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Popular Trigonometry >

cosh(θ)= 29/8 \land θ<0,sinh(θ)

Solution

cosh (θ) = \frac{29}{8} and θ < 0, sinh (θ)

Solution

θ = ln (\frac{29 - 777}{8})

Decimal

θ = - 1.96140 \dots

Solution steps

cosh (θ) = \frac{29}{8} and θ < 0

cosh (θ) = \frac{29}{8} : θ = ln (\frac{29 + 777}{8}), θ = ln (\frac{29 - 777}{8})

cosh (θ) = \frac{29}{8}

Rewrite using trig identities

cosh (θ) = \frac{29}{8}

Use the Hyperbolic identity:

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

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{29}{8}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{29}{8}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{29}{8} : θ = ln (\frac{29 + 777}{8}), θ = ln (\frac{29 - 777}{8})

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{29}{8}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{θ} + e^{- θ}) \cdot 8 = 2 \cdot 29

Simplify

(e^{θ} + e^{- θ}) \cdot 8 = 58

Apply exponent rules

(e^{θ} + e^{- θ}) \cdot 8 = 58

Apply exponent rule:

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

e^{- θ} = (e^{θ})^{- 1}

(e^{θ} + (e^{θ})^{- 1}) \cdot 8 = 58

(e^{θ} + (e^{θ})^{- 1}) \cdot 8 = 58

Rewrite the equation with

e^{θ} = u

(u + (u)^{- 1}) \cdot 8 = 58

Solve

(u + u^{- 1}) \cdot 8 = 58 : u = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

(u + u^{- 1}) \cdot 8 = 58

Refine

(u + \frac{1}{u}) \cdot 8 = 58

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}) = 58

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} = 58

Multiply both sides by

u

8 u + \frac{8}{u} = 58

Multiply both sides by

u

8 uu + \frac{8}{u} u = 58 u

Simplify

8 uu + \frac{8}{u} u = 58 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 = 58 u

8 u^{2} + 8 = 58 u

8 u^{2} + 8 = 58 u

Solve

8 u^{2} + 8 = 58 u : u = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

8 u^{2} + 8 = 58 u

Move

58 u

to the left side

8 u^{2} + 8 = 58 u

Subtract

58 u

from both sides

8 u^{2} + 8 - 58 u = 58 u - 58 u

Simplify

8 u^{2} + 8 - 58 u = 0

8 u^{2} + 8 - 58 u = 0

Write in the standard form

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

8 u^{2} - 58 u + 8 = 0

Solve with the quadratic formula

8 u^{2} - 58 u + 8 = 0

Quadratic Equation Formula:

For

a = 8, b = - 58, c = 8

u_{1, 2} = \frac{- ( - 58 ) \pm ( - 58 ) ^{2} - 4 \cdot 8 \cdot 8}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 58 ) \pm ( - 58 ) ^{2} - 4 \cdot 8 \cdot 8}{2 \cdot 8}

(- 58)^{2} - 4 \cdot 8 \cdot 8 = 2777

(- 58)^{2} - 4 \cdot 8 \cdot 8

Apply exponent rule:

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

n

is even

(- 58)^{2} = 5 8^{2}

= 5 8^{2} - 4 \cdot 8 \cdot 8

Multiply the numbers:

4 \cdot 8 \cdot 8 = 256

= 5 8^{2} - 256

5 8^{2} = 3364

= 3364 - 256

Subtract the numbers:

3364 - 256 = 3108

= 3108

Prime factorization of

3108 : 2^{2} \cdot 3 \cdot 7 \cdot 37

3108

3108

divides by

23108 = 1554 \cdot 2

= 2 \cdot 1554

1554

divides by

21554 = 777 \cdot 2

= 2 \cdot 2 \cdot 777

777

divides by

3777 = 259 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 259

259

divides by

7259 = 37 \cdot 7

= 2 \cdot 2 \cdot 3 \cdot 7 \cdot 37

2, 3, 7, 37

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 7 \cdot 37

= 2^{2} \cdot 3 \cdot 7 \cdot 37

= 2^{2} \cdot 3 \cdot 7 \cdot 37

Apply radical rule:

= 2^{2} 3 \cdot 7 \cdot 37

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 7 \cdot 37

Refine

= 2777

u_{1, 2} = \frac{- ( - 58 ) \pm 2 777}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( - 58 ) + 2 777}{2 \cdot 8}, u_{2} = \frac{- ( - 58 ) - 2 777}{2 \cdot 8}

u = \frac{- ( - 58 ) + 2 777}{2 \cdot 8} : \frac{29 + 777}{8}

\frac{- ( - 58 ) + 2 777}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{58 + 2 777}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{58 + 2 777}{16}

Factor

58 + 2777 : 2 (29 + 777)

58 + 2777

Rewrite as

= 2 \cdot 29 + 2777

Factor out common term

2

= 2 (29 + 777)

= \frac{2 ( 29 + 777 )}{16}

Cancel the common factor:

2

= \frac{29 + 777}{8}

u = \frac{- ( - 58 ) - 2 777}{2 \cdot 8} : \frac{29 - 777}{8}

\frac{- ( - 58 ) - 2 777}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{58 - 2 777}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{58 - 2 777}{16}

Factor

58 - 2777 : 2 (29 - 777)

58 - 2777

Rewrite as

= 2 \cdot 29 - 2777

Factor out common term

2

= 2 (29 - 777)

= \frac{2 ( 29 - 777 )}{16}

Cancel the common factor:

2

= \frac{29 - 777}{8}

The solutions to the quadratic equation are:

u = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

u = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

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 = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

u = \frac{29 + 777}{8}, u = \frac{29 - 777}{8}

Substitute back

u = e^{θ},

solve for

θ

Solve

e^{θ} = \frac{29 + 777}{8} : θ = ln (\frac{29 + 777}{8})

e^{θ} = \frac{29 + 777}{8}

Apply exponent rules

e^{θ} = \frac{29 + 777}{8}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{29 + 777}{8})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{29 + 777}{8})

θ = ln (\frac{29 + 777}{8})

Solve

e^{θ} = \frac{29 - 777}{8} : θ = ln (\frac{29 - 777}{8})

e^{θ} = \frac{29 - 777}{8}

Apply exponent rules

e^{θ} = \frac{29 - 777}{8}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{29 - 777}{8})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{29 - 777}{8})

θ = ln (\frac{29 - 777}{8})

θ = ln (\frac{29 + 777}{8}), θ = ln (\frac{29 - 777}{8})

θ = ln (\frac{29 + 777}{8}), θ = ln (\frac{29 - 777}{8})

Combine the intervals

(θ = ln (\frac{29 - 777}{8}) or θ = ln (\frac{29 + 777}{8})) and θ < 0

Merge Overlapping Intervals

θ = ln (\frac{29 - 777}{8}) or θ = ln (\frac{29 + 777}{8}) and θ < 0

The intersection of two intervals is the set of numbers which are in both intervals

θ = ln (\frac{29 - 777}{8}) or θ = ln (\frac{29 + 777}{8})

and

θ < 0

θ = ln (\frac{29 - 777}{8})

θ = ln (\frac{29 - 777}{8})

Graph

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