Trignometry to non-right angled triangles?

[CN495]

I learned trignometry on the basis of right-angled triangles.Is it applicable to all(acute and obtuse) triangles?and how do you explian cos or sin(<=90),since it's no longer associated with right angled triangle?.can anyone please clarify.

 

[Zubeen]

Well, the basic definition of trigonometric functions use the word "hypotenuse" nd hypotenuse is associated to only right triangles.
Hence we can't directly apply trigonometric functions on other triangles but if some how you convert the given triangle into a right triangle then you can apply it and use it.
A Larger part of Maths and Physics (numerical questions) are just about transformations, transformations of the unknown into the form you know well.
And so if you want to apply trigonometric functions on other triangles, you need to convert a part of them into right triangles without breaking any law.

 

[phyzguy]

You might look up the Law of Cosines, and the Law of Sines, which are trigonometric identities which apply to any shape triangle.

 

[Zubeen]

Though Law of cosines and sines apply to all shapes but there derivation is based on right triangles only.
Even in there derivation a part of the triangle is made to be right angled as i mentioned in my earlier post, and then the laws have been derived.

 

[DonAntonio]

I learned trignometry on the basis of right-angled triangles.Is it applicable to all(acute and obtuse) triangles?and how do you explian cos or sin(<=90),since it's no longer associated with right angled triangle?.can anyone please clarify.

You may want to to read here: http://en.wikipedia.org/wiki/Trigonometric_functions , the section "Unit-Circle Definitions", where the basic trig. functions are extended to real functions.

You will have to learn about angles measured in radians and some other stuff (trig. identites, trig. equations, calculus on trig. functions, etc.) , and this is one of the most beautiful and important stuff high school students study in mathematics, imo.

DonAntonio

 

[Zubeen]

Even after having a look on that link i would say the same, the geometrical meaning of trigonometry is associated with only right triangles.

 

[Stephen Tashi]

Even after having a look on that link i would say the same, the geometrical meaning of trigonometry is associated with only right triangles.

When applied to non-right triangles, the Law Of Sines and Law Of Cosines are used. One way that you can derive these "Laws" is to break the non-right triangle into right triangles and compute the relations among the pieces. Although the definitions of sine and cosine involve only right triangles, they can be applied to more complex situations that can be broken into right triangles.

 

[arildno]

It is perfectly possible to derive the properties of the sine and cosine functions based on definitions wholly independent of the concdept of "right angles".

For example, they can be regarded as the solutions of a second-order Sturm-Liouville differential equation; the periodicity of the functions is in this case a derivable property, rather than an "axiomatic" property.

 

[DonAntonio]

It is perfectly possible to derive the properties of the sine and cosine functions based on definitions wholly independent of the concdept of "right angles".

For example, they can be regarded as the solutions of a second-order Sturm-Liouville differential equation; the periodicity of the functions is in this case a derivable property, rather than an "axiomatic" property.

Not only that: these functions can be defined by infinite power (complex or real) series, and they also can be defined by means of the complex exponential function...

DonAntonio