Hyperbolic Functions

Question 1

What does inverse sinh look like

Answer 1

A slanted s

Question 2

What’s the domain and range of arsinh

Answer 2

Domain {x: x all real numbers}
Range {y: y all real numbers}

Question 3

What does arcosh

Answer 3

A half semi circle starting at x = 1

Question 4

What is the domain and range of arcosh

Answer 4

Domain {x: x all real number x>=1}
Range { y: y>=0}

Question 5

How to derive the logarithmic form of arsinh

Answer 5

Swap the x and y
Multiply through by 2
Multiply by e^y
Solve the quadratic
E^y is always positive and the square root of x^2 + 1 is always bigger then x so there is no negative

Question 6

If cosh x = k k>1

Answer 6

Then the solutions are negatives of each other

Question 7

How to find the logarithmic form of arctan

Answer 7

Swap the x and y
Multiply through by 2
Multiply by e^y
Solve the quadratic
Cosby is a many to one function so in order to have an inverse you need to reduce the domain x>0

Question 8

How to differentiate arsinh

Answer 8

Swap x and y
Differentiate implicitly
Use cosh^2 x -sinh^2x = 1
The gradient is always positive

Question 9

How to differentiate arctanh

Answer 9

Do the same method but replace tanhx with sinhx/coshx

Question 10

When u differentiate arc sinh and cosh

Answer 10

1/ root x^2 + a^2

1/ root x^2 - a^2

Question 11

Sinh2x

Answer 11

2sinhxcoshx

Question 12

Cosh2x

Answer 12

Cosh^2x + sinh^2x

Question 13

What’s the main hyperbolic identity

Answer 13

Cosh^2x - sinh^2x = 1