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Popular Trigonometry >

cosh(θ)= 12/7 \land θ<0,sinh(θ)

Solution

cosh (θ) = \frac{12}{7} and θ < 0, sinh (θ)

Solution

θ = ln (\frac{12 - 95}{7})

Decimal

θ = - 1.13355 \dots

Solution steps

cosh (θ) = \frac{12}{7} and θ < 0

cosh (θ) = \frac{12}{7} : θ = ln (\frac{12 + 95}{7}), θ = ln (\frac{12 - 95}{7})

cosh (θ) = \frac{12}{7}

Rewrite using trig identities

cosh (θ) = \frac{12}{7}

Use the Hyperbolic identity:

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

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{12}{7}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{12}{7}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{12}{7} : θ = ln (\frac{12 + 95}{7}), θ = ln (\frac{12 - 95}{7})

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{12}{7}

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

(e^{θ} + e^{- θ}) \cdot 7 = 2 \cdot 12

Simplify

(e^{θ} + e^{- θ}) \cdot 7 = 24

Apply exponent rules

(e^{θ} + e^{- θ}) \cdot 7 = 24

Apply exponent rule:

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

e^{- θ} = (e^{θ})^{- 1}

(e^{θ} + (e^{θ})^{- 1}) \cdot 7 = 24

(e^{θ} + (e^{θ})^{- 1}) \cdot 7 = 24

Rewrite the equation with

e^{θ} = u

(u + (u)^{- 1}) \cdot 7 = 24

Solve

(u + u^{- 1}) \cdot 7 = 24 : u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

(u + u^{- 1}) \cdot 7 = 24

Refine

(u + \frac{1}{u}) \cdot 7 = 24

Simplify

(u + \frac{1}{u}) \cdot 7 : 7 (u + \frac{1}{u})

(u + \frac{1}{u}) \cdot 7

Apply the commutative law:

(u + \frac{1}{u}) \cdot 7 = 7 (u + \frac{1}{u})

7 (u + \frac{1}{u})

7 (u + \frac{1}{u}) = 24

Expand

7 (u + \frac{1}{u}) : 7 u + \frac{7}{u}

7 (u + \frac{1}{u})

Apply the distributive law:

a (b + c) = ab + a c

a = 7, b = u, c = \frac{1}{u}

= 7 u + 7 \cdot \frac{1}{u}

7 \cdot \frac{1}{u} = \frac{7}{u}

7 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 7}{u}

Multiply the numbers:

1 \cdot 7 = 7

= \frac{7}{u}

= 7 u + \frac{7}{u}

7 u + \frac{7}{u} = 24

Multiply both sides by

u

7 u + \frac{7}{u} = 24

Multiply both sides by

u

7 uu + \frac{7}{u} u = 24 u

Simplify

7 uu + \frac{7}{u} u = 24 u

Simplify

7 uu : 7 u^{2}

7 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 7 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 7 u^{2}

Simplify

\frac{7}{u} u : 7

\frac{7}{u} u

Multiply fractions:

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

= \frac{7 u}{u}

Cancel the common factor:

u

= 7

7 u^{2} + 7 = 24 u

7 u^{2} + 7 = 24 u

7 u^{2} + 7 = 24 u

Solve

7 u^{2} + 7 = 24 u : u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

7 u^{2} + 7 = 24 u

Move

24 u

to the left side

7 u^{2} + 7 = 24 u

Subtract

24 u

from both sides

7 u^{2} + 7 - 24 u = 24 u - 24 u

Simplify

7 u^{2} + 7 - 24 u = 0

7 u^{2} + 7 - 24 u = 0

Write in the standard form

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

7 u^{2} - 24 u + 7 = 0

Solve with the quadratic formula

7 u^{2} - 24 u + 7 = 0

Quadratic Equation Formula:

For

a = 7, b = - 24, c = 7

u_{1, 2} = \frac{- ( - 24 ) \pm ( - 24 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 24 ) \pm ( - 24 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

(- 24)^{2} - 4 \cdot 7 \cdot 7 = 295

(- 24)^{2} - 4 \cdot 7 \cdot 7

Apply exponent rule:

