How do we define trigonometric functions?

[Mr Davis 97]

I'm having a problem understanding exactly why trig functions are defined the way they are. Of course, the definition in terms of 0 to 90 degree angles within right triangles is easy: the functions just give the ratio of the sides given the angle. However, I don't understand how or why trig functions are defined for angles greater than 90 degrees. How does this relate to right angles? Why is it useful to define the trig functions as the points on the unit circle? That definition seems arbitrary, and not useful. Once we go beyond 90 degrees, why is the subject even called trigonometry, if it mostly only relates to the points on the unit circle? Why did we decide to define trig functions this way after defining them as ratios of the sides of right triangles? I hope that somebody can answer these question to put my mind at ease, because as of now, this doesn't make much sense to me.

 

[gopher_p]

I made this post in a related topic a while ago. I think it's relevant here.

One thing that doesn't get sufficient coverage, in my opinion, is the fact that there are actually (at least) three different versions of the trigonometric functions;

There are right-triangle trig functions that are defined in terms of ratios of side lengths of right triangles. The arguments (domain) of these functions are angles; specifically angles between 0 and 90 degrees (whether or not we include 0 and 90 depends on how you want to define "right triangle"). As ratios of side lengths, the outputs (range) of the right-triangle trig functions are positive real numbers.

Then there are the "unit-circle" trig functions which are defined in terms of coordinates of points on the unit circle. The arguments of these functions are technically arc lengths, and the outputs are, again, real numbers; e.g. for the unit-circle version of $\sin$, $\sin a$ is the $y$-coordinate of the point on the unit circle that is $a$ counter-clockwise units around the circle.

Finally, there are the analytic trig functions, which have power series definitions. The domains and ranges of these functions are real (or complex) numbers. These are the functions of primary interest in a calculus course.

Now all of the versions of the trig functions can be understood in terms of the others - e.g. it is common for students to use "reference angles" and right-triangle trig functions to aid in their understanding of the unit-circle trig functions - so they are often considered to be the same. But I believe there is value in realizing that they're fundamentally different kinds of functions that happen to be comparable.

*Remark: It's reasonable to consider a fourth class of trig functions, which I would call the "rotational" trig functions, but one could argue that this class is the same as the "unit-circle" class.

 

[Mr Davis 97]

I made this post in a related topic a while ago. I think it's relevant here.

Thank you, that helps a lot. Knowing that they are not fundamentally the same thing helps me understand why we define them the way we do. But following this, I have another question. What compelled us to define functions in terms of the unit circle? I see how the right-triangle defined functions have a lot of practical applicability, but I don't see the reason that went on to define trig functions in the way of the unit circle in the first place.

 

[bhillyard]

Technically it doesn't have to be a unit circle. But if it isn't the resulting definitions will require the radius of the circle to be considered. The reason for considering a definition in terms of a circle is that once you are considering angles greater than 90^o then we want a definition that, for angles less than 90, is equivalent to the definitions in terms of right angled triangles.

 

[CellCoree]

I can understand your confusion regarding the definition of trigonometric functions. Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are used to solve problems in geometry, physics, engineering, and other fields.

The most common definition of trigonometric functions involves right triangles, where the functions are defined as ratios of the lengths of the sides of the triangle. This definition is based on the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This definition is straightforward and intuitive, as it directly relates the angles to the sides of the triangle.

However, trigonometric functions can also be defined for angles greater than 90 degrees, and this is where the concept of the unit circle comes into play. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system. The trigonometric functions can be defined as the coordinates of points on the unit circle, where the angle formed by the point and the positive x-axis is equal to the angle being evaluated. This definition may seem arbitrary, but it has many useful applications in mathematics and physics, such as in the study of periodic functions and complex numbers.

The reason why trigonometry is still called trigonometry, even though it involves more than just right triangles, is because the fundamental concept of relating angles to sides of a triangle still applies. The unit circle can be seen as a way to extend this concept to angles greater than 90 degrees.

In conclusion, the definition of trigonometric functions may seem confusing at first, but it is based on fundamental mathematical concepts and has many useful applications. The use of the unit circle to define trigonometric functions beyond 90 degrees may seem arbitrary, but it provides a powerful tool for solving various mathematical problems. I hope this explanation has helped clarify some of your questions and put your mind at ease.