Why do we study right triangles in trigonometry?

[Avichal]

Why do we study so much about right triangles like trigonometry. We could define sin and cos like functions in a 70 degree triangle too.
I also know right triangle is something special but i don't know what is it. Also why won't trigonometry on other type of triangles be not so good

 

[arildno]

Because right-angled triangles are simplest to start out with.
Furthermore, trigonometry based on that is sufficient to handle all questions you would like to ask non-right-angled triangles, as well. Thus, developing other types of plane trigonometry is unnecessary and a waste of time.

 

[klausas]

If you find yourself working with a lot of 70 degree triangles, or any other number of degrees for that matter, feel free to make a trig table based on that if you feel it will save you time.

Right triangles are preferable as the basis for trigonometry because any other triangle can be cut into two right triangles, by drawing a line through one vertex perpendicular to its opposite side. While you could, in this day and age of analytic geometry, draw a 70 degree angled line just as well as a perpendicular, the two triangles you got from that would have one with a 70 degree angle and the other with a 110 degree angle. So, you would need two trig tables to sort it all out (or a messy set of successive approximations).

Also, when you get to unit circle trig (if you haven't already), the right triangle definition correlates with the coordinates of a point on a grid in a way that is very neat and easy to see and use. You could come up with a formula based on other angles, but it would be much messier and harder to use.

 

[mathwonk]

sin and cos have not so much to do with triangles, they are the coordinates of points on the circle. the right triangles come in because the coordinate axes are perpendicular to each other.

The reason we use perpendicular axes, or right triangle trig, is the pythagorean theorem. The pythagorean theorem for other triangles is called the law of cosines and is more complicated.

 

[CellCoree]

We study right triangles in trigonometry because they have special properties that make them useful in solving real-world problems. Right triangles have one angle that measures 90 degrees, which allows us to use the Pythagorean theorem to find the length of any side if we know the lengths of the other two sides. This relationship between the sides of a right triangle is essential in fields such as engineering, architecture, and navigation.

Additionally, the ratios of the sides in a right triangle, known as sine, cosine, and tangent, are fundamental trigonometric functions that have many applications in mathematics and science. They can be used to calculate the height of a building, the distance between two points, or the slope of a hill.

While we can define these trigonometric functions for any angle, right triangles provide a simple and clear way to understand and apply them. The 90-degree angle allows us to easily visualize and measure the sides and angles, making calculations more straightforward.

Furthermore, other types of triangles may not have the same predictable relationships between their sides and angles, making it more challenging to apply trigonometric functions. Right triangles also have the advantage of being the most common type of triangle in real-world scenarios, making them a practical focus of study.

In summary, studying right triangles in trigonometry allows us to understand and apply fundamental mathematical principles to solve real-world problems efficiently. While other types of triangles have their unique properties, right triangles offer a simple and practical approach to learning and using trigonometry.