Mathematics · Class 11 Science
Ch 13Hyperbolic Functions — Class 11 Mathematics, concept-first.
We build the hyperbolic functions directly from the exponential function , in the same way circular (trigonometric) functions can be pictured on the unit circle. For any real number , we define
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Hyperbolic Identities and Addition Formulas
The master identity of this chapter is , which follows at once from expanding both squares in terms of and and watching every cross-term cancel.
Most relevant Q&A
- Prove that $\cosh 2x = 1 + 2\sinh^2 x$ for all real $x$, using the exponential definitions of $\sinh x$ and $\cosh x$.Free
- If $\sinh x = \dfrac{3}{4}$, find the value of $\cosh 2x$.Free
- Using the addition formulas for hyperbolic sine, prove that $\sinh(x+y) + \sinh(x-y) = 2\sinh x \cosh y$.Preview
- If $\tanh x = \dfrac{5}{13}$, find the values of $\cosh 2x$ and $\sinh 2x$.Preview
- Show that: $(\cosh x + \sinh x)^n = \cosh(nx) + \sinh(nx)$, for any $n \in R$.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Definitions of Hyperbolic Functions
We build the hyperbolic functions directly from the exponential function , in the same way circular (trigonometric) functions can be pictured on the unit circle. For any real number , we define
Domain, Range and Graphs of Hyperbolic Functions
Because each hyperbolic function is built from and , their domains and ranges follow from the behaviour of the exponential function rather than from any periodicity — hyperbolic functions are not peri…
Inverse Hyperbolic Functions and Their Graphs
Since is continuous and strictly increasing on all of with range , it is a bijection and therefore has a genuine inverse defined for every real , with range .
Identities, Addition Formulas and Logarithmic Forms
Fundamental identities. From the definitions, for every real — the hyperbolic analogue of , but with a minus sign, which is exactly why these are called hyperbolic functions: the point traces the righ…
+−Exercise 9(a)i8 questions
- Q1Prove that $\cosh 2x = 1 + 2\sinh^2 x$ for all real $x$, using the exponential definitions of $\sinh x$ and $\cosh x$.Free
- Q2If $\sinh x = \dfrac{3}{4}$, find the value of $\cosh 2x$.Free
- Q3Express $\cosh^{-1}\left(\dfrac{5}{4}\right)$ in logarithmic form and hence find its value.Free
- Q4Solve the equation $\cosh x = 2$ for $x$, giving the answer in logarithmic form.Preview
- Q5Using the addition formulas for hyperbolic sine, prove that $\sinh(x+y) + \sinh(x-y) = 2\sinh x \cosh y$.Preview
- Q6Express $\tanh^{-1}\left(\dfrac{1}{2}\right)$ in logarithmic form and hence evaluate it.Preview
- Q7Starting from the definitions of $\tanh x$ and $\operatorname{sech} x$ in terms of $\sinh x$ and $\cosh x$, prove that $1 - \tanh^2 x = \ope…Preview
- Q8If $\tanh x = \dfrac{5}{13}$, find the values of $\cosh 2x$ and $\sinh 2x$.Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 8 questionsHide questions8 questions
- Q1Show that: $(\cosh x + \sinh x)^n = \cosh(nx) + \sinh(nx)$, for any $n \in R$.Preview
- Q2For any $x \in R$, prove that $\cosh^4 x - \sinh^4 x = \cosh(2x)$.Preview
- Q3If $\cosh x = \sec\theta$, then prove that $\tanh^2 \dfrac{x}{2} = \tan^2 \dfrac{\theta}{2}$.Preview
- Q4If $\sinh x = \dfrac{3}{4}$, then find $\sinh(2x)$.Preview
- Q5Prove that for any $x \in R$, $\sinh(3x) = 3\sinh x + 4\sinh^3 x$.Preview
- Q6Show that $\tanh^{-1}\left(\dfrac{1}{2}\right) = \dfrac{1}{2}\log_e 3$Preview
- Q7If $\sinh x = 3$ then show that $x = \log_e\left(3 + \sqrt{10}\right)$.Preview
- Q8For any $x \in R$, prove that $\cosh^4 x - \sinh^4 x = \cosh(2x)$.Preview