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Popular Trigonometry >

cosh(θ)= 26/7 \land θ<0,sinh(θ)

Solution

cosh (θ) = \frac{26}{7} and θ < 0, sinh (θ)

Solution

θ = ln (\frac{26 - 627}{7})

Decimal

θ = - 1.98669 \dots

Solution steps

cosh (θ) = \frac{26}{7} and θ < 0

cosh (θ) = \frac{26}{7} : θ = ln (\frac{26 + 627}{7}), θ = ln (\frac{26 - 627}{7})

cosh (θ) = \frac{26}{7}

Rewrite using trig identities

cosh (θ) = \frac{26}{7}

Use the Hyperbolic identity:

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

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{26}{7}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{26}{7}

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{26}{7} : θ = ln (\frac{26 + 627}{7}), θ = ln (\frac{26 - 627}{7})

\frac{e ^{θ} + e ^{- θ}}{2} = \frac{26}{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 26

Simplify

(e^{θ} + e^{- θ}) \cdot 7 = 52

Apply exponent rules

(e^{θ} + e^{- θ}) \cdot 7 = 52

Apply exponent rule:

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

e^{- θ} = (e^{θ})^{- 1}

(e^{θ} + (e^{θ})^{- 1}) \cdot 7 = 52

(e^{θ} + (e^{θ})^{- 1}) \cdot 7 = 52

Rewrite the equation with

e^{θ} = u

(u + (u)^{- 1}) \cdot 7 = 52

Solve

(u + u^{- 1}) \cdot 7 = 52 : u = \frac{26 + 627}{7}, u = \frac{26 - 627}{7}

(u + u^{- 1}) \cdot 7 = 52

Refine

(u + \frac{1}{u}) \cdot 7 = 52

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}) = 52

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} = 52

Multiply both sides by

u

7 u + \frac{7}{u} = 52

Multiply both sides by

u

7 uu + \frac{7}{u} u = 52 u

Simplify

7 uu + \frac{7}{u} u = 52 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 = 52 u

7 u^{2} + 7 = 52 u

7 u^{2} + 7 = 52 u

Solve

7 u^{2} + 7 = 52 u : u = \frac{26 + 627}{7}, u = \frac{26 - 627}{7}

7 u^{2} + 7 = 52 u

Move

52 u

to the left side

7 u^{2} + 7 = 52 u

Subtract

52 u

from both sides

7 u^{2} + 7 - 52 u = 52 u - 52 u

Simplify

7 u^{2} + 7 - 52 u = 0

7 u^{2} + 7 - 52 u = 0

Write in the standard form

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

7 u^{2} - 52 u + 7 = 0

Solve with the quadratic formula

7 u^{2} - 52 u + 7 = 0

Quadratic Equation Formula:

For

a = 7, b = - 52, c = 7

u_{1, 2} = \frac{- ( - 52 ) \pm ( - 52 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 52 ) \pm ( - 52 ) ^{2} - 4 \cdot 7 \cdot 7}{2 \cdot 7}

(- 52)^{2} - 4 \cdot 7 \cdot 7 = 2627

(- 52)^{2} - 4 \cdot 7 \cdot 7

Apply exponent rule:

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

n

is even

(- 52)^{2} = 5 2^{2}

= 5 2^{2} - 4 \cdot 7 \cdot 7

Multiply the numbers:

4 \cdot 7 \cdot 7 = 196

= 5 2^{2} - 196

5 2^{2} = 2704

= 2704 - 196

Subtract the numbers:

2704 - 196 = 2508

= 2508

Prime factorization of

2508 : 2^{2} \cdot 3 \cdot 11 \cdot 19

2508

2508

divides by

22508 = 1254 \cdot 2

= 2 \cdot 1254

1254

divides by

21254 = 627 \cdot 2

= 2 \cdot 2 \cdot 627

627

divides by

3627 = 209 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 209

209

divides by

11209 = 19 \cdot 11

= 2 \cdot 2 \cdot 3 \cdot 11 \cdot 19

2, 3, 11, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 11 \cdot 19

= 2^{2} \cdot 3 \cdot 11 \cdot 19

= 2^{2} \cdot 3 \cdot 11 \cdot 19

Apply radical rule:

= 2^{2} 3 \cdot 11 \cdot 19

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 11 \cdot 19

Refine

= 2627

u_{1, 2} = \frac{- ( - 52 ) \pm 2 627}{2 \cdot 7}

Separate the solutions

u_{1} = \frac{- ( - 52 ) + 2 627}{2 \cdot 7}, u_{2} = \frac{- ( - 52 ) - 2 627}{2 \cdot 7}

u = \frac{- ( - 52 ) + 2 627}{2 \cdot 7} : \frac{26 + 627}{7}

\frac{- ( - 52 ) + 2 627}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{52 + 2 627}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{52 + 2 627}{14}

Factor

52 + 2627 : 2 (26 + 627)

52 + 2627

Rewrite as

= 2 \cdot 26 + 2627

Factor out common term

2

= 2 (26 + 627)

= \frac{2 ( 26 + 627 )}{14}

Cancel the common factor:

2

= \frac{26 + 627}{7}

u = \frac{- ( - 52 ) - 2 627}{2 \cdot 7} : \frac{26 - 627}{7}

\frac{- ( - 52 ) - 2 627}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{52 - 2 627}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{52 - 2 627}{14}

Factor

52 - 2627 : 2 (26 - 627)

52 - 2627

Rewrite as

= 2 \cdot 26 - 2627

Factor out common term

2

= 2 (26 - 627)

= \frac{2 ( 26 - 627 )}{14}

Cancel the common factor:

2

= \frac{26 - 627}{7}

The solutions to the quadratic equation are:

u = \frac{26 + 627}{7}, u = \frac{26 - 627}{7}

u = \frac{26 + 627}{7}, u = \frac{26 - 627}{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{26 + 627}{7}, u = \frac{26 - 627}{7}

u = \frac{26 + 627}{7}, u = \frac{26 - 627}{7}

Substitute back

u = e^{θ},

solve for

θ

Solve

e^{θ} = \frac{26 + 627}{7} : θ = ln (\frac{26 + 627}{7})

e^{θ} = \frac{26 + 627}{7}

Apply exponent rules

e^{θ} = \frac{26 + 627}{7}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{26 + 627}{7})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{26 + 627}{7})

θ = ln (\frac{26 + 627}{7})

Solve

e^{θ} = \frac{26 - 627}{7} : θ = ln (\frac{26 - 627}{7})

e^{θ} = \frac{26 - 627}{7}

Apply exponent rules

e^{θ} = \frac{26 - 627}{7}

f (x) = g (x)

, then

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

ln (e^{θ}) = ln (\frac{26 - 627}{7})

Apply log rule:

ln (e^{a}) = a

ln (e^{θ}) = θ

θ = ln (\frac{26 - 627}{7})

θ = ln (\frac{26 - 627}{7})

θ = ln (\frac{26 + 627}{7}), θ = ln (\frac{26 - 627}{7})

θ = ln (\frac{26 + 627}{7}), θ = ln (\frac{26 - 627}{7})

Combine the intervals

(θ = ln (\frac{26 - 627}{7}) or θ = ln (\frac{26 + 627}{7})) and θ < 0

Merge Overlapping Intervals

θ = ln (\frac{26 - 627}{7}) or θ = ln (\frac{26 + 627}{7}) and θ < 0

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

θ = ln (\frac{26 - 627}{7}) or θ = ln (\frac{26 + 627}{7})

and

θ < 0

θ = ln (\frac{26 - 627}{7})

θ = ln (\frac{26 - 627}{7})

Graph

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