Question

(i) If 4𝑒 𝑥 − 3𝑒 −𝑥 ≡ 𝑃𝑠𝑖𝑛ℎ 𝑥 + 𝑄𝑐𝑜𝑠ℎ 𝑥, determine the values of 𝑃 and 𝑄. (ii) Solve the equation 𝑐𝑜𝑠𝑒𝑐ℎ 𝑦 = −0.4458, correct to 4 significant figures

Answer 1

We have

$sinh(x)=\frac{e^x-e^{-x}}{2}\\ cosh(x)=\frac{e^x-+e^{-x}}{2}\\ P\cdot sinh(x)+Q\cdot xosh(x)=P\cdot \left( \frac{e^x-e^{-x}}{2} \right)+ Q\cdot \left( \frac{e^+-e^{-x}}{2} \right)=\\ e^x\cdot \left( \frac{P+Q}{2} \right)+ e^{-x}\cdot \left( \frac{-P+Q}{2} \right)=4\cdot e^x-3\cdot e^{-x}$

Therefore

$\begin{cases} P+Q=8 \\ -P+Q=-6 \end{cases}$

Adding equatioms we have 2Q=2=>Q=1.

Inserting this value to the first equation have P=8-1=7;

Answer: P=7,Q=1

2)cosech(x)=1/sinh(x)=

$\frac{2}{e^x-e^{-x}}=-0.4458;\\ e^x-e^{-x}=\frac{2}{-0.4458}=-4.486\\$

Let be $e^x=t$

Then we have an equation

$t-\frac{1}{t}=-4.486\\ t^2+4.486\cdot t-1=0;\\ t=\frac{-4.486+\sqrt{4.486^2+4}}{2}=0.2128$

$e^x=0.2128$

x=ln(0.2128)=-1.5474

Answer: x=-1.5474 (or y=-1.5474)