Edexcel IAL Mathematics FP3.1.2 inverse hyperbolic functions

Only two direct questions exist, so keep the transfer narrow: derive the correct logarithmic branch, then use sinh identities to give exact intersection coordinates.

Exam points

Derive inverse hyperbolic log forms by solving the exponential quadratic branch.
Use sinh 3x identities to find intersections and express coordinates as logs.

[Maximum number: 6]

Given that

\cosh y=x \quad \text { and } \quad y<0

use the definition of $\cosh y$ in terms of exponential functions to prove that

y=\ln \left(x-\sqrt{x^{2}-1}\right)

$\cosh y=x,\quad y<0,\quad x=\frac{e^y+e^{-y}}{2}$ $2xe^y=e^{2y}+1 \Rightarrow e^{2y}-2xe^y+1=0$ $e^y=\frac{2x\pm\sqrt{(2x)^2-4}}{2} =x\pm\sqrt{x^2-1}$ $y=\ln\left(x-\sqrt{x^2-1}\right)$

Since y<0, $e^y<1$ , so the negative root is required.

B1 M1 M1 A1 A1* B1

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FP3.1.2 - Inverse hyperbolic functions question 1