Basic hyperbolic identitiesAQA A-Level Further Maths: Subtopic test

1. 1
   A real number $x$ satisfies $\sinh x=\frac34$.

   (a)

   Find the value of $\cosh x$.

   [1 mark]

   • A$\frac{\sqrt7}{4}$
   • B$\frac54$
   • C$\frac{25}{16}$
   • D$-\frac54$

   (b)

   Find the value of $\tanh x$.

   [1 mark]

   • A$\frac35$
   • B$\frac53$
   • C$\frac34$
   • D$\frac45$

   (c)

   Find the exact value of $\cosh^2x+\sinh^2x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A real number $x>0$ satisfies $\tanh x=\frac45$.

   (a)

   Find the value of $\cosh x$.

   [1 mark]

   • A$\frac35$
   • B$\frac{25}{9}$
   • C$\frac53$
   • D$\frac{\sqrt{41}}{5}$

   (b)

   Find the value of $\sinh x$.

   [1 mark]

   • A$\frac53$
   • B$\frac34$
   • C$\frac45$
   • D$\frac43$

   (c)

   Hence find the exact value of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Hyperbolic functions are defined by $\cosh x=\frac12(e^x+e^{-x})$ and $\sinh x=\frac12(e^x-e^{-x})$.

   (a)

   Prove that $\cosh^2x-\sinh^2x=1$ for all real $x$.

   [3 marks]

   (b)

   Show that $\tanh^2x=\frac{\sinh^2x}{1+\sinh^2x}$, and hence find the exact value of $\tanh x$ when $x>0$ and $\sinh x=2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For real $x$, let $p=\cosh x+\sinh x$ and $q=\cosh x-\sinh x$.

   (a)

   (i) Show that $p=e^x$ and $q=e^{-x}$.
   (ii) Hence show that $pq=1$, and explain how this proves that $\cosh^2x-\sinh^2x=1$.
   (iii) Given that $p=4$, find the exact values of $\cosh x$ and $\sinh x$.

   [6 marks]

   (b)

   Given that $x>0$ and $\tanh x=\frac{5}{13}$:
   (i) show that $1-\tanh^2x=\frac{1}{\cosh^2x}$;
   (ii) hence find the exact values of $\cosh x$ and $\sinh x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions