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Popular Trigonometry >

sinh(x)= 6/5

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Solution

sinh(x)=56​

Solution

x=ln(56+61​​)
+1
Degrees
x=58.21097…∘
Solution steps
sinh(x)=56​
Rewrite using trig identities
sinh(x)=56​
Use the Hyperbolic identity: sinh(x)=2ex−e−x​2ex−e−x​=56​
2ex−e−x​=56​
2ex−e−x​=56​:x=ln(56+61​​)
2ex−e−x​=56​
Apply fraction cross multiply: if ba​=dc​ then a⋅d=b⋅c(ex−e−x)⋅5=2⋅6
Simplify(ex−e−x)⋅5=12
Apply exponent rules
(ex−e−x)⋅5=12
Apply exponent rule: abc=(ab)ce−x=(ex)−1(ex−(ex)−1)⋅5=12
(ex−(ex)−1)⋅5=12
Rewrite the equation with ex=u(u−(u)−1)⋅5=12
Solve (u−u−1)⋅5=12:u=56+61​​,u=56−61​​
(u−u−1)⋅5=12
Refine(u−u1​)⋅5=12
Simplify (u−u1​)⋅5:5(u−u1​)
(u−u1​)⋅5
Apply the commutative law: (u−u1​)⋅5=5(u−u1​)5(u−u1​)
5(u−u1​)=12
Expand 5(u−u1​):5u−u5​
5(u−u1​)
Apply the distributive law: a(b−c)=ab−aca=5,b=u,c=u1​=5u−5⋅u1​
5⋅u1​=u5​
5⋅u1​
Multiply fractions: a⋅cb​=ca⋅b​=u1⋅5​
Multiply the numbers: 1⋅5=5=u5​
=5u−u5​
5u−u5​=12
Multiply both sides by u
5u−u5​=12
Multiply both sides by u5uu−u5​u=12u
Simplify
5uu−u5​u=12u
Simplify 5uu:5u2
5uu
Apply exponent rule: ab⋅ac=ab+cuu=u1+1=5u1+1
Add the numbers: 1+1=2=5u2
Simplify −u5​u:−5
−u5​u
Multiply fractions: a⋅cb​=ca⋅b​=−u5u​
Cancel the common factor: u=−5
5u2−5=12u
5u2−5=12u
5u2−5=12u
Solve 5u2−5=12u:u=56+61​​,u=56−61​​
5u2−5=12u
Move 12uto the left side
5u2−5=12u
Subtract 12u from both sides5u2−5−12u=12u−12u
Simplify5u2−5−12u=0
5u2−5−12u=0
Write in the standard form ax2+bx+c=05u2−12u−5=0
Solve with the quadratic formula
5u2−12u−5=0
Quadratic Equation Formula:
For a=5,b=−12,c=−5u1,2​=2⋅5−(−12)±(−12)2−4⋅5(−5)​​
u1,2​=2⋅5−(−12)±(−12)2−4⋅5(−5)​​
(−12)2−4⋅5(−5)​=261​
(−12)2−4⋅5(−5)​
Apply rule −(−a)=a=(−12)2+4⋅5⋅5​
Apply exponent rule: (−a)n=an,if n is even(−12)2=122=122+4⋅5⋅5​
Multiply the numbers: 4⋅5⋅5=100=122+100​
122=144=144+100​
Add the numbers: 144+100=244=244​
Prime factorization of 244:22⋅61
244
244divides by 2244=122⋅2=2⋅122
122divides by 2122=61⋅2=2⋅2⋅61
2,61 are all prime numbers, therefore no further factorization is possible=2⋅2⋅61
=22⋅61
=22⋅61​
Apply radical rule: =61​22​
Apply radical rule: 22​=2=261​
u1,2​=2⋅5−(−12)±261​​
Separate the solutionsu1​=2⋅5−(−12)+261​​,u2​=2⋅5−(−12)−261​​
u=2⋅5−(−12)+261​​:56+61​​
2⋅5−(−12)+261​​
Apply rule −(−a)=a=2⋅512+261​​
Multiply the numbers: 2⋅5=10=1012+261​​
Factor 12+261​:2(6+61​)
12+261​
Rewrite as=2⋅6+261​
Factor out common term 2=2(6+61​)
=102(6+61​)​
Cancel the common factor: 2=56+61​​
u=2⋅5−(−12)−261​​:56−61​​
2⋅5−(−12)−261​​
Apply rule −(−a)=a=2⋅512−261​​
Multiply the numbers: 2⋅5=10=1012−261​​
Factor 12−261​:2(6−61​)
12−261​
Rewrite as=2⋅6−261​
Factor out common term 2=2(6−61​)
=102(6−61​)​
Cancel the common factor: 2=56−61​​
The solutions to the quadratic equation are:u=56+61​​,u=56−61​​
u=56+61​​,u=56−61​​
Verify Solutions
Find undefined (singularity) points:u=0
Take the denominator(s) of (u−u−1)5 and compare to zero
u=0
The following points are undefinedu=0
Combine undefined points with solutions:
u=56+61​​,u=56−61​​
u=56+61​​,u=56−61​​
Substitute back u=ex,solve for x
Solve ex=56+61​​:x=ln(56+61​​)
ex=56+61​​
Apply exponent rules
ex=56+61​​
If f(x)=g(x), then ln(f(x))=ln(g(x))ln(ex)=ln(56+61​​)
Apply log rule: ln(ea)=aln(ex)=xx=ln(56+61​​)
x=ln(56+61​​)
Solve ex=56−61​​:No Solution for x∈R
ex=56−61​​
af(x) cannot be zero or negative for x∈RNoSolutionforx∈R
x=ln(56+61​​)
x=ln(56+61​​)

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tan(2x)tan(x)=1(csc^2(x))/4 =4sin^2(x)6sec^2(x)-3cos(x)-10=sec(x)tan(x)-sec(x)=sqrt(3)sin^2(x)+cos(2x)=1

Frequently Asked Questions (FAQ)

  • What is the general solution for sinh(x)= 6/5 ?

    The general solution for sinh(x)= 6/5 is x=ln((6+sqrt(61))/5)
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