Find the exact value of ((2)).

Write down the expression to be evaluated: ((2)) Quote the exponential definition: (x)=e^x-e^-x2 State the exact value: ((2))=(3)/(4) Answer: (3)/(4)

Find the exact value of ((3)).

Write down the expression to be evaluated: ((3)) Quote the exponential definition: (x)=e^x+e^-x2 Substitute the argument into the definition: ((3))=e^(3)+e^-(3)2 State the exact value: ((3))=(5)/(3) Answer: (5)/(3)

Find the exact value of ((2)).

Write down the expression to be evaluated: ((2)) Quote the exponential definition: (x)=e^x-e^-xe^x+e^-x State the exact value: ((2))=(3)/(5) Answer: (3)/(5)

Find the exact value of ((5)).

Write down the expression to be evaluated: ((5)) Quote the exponential definition: (x)=e^x+e^-x2 State the exact value: ((5))=(13)/(5) Answer: (13)/(5)

Find the exact value of ((3)).

Write down the expression to be evaluated: ((3)) Quote the exponential definition: (x)=e^x-e^-xe^x+e^-x Substitute the argument into the definition: ((3))=e^(3)-e^-(3)e^(3)+e^-(3) State the exact value: ((3))=(4)/(5) Answer: (4)/(5)

Hyperbolic functions Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Hyperbolic functions questions. See exactly how to solve problems on hyperbolic-functions, exponential-definitions, identities, osborns-rule.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Find the exact value of

\sinh{\left(\ln{\left(2\right)}\right)}

Worked solution

Write down the expression to be evaluated
$\sinh{\left(\ln{\left(2\right)}\right)}$
Start from the expression given in the question.
Quote the exponential definition
$\sinh(x)=\frac{e^{x}-e^{-x}}{2}$
Every hyperbolic function is defined directly in terms of $e^{x}$ .
State the exact value
$\sinh{\left(\ln{\left(2\right)}\right)}=\frac{3}{4}$
This is the exact value of the expression.

Answer

\frac{3}{4}

Question 2

2 markseasy

Find the exact value of

\cosh{\left(\ln{\left(3\right)}\right)}

Worked solution

Write down the expression to be evaluated
$\cosh{\left(\ln{\left(3\right)}\right)}$
Start from the expression given in the question.
Quote the exponential definition
$\cosh(x)=\frac{e^{x}+e^{-x}}{2}$
Every hyperbolic function is defined directly in terms of $e^{x}$ .
Substitute the argument into the definition
$\cosh{\left(\ln{\left(3\right)}\right)}=\frac{e^{\ln{\left(3\right)}}+e^{-\ln{\left(3\right)}}}{2}$
Replace $x$ by the argument given in the question.
State the exact value
$\cosh{\left(\ln{\left(3\right)}\right)}=\frac{5}{3}$
This is the exact value of the expression.

Answer

\frac{5}{3}

Question 3

2 markseasy

Find the exact value of

\tanh{\left(\ln{\left(2\right)}\right)}

Worked solution

Write down the expression to be evaluated
$\tanh{\left(\ln{\left(2\right)}\right)}$
Start from the expression given in the question.
Quote the exponential definition
$\tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$
Every hyperbolic function is defined directly in terms of $e^{x}$ .
State the exact value
$\tanh{\left(\ln{\left(2\right)}\right)}=\frac{3}{5}$
This is the exact value of the expression.

Answer

\frac{3}{5}

Question 4

2 markseasy

Find the exact value of

\cosh{\left(\ln{\left(5\right)}\right)}

Worked solution

Write down the expression to be evaluated
$\cosh{\left(\ln{\left(5\right)}\right)}$
Start from the expression given in the question.
Quote the exponential definition
$\cosh(x)=\frac{e^{x}+e^{-x}}{2}$
Every hyperbolic function is defined directly in terms of $e^{x}$ .
State the exact value
$\cosh{\left(\ln{\left(5\right)}\right)}=\frac{13}{5}$
This is the exact value of the expression.

Answer

\frac{13}{5}

Question 5

2 markseasy

Find the exact value of

\tanh{\left(\ln{\left(3\right)}\right)}

Worked solution

Write down the expression to be evaluated
$\tanh{\left(\ln{\left(3\right)}\right)}$
Start from the expression given in the question.
Quote the exponential definition
$\tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$
Every hyperbolic function is defined directly in terms of $e^{x}$ .
Substitute the argument into the definition
$\tanh{\left(\ln{\left(3\right)}\right)}=\frac{e^{\ln{\left(3\right)}}-e^{-\ln{\left(3\right)}}}{e^{\ln{\left(3\right)}}+e^{-\ln{\left(3\right)}}}$
Replace $x$ by the argument given in the question.
State the exact value
$\tanh{\left(\ln{\left(3\right)}\right)}=\frac{4}{5}$
This is the exact value of the expression.

Answer

\frac{4}{5}

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