What is the general solution for 2-2cosh(t-2)=0 ?

The general solution for 2-2cosh(t-2)=0 is t=2

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2-2cosh(t-2)=0

Solution

2 - 2 cosh (t - 2) = 0

Solution

t = 2

Degrees

t = 114.59155 \dots^{\circ}

Solution steps

2 - 2 cosh (t - 2) = 0

Rewrite using trig identities

2 - 2 cosh (t - 2) = 0

Use the Hyperbolic identity:

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

2 - 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} = 0

2 - 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} = 0

2 - 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} = 0 : t = 2

2 - 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} = 0

Add

2 \frac{e ^{t - 2} + e ^{- (t - 2)}}{2}

to both sides

2 - 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} + 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2} = 0 + 2 \cdot \frac{e ^{t - 2} + e ^{- (t - 2)}}{2}

Simplify

2 = e^{t - 2} + e^{- (t - 2)}

Apply exponent rules

2 = e^{t - 2} + e^{- (t - 2)}

Apply exponent rule:

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

e^{t - 2} = e^{t} e^{- 2}, e^{- (t - 2)} = e^{- 1 t} e^{2}

2 = e^{t} e^{- 2} + e^{- 1 \cdot t} e^{2}

Apply exponent rule:

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

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

2 = e^{t} e^{- 2} + (e^{t})^{- 1} e^{2}

2 = e^{t} e^{- 2} + (e^{t})^{- 1} e^{2}

Rewrite the equation with

e^{t} = u

2 = u e^{- 2} + (u)^{- 1} e^{2}

Solve

2 = u e^{- 2} + u^{- 1} e^{2} : u = e^{2}

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

Refine

2 = \frac{1}{e ^{2}} u + \frac{e ^{2}}{u}

Multiply by LCM

2 = \frac{1}{e ^{2}} u + \frac{e ^{2}}{u}

Find Least Common Multiplier of

e^{2}, u : e^{2} u

e^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

e^{2}

u

= e^{2} u

Multiply by LCM=

e^{2} u

2 e^{2} u = \frac{1}{e ^{2}} u e^{2} u + \frac{e ^{2}}{u} e^{2} u

Simplify

2 e^{2} u = \frac{1}{e ^{2}} u e^{2} u + \frac{e ^{2}}{u} e^{2} u

Simplify

\frac{1}{e ^{2}} u e^{2} u : u^{2}

\frac{1}{e ^{2}} u e^{2} u

Apply exponent rule:

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

uu = u^{1 + 1}

= \frac{1}{e ^{2}} e^{2} u^{1 + 1}

Add the numbers:

1 + 1 = 2

= \frac{1}{e ^{2}} e^{2} u^{2}

Multiply fractions:

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

= \frac{1 \cdot e ^{2}}{e ^{2}} u^{2}

Cancel the common factor:

e^{2}

= u^{2} \cdot 1

Multiply:

u^{2} \cdot 1 = u^{2}

= u^{2}

Simplify

\frac{e ^{2}}{u} e^{2} u : e^{4}

\frac{e ^{2}}{u} e^{2} u

Multiply fractions:

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

= \frac{e ^{2} e ^{2} u}{u}

Cancel the common factor:

u

= e^{2} e^{2}

Apply exponent rule:

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

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

2 e^{2} u = u^{2} + e^{4}

2 e^{2} u = u^{2} + e^{4}

2 e^{2} u = u^{2} + e^{4}

Solve

2 e^{2} u = u^{2} + e^{4} : u = e^{2}

2 e^{2} u = u^{2} + e^{4}

Switch sides

u^{2} + e^{4} = 2 e^{2} u

Move

2 e^{2} u

to the left side

u^{2} + e^{4} = 2 e^{2} u

Subtract

2 e^{2} u

from both sides

u^{2} + e^{4} - 2 e^{2} u = 2 e^{2} u - 2 e^{2} u

Simplify

u^{2} + e^{4} - 2 e^{2} u = 0

u^{2} + e^{4} - 2 e^{2} u = 0

Write in the standard form

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

u^{2} - 2 e^{2} u + e^{4} = 0

Solve with the quadratic formula

u^{2} - 2 e^{2} u + e^{4} = 0

Quadratic Equation Formula:

For

a = 1, b = - 2 e^{2}, c = e^{4}

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

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

(- 2 e^{2})^{2} - 4 \cdot 1 \cdot e^{4} = 0

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

(- 2 e^{2})^{2} = 2^{2} e^{4}

(- 2 e^{2})^{2}

Apply exponent rule:

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

n

is even

(- 2 e^{2})^{2} = (2 e^{2})^{2}

= (2 e^{2})^{2}

Apply exponent rule:

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

= 2^{2} (e^{2})^{2}

(e^{2})^{2} : e^{4}

Apply exponent rule:

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

= e^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= e^{4}

= 2^{2} e^{4}

4 \cdot 1 \cdot e^{4} = 4 e^{4}

4 \cdot 1 \cdot e^{4}

Multiply the numbers:

4 \cdot 1 = 4

= 4 e^{4}

= 2^{2} e^{4} - 4 e^{4}

2^{2} = 4

= 4 e^{4} - 4 e^{4}

Add similar elements:

4 e^{4} - 4 e^{4} = 0

= 0

u_{1, 2} = \frac{- ( - 2 e ^{2} ) \pm 0}{2 \cdot 1}

u = \frac{- ( - 2 e ^{2} )}{2 \cdot 1}

\frac{- ( - 2 e ^{2} )}{2 \cdot 1} = e^{2}

\frac{- ( - 2 e ^{2} )}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 e ^{2}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 e ^{2}}{2}

Divide the numbers:

\frac{2}{2} = 1

= e^{2}

u = e^{2}

The solution to the quadratic equation is:

u = e^{2}

u = e^{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u e^{- 2} + u^{- 1} e^{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = e^{2}

u = e^{2}

Substitute back

u = e^{t},

solve for

t

Solve

e^{t} = e^{2} : t = 2

e^{t} = e^{2}

Apply exponent rules

e^{t} = e^{2}

a^{f (x)} = a^{g (x)}

, then

f (x) = g (x)

t = 2

t = 2

t = 2

t = 2

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Frequently Asked Questions (FAQ)

What is the general solution for 2-2cosh(t-2)=0 ?
The general solution for 2-2cosh(t-2)=0 is t=2