Further hyperbolic identities and proofsAQA A-Level Further Maths: Flashcards

What these 13 flashcards ask

State the fundamental hyperbolic identity.
State the identity for \operatorname{sech}^2x.
State the identity for \operatorname{cosech}^2x.
How do you obtain \operatorname{sech}^2x=1-\tanh^2x?
How do you obtain \operatorname{cosech}^2x=\coth^2x-1?
State the double-angle formula for \sinh 2x.
State the double-angle formula for \cosh 2x.
Write \cosh 2x in terms of \sinh x only.
Write \cosh 2x in terms of \cosh x only.
State \tanh 2x in terms of \tanh x.
What is the main sign difference from trigonometry?
Name two ways to prove a hyperbolic identity.
What range checks rule out invalid roots?

Exam questions on Further hyperbolic identities and proofs

Given that $\tanh x=\frac{3}{5}$ and $x>0$ .
Find the exact value of $\sinh 2x$ .2 marks
Given that $\coth x=\frac{5}{4}$ and $x>0$ .
Find the exact value of $\cosh 2x$ .2 marks
You may assume $\cosh^2x-\sinh^2x=1$ and $\cosh 2x=\cosh^2x+\sinh^2x$ .
Show that $\cosh 2x\equiv1+2\sinh^2x$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).