution for cosh(θ)=4-j ?

The general solution for cosh(θ)=4-j is θ=ln(-j+4+sqrt(j^2-8j+15)),θ=ln(-j+4-sqrt(j^2-8j+15))

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cosh(θ)=4-j

Solution

cosh (θ) = 4 - j

Solution

θ = ln (- j + 4 + j^{2} - 8 j + 15), θ = ln (- j + 4 - j^{2} - 8 j + 15)

Solution steps

cosh (θ) = 4 - j

Rewrite using trig identities

cosh (θ) = 4 - j

Use the Hyperbolic identity:

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

\frac{e ^{θ} + e ^{- θ}}{2} = 4 - j

\frac{e ^{θ} + e ^{- θ}}{2} = 4 - j

\frac{e ^{θ} + e ^{- θ}}{2} = 4 - j : θ = ln (- j + 4 + j^{2} - 8 j + 15), θ = ln (- j + 4 - j^{2} - 8 j + 15)

\frac{e ^{θ} + e ^{- θ}}{2} = 4 - j

Multiply both sides by

2

\frac{e ^{θ} + e ^{- θ}}{2} \cdot 2 = 4 \cdot 2 - j \cdot 2

Simplify

e^{θ} + e^{- θ} = 8 - 2 j

Apply exponent rules

e^{θ} + e^{- θ} = 8 - 2 j

Apply exponent rule:

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

e^{- θ} = (e^{θ})^{- 1}

e^{θ} + (e^{θ})^{- 1} = 8 - 2 j

e^{θ} + (e^{θ})^{- 1} = 8 - 2 j

Rewrite the equation with

e^{θ} = u

u + (u)^{- 1} = 8 - 2 j

Solve

u + u^{- 1} = 8 - 2 j : u = - j + 4 + j^{2} - 8 j + 15, u = - j + 4 - j^{2} - 8 j + 15

u + u^{- 1} = 8 - 2 j

Refine

u + \frac{1}{u} = 8 - 2 j

Multiply both sides by

u

u + \frac{1}{u} = 8 - 2 j

Multiply both sides by

u

uu + \frac{1}{u} u = 8 u - 2 j u

Simplify

uu + \frac{1}{u} u = 8 u - 2 j 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 = 8 u - 2 j u

u^{2} + 1 = 8 u - 2 j u

u^{2} + 1 = 8 u - 2 j u

Solve

u^{2} + 1 = 8 u - 2 j u : u = - j + 4 + j^{2} - 8 j + 15, u = - j + 4 - j^{2} - 8 j + 15

u^{2} + 1 = 8 u - 2 j u

Move

2 j u

to the left side

u^{2} + 1 = 8 u - 2 j u

Add

2 j u

to both sides

u^{2} + 1 + 2 j u = 8 u - 2 j u + 2 j u

Simplify

u^{2} + 1 + 2 j u = 8 u

u^{2} + 1 + 2 j u = 8 u

Move

8 u

to the left side

u^{2} + 1 + 2 j u = 8 u

Subtract

8 u

from both sides

u^{2} + 1 + 2 j u - 8 u = 8 u - 8 u

Simplify

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

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

Write in the standard form

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

u^{2} + (2 j - 8) u + 1 = 0

Solve with the quadratic formula

u^{2} + (2 j - 8) u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 2 j - 8, c = 1

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

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

Simplify

(2 j - 8)^{2} - 4 \cdot 1 \cdot 1 : 2 j^{2} - 8 j + 15

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= (2 j - 8)^{2} - 4

Expand

(2 j - 8)^{2} - 4 : 4 j^{2} - 32 j + 60

(2 j - 8)^{2} - 4

(2 j - 8)^{2} : 4 j^{2} - 32 j + 64

Apply Perfect Square Formula:

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

a = 2 j, b = 8

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

Simplify

(2 j)^{2} - 2 \cdot 2 j \cdot 8 + 8^{2} : 4 j^{2} - 32 j + 64

(2 j)^{2} - 2 \cdot 2 j \cdot 8 + 8^{2}

(2 j)^{2} = 4 j^{2}

(2 j)^{2}

Apply exponent rule:

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

= 2^{2} j^{2}

2^{2} = 4

= 4 j^{2}

2 \cdot 2 j \cdot 8 = 32 j

2 \cdot 2 j \cdot 8

Multiply the numbers:

2 \cdot 2 \cdot 8 = 32

= 32 j

8^{2} = 64

8^{2}

8^{2} = 64

= 64

= 4 j^{2} - 32 j + 64

= 4 j^{2} - 32 j + 64

= 4 j^{2} - 32 j + 64 - 4

Add/Subtract the numbers:

64 - 4 = 60

= 4 j^{2} - 32 j + 60

= 4 j^{2} - 32 j + 60

Factor

4 j^{2} - 32 j + 60 : 4 (j^{2} - 8 j + 15)

4 j^{2} - 32 j + 60

Rewrite as

= 4 j^{2} - 4 \cdot 8 j + 4 \cdot 15

Factor out common term

4

= 4 (j^{2} - 8 j + 15)

= 4 (j^{2} - 8 j + 15)

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 4 j^{2} - 8 j + 15

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= 2 j^{2} - 8 j + 15

u_{1, 2} = \frac{- ( 2 j - 8 ) \pm 2 j ^{2} - 8 j + 15}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( 2 j - 8 ) + 2 j ^{2} - 8 j + 15}{2 \cdot 1}, u_{2} = \frac{- ( 2 j - 8 ) - 2 j ^{2} - 8 j + 15}{2 \cdot 1}

u = \frac{- ( 2 j - 8 ) + 2 j ^{2} - 8 j + 15}{2 \cdot 1} : - j + 4 + j^{2} - 8 j + 15

\frac{- ( 2 j - 8 ) + 2 j ^{2} - 8 j + 15}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- ( 2 j - 8 ) + 2 j ^{2} - 8 j + 15}{2}

- (2 j - 8) : - 2 j + 8

- (2 j - 8)

Distribute parentheses

= - (2 j) - (- 8)

Apply minus-plus rules

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

= - 2 j + 8

= \frac{- 2 j + 8 + 2 j ^{2} - 8 j + 15}{2}

Factor

- 2 j + 8 + 2 j^{2} - 8 j + 15 : 2 (- j + 4 + j^{2} + 15 - 8 j)

- 2 j + 8 + 2 j^{2} - 8 j + 15

Rewrite as

= - 2 j + 2 \cdot 4 + 2 j^{2} + 15 - 8 j

Factor out common term

2

= 2 (- j + 4 + j^{2} + 15 - 8 j)

= \frac{2 ( - j + 4 + j ^{2} + 15 - 8 j )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - j + 4 + j^{2} - 8 j + 15

u = \frac{- ( 2 j - 8 ) - 2 j ^{2} - 8 j + 15}{2 \cdot 1} : - j + 4 - j^{2} - 8 j + 15

\frac{- ( 2 j - 8 ) - 2 j ^{2} - 8 j + 15}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- ( 2 j - 8 ) - 2 j ^{2} - 8 j + 15}{2}

- (2 j - 8) : - 2 j + 8

- (2 j - 8)

Distribute parentheses

= - (2 j) - (- 8)

Apply minus-plus rules

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

= - 2 j + 8

= \frac{- 2 j + 8 - 2 j ^{2} - 8 j + 15}{2}

Factor

- 2 j + 8 - 2 j^{2} - 8 j + 15 : 2 (- j + 4 - j^{2} + 15 - 8 j)

- 2 j + 8 - 2 j^{2} - 8 j + 15

Rewrite as

= - 2 j + 2 \cdot 4 - 2 j^{2} + 15 - 8 j

Factor out common term

2

= 2 (- j + 4 - j^{2} + 15 - 8 j)

= \frac{2 ( - j + 4 - j ^{2} + 15 - 8 j )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - j + 4 - j^{2} - 8 j + 15

The solutions to the quadratic equation are:

u = - j + 4 + j^{2} - 8 j + 15, u = - j + 4 - j^{2} - 8 j + 15

u = - j + 4 + j^{2} - 8 j + 15, u = - j + 4 - j^{2} - 8 j + 15

u = - j + 4 + j^{2} - 8 j + 15, u = - j + 4 - j^{2} - 8 j + 15

Substitute back

u = e^{θ},

solve for

θ

Solve

e^{θ} = - j + 4 + j^{2} - 8 j + 15 : θ = ln (- j + 4 + j^{2} - 8 j + 15)

e^{θ} = - j + 4 + j^{2} - 8 j + 15

Apply exponent rules

e^{θ} = - j + 4 + j^{2} - 8 j + 15

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (- j + 4 + j^{2} - 8 j + 15)

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (- j + 4 + j^{2} - 8 j + 15)

θ = ln (- j + 4 + j^{2} - 8 j + 15)

Solve

e^{θ} = - j + 4 - j^{2} - 8 j + 15 : θ = ln (- j + 4 - j^{2} - 8 j + 15)

e^{θ} = - j + 4 - j^{2} - 8 j + 15

Apply exponent rules

e^{θ} = - j + 4 - j^{2} - 8 j + 15

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (- j + 4 - j^{2} - 8 j + 15)

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (- j + 4 - j^{2} - 8 j + 15)

θ = ln (- j + 4 - j^{2} - 8 j + 15)

θ = ln (- j + 4 + j^{2} - 8 j + 15), θ = ln (- j + 4 - j^{2} - 8 j + 15)

θ = ln (- j + 4 + j^{2} - 8 j + 15), θ = ln (- j + 4 - j^{2} - 8 j + 15)

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cosh(θ)=4-j ?
The general solution for cosh(θ)=4-j is θ=ln(-j+4+sqrt(j^2-8j+15)),θ=ln(-j+4-sqrt(j^2-8j+15))