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

n

is even

(- 24)^{2} = 2 4^{2}

= 2 4^{2} - 4 \cdot 7 \cdot 7

Multiply the numbers:

4 \cdot 7 \cdot 7 = 196

= 2 4^{2} - 196

2 4^{2} = 576

= 576 - 196

Subtract the numbers:

576 - 196 = 380

= 380

Prime factorization of

380 : 2^{2} \cdot 5 \cdot 19

380

380

divides by

2380 = 190 \cdot 2

= 2 \cdot 190

190

divides by

2190 = 95 \cdot 2

= 2 \cdot 2 \cdot 95

95

divides by

595 = 19 \cdot 5

= 2 \cdot 2 \cdot 5 \cdot 19

2, 5, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5 \cdot 19

= 2^{2} \cdot 5 \cdot 19

= 2^{2} \cdot 5 \cdot 19

Apply radical rule:

n ab = n a n b

= 2^{2} 5 \cdot 19

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2 5 \cdot 19

Refine

= 295

u_{1, 2} = \frac{- ( - 24 ) \pm 2 95}{2 \cdot 7}

Separate the solutions

u_{1} = \frac{- ( - 24 ) + 2 95}{2 \cdot 7}, u_{2} = \frac{- ( - 24 ) - 2 95}{2 \cdot 7}

u = \frac{- ( - 24 ) + 2 95}{2 \cdot 7} : \frac{12 + 95}{7}

\frac{- ( - 24 ) + 2 95}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{24 + 2 95}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{24 + 2 95}{14}

Factor

24 + 295 : 2 (12 + 95)

24 + 295

Rewrite as

= 2 \cdot 12 + 295

Factor out common term

2

= 2 (12 + 95)

= \frac{2 ( 12 + 95 )}{14}

Cancel the common factor:

2

= \frac{12 + 95}{7}

u = \frac{- ( - 24 ) - 2 95}{2 \cdot 7} : \frac{12 - 95}{7}

\frac{- ( - 24 ) - 2 95}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{24 - 2 95}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{24 - 2 95}{14}

Factor

24 - 295 : 2 (12 - 95)

24 - 295

Rewrite as

= 2 \cdot 12 - 295

Factor out common term

2

= 2 (12 - 95)

= \frac{2 ( 12 - 95 )}{14}

Cancel the common factor:

2

= \frac{12 - 95}{7}

The solutions to the quadratic equation are:

u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(u + u^{- 1}) 7

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

u = \frac{12 + 95}{7}, u = \frac{12 - 95}{7}

Substitute back

u = e^{θ},

solve for

θ

Solve

e^{θ} = \frac{12 + 95}{7} : θ = ln (\frac{12 + 95}{7})

e^{θ} = \frac{12 + 95}{7}

Apply exponent rules

e^{θ} = \frac{12 + 95}{7}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{12 + 95}{7})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{12 + 95}{7})

θ = ln (\frac{12 + 95}{7})

Solve

e^{θ} = \frac{12 - 95}{7} : θ = ln (\frac{12 - 95}{7})

e^{θ} = \frac{12 - 95}{7}

Apply exponent rules

e^{θ} = \frac{12 - 95}{7}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{12 - 95}{7})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{12 - 95}{7})

θ = ln (\frac{12 - 95}{7})

θ = ln (\frac{12 + 95}{7}), θ = ln (\frac{12 - 95}{7})

θ = ln (\frac{12 + 95}{7}), θ = ln (\frac{12 - 95}{7})

Combine the intervals

(θ = ln (\frac{12 - 95}{7}) or θ = ln (\frac{12 + 95}{7})) and θ < 0

Merge Overlapping Intervals

θ = ln (\frac{12 - 95}{7}) or θ = ln (\frac{12 + 95}{7}) and θ < 0

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

θ = ln (\frac{12 - 95}{7}) or θ = ln (\frac{12 + 95}{7})

and

θ < 0

θ = ln (\frac{12 - 95}{7})

θ = ln (\frac{12 - 95}{7})

Graph

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