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Find the numerical value of each expression.

(a) $ \sinh 1 $

(b) $ \sinh^{-1} 1 $

$$

\sin h^{-1} 1 \approx 0.881

$$

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Missouri State University

Campbell University

Oregon State University

to find the hyperbolic sine of one. We note that the hyperbolic sine of X. This is equal to erase the x minus erase the negative X over two. And so if Taxes one We have Hyperbolic Sine of one. This is equal to erase to one minus erased negative 1/2. That's the same as 1/2 times E minus one over E. Or this is equal to 1/2 times t squared minus one over E. Or that's the same as E squared minus one over to E. Now for the inverse of the hyperbolic sine suppose we let this as X. And so from here we have one equal to hyperbolic sine of X. Now hyperbolic side effects. This is suggests erased X minus erase the negative X over two. And so from here we have one equal to e. race to X erase the negative X over two. And then from here we will solve for X. Multiplying both sides by two. We have to equal to the race to x minus one over e. to the zero equal to erase x squared -2 E. raised X -1. From here. We want to apply quadratic formula and we have erase two X. This is equal to negative B plus or minus the square root of B squared minus four. A. C over two A. In which it is just one And then B is -2 & C is -1. So we have erase excess is equal to Negative of -2. That's positive too plus or minus. You have negative two square. That's for minus four times one times negative one Over two times 1. This is just too Plus or minus the square it of eight over to And it's just just equal to two plus or minus two squared of 2/2. Or that's just one plus or minus squared of two. Now, if this is our erased X. Then applying Ellen on both sides, we have L. N. Of one plus or minus the square root of two. This is equal to X. Therefore the hyperbolic sine or the inverse of the hyperbolic sine of one. This is equal to Ln of one plus the square root of two. Not that we cannot include the negative because 1- Creative to will be a negative value inside the natural log. So this is just the value of the inverse of the hyperbolic sine.