Comparing Hyperbolic and Cartesian Trig Properties

In summary, the conversation discusses the similarity between hyperbolic trigonometry properties and cartesian trigonometry properties. It also suggests using the chain rule to solve the problem at hand.
  • #1
chwala
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Homework Statement
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hyperbolic trig. properties
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I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties...

My approach on this;

##\tanh x = \sinh y##

...just follows from

##y=\sin^{-1}(\tan x)##

##\tan x = \sin y##

Therefore continuing with our problem;

##\sech^{2}x= \cosh y \dfrac{dy}{dx}##

##⇒\dfrac{dy}{dx}= \dfrac{\sech^{2}x}{\cosh y}##

We know that;

##\cosh^2 y - \sinh^2y =1##

Therefore,

##\dfrac{dy}{dx}= \dfrac{\sech^{2}x}{\sqrt{1+\sinh^2y}}##

which gives us;

##\dfrac{dy}{dx}= \dfrac{\sech^{2}x}{\sqrt{1+\tanh^2x}}##

would appreciate insight or any other approach...
 
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  • #2
Why don't you simply use the chain rule?
 
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Related to Comparing Hyperbolic and Cartesian Trig Properties

1. What is the difference between hyperbolic and Cartesian trigonometric functions?

Hyperbolic and Cartesian trigonometric functions are different types of mathematical functions used to describe the relationship between the sides and angles of a triangle. Hyperbolic functions are based on the hyperbola, while Cartesian functions are based on the Cartesian coordinate system. This results in different equations and properties for each type of function.

2. How are the graphs of hyperbolic and Cartesian trigonometric functions different?

The graphs of hyperbolic and Cartesian trigonometric functions have different shapes and characteristics. Hyperbolic functions have a more curved shape, while Cartesian functions have a more angular shape. Additionally, the range and asymptotes of the graphs differ between the two types of functions.

3. Can hyperbolic and Cartesian trigonometric functions be used interchangeably?

No, hyperbolic and Cartesian trigonometric functions cannot be used interchangeably. While they both describe the relationship between sides and angles of a triangle, they have different equations and properties. Attempting to use them interchangeably can lead to incorrect calculations and results.

4. What are some real-world applications of hyperbolic and Cartesian trigonometric functions?

Hyperbolic and Cartesian trigonometric functions have various real-world applications, including in physics, engineering, and astronomy. They are used to calculate the motion of objects, the shape of bridges and buildings, and the orbits of planets and satellites.

5. Are there any similarities between hyperbolic and Cartesian trigonometric functions?

Yes, there are some similarities between hyperbolic and Cartesian trigonometric functions. Both types of functions involve the use of triangles and have similar relationships between their sides and angles. Additionally, some of their properties, such as symmetry and periodicity, are also similar.

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