Answer to Question #242758 in Algebra for JaytheCreator

Question #242758

(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


1
Expert's answer
2021-09-29T00:01:34-0400

We have

"sinh(x)=\\frac{e^x-e^{-x}}{2}\\\\\ncosh(x)=\\frac{e^x-+e^{-x}}{2}\\\\\nP\\cdot sinh(x)+Q\\cdot xosh(x)=P\\cdot \\left( \\frac{e^x-e^{-x}}{2} \\right)+\nQ\\cdot \\left( \\frac{e^+-e^{-x}}{2} \\right)=\\\\\ne^x\\cdot \\left( \\frac{P+Q}{2} \\right)+\ne^{-x}\\cdot \\left( \\frac{-P+Q}{2} \\right)=4\\cdot e^x-3\\cdot e^{-x}"

Therefore

"\\begin{cases}\n P+Q=8 \\\\\n -P+Q=-6\n\\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;\\\\\ne^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\\\\\nt^2+4.486\\cdot t-1=0;\\\\\nt=\\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)


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