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I have this equation

  • $6 + \sinh(x) = \sinh(3x)$

I know that I have to use this equation

  • $\sinh(3x) = 3\sinh(x) + 4\sinh^3(x)$

  • and substitution

Can anybody please help me? thx

Alex
  • 4,873
naruto25
  • 461
  • already edited :) – naruto25 Nov 10 '19 at 23:20
  • You say you know what $\sinh(3x)$ is. What's stopping you from using that? And you say you have to use substitution. What substitution seems sensible to you? What's stopping you from doing that substitution? – Arthur Nov 10 '19 at 23:30
  • What have you tried? An immediate method comes to mind, but I want to know if you’ve tried it. – mdave16 Nov 10 '19 at 23:30

1 Answers1

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Let $t=\sinh(x)$. Then $\sinh(3x)=4t^3+3t$. So your equation becomes $$ 6+t = 4t^3+3t, $$ $$ 4t^3+2t-6=0, $$ $$ (2t-2)(2t^2+2t+3)=0. $$ If a product of two factors is zero, then at least one of the factors must be zero: $$ 2t-2=0 \quad\mbox{ or }\quad 2t^2+2t+3=0. $$ The first factor gives us a real solution $$t=1, \quad \sinh x=1, \quad x=\sinh^{-1}(1)=\ln(1+\sqrt{2})\approx0.88137. $$ The second factor does not give us real solutions t (but there are complex ones).

Alex
  • 4,873
  • Yea, I got exactly there and then I do not know what else to do – naruto25 Nov 10 '19 at 23:35
  • How did you come to those two factors? That is my obstacle. – naruto25 Nov 11 '19 at 15:19
  • You can always use Wolfram Alpha – Alex Nov 11 '19 at 16:01
  • yea but if you have to write on paper without any helper :) – naruto25 Nov 11 '19 at 17:38
  • Please refer to SparkNotes explanation how to factor a polynomial to (x-root1)(x-root2)... "by hand" – Alex Nov 11 '19 at 17:45
  • yea, now I see, same technique which is used in linear algebra - horner's scheme, didnt realize I can use it there too :-) – naruto25 Nov 11 '19 at 18:01
  • I have one (maybe stupid) question. If I take prescription for sinh(x), put in 1 and then make its exponent -1 and finally get 0.850 Why it is not similar to derived prescription of argsinh(x)? – naruto25 Nov 11 '19 at 23:07
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    The exponent -1 in f^{-1}(x) means: the inverse function of f. This notation is actually used; e.g. WolframAlpha understands both sinh^(-1)(1) and arsinh(1). But a special thing about hyperbolic functions sinh, cosh, ... is that they are expressed in terms of exp(x) - and their inverses are expressed in terms of ln(...). For example, in this question, arsinh(1) = ln(1+sqrt(2)) – Alex Nov 11 '19 at 23:23