I have studied trigonometry for two years. In my current class, we are being taught trigonometric functions (also known as circular functions) in which a (unit) circle is involved whereas in my previous classes we were taught trigonometric ratios in which a right angled triangle was involved. The author states at the beginning of the chapter that: there are two approaches to trigonometry, one using right angled triangle and another using a unit circle and in this class we will study trigonometry using the (unit) circle approach. He writes that the unit circle approach helps us define the trigonometric functions of real numbers which is required in calculus and from there onwards the chapter continues.
My question: What is the difference between the two approaches to trigonometry? Why were we taught the right angled triangle approach at the starting and not the unit circle approach if the latter is comparatively more useful?
Best Answer
The two approaches are just that - different ways to get at the same underlying mathematics. That mathematics has several different uses. One is to "solve triangles" - figure out some sides and angles when you know the values of other sides and angles. You start with right triangles but then move on to more general ones. That's where you began, and where most studies of trigonometry begin.
It turns out that those same trigonometric functions are also useful when you want to study periodic phenomena, like the length of the day in the year, or springs bobbing up and down, or old-fashioned phonograph turntables. Then it's the approach using the unit circle that gets you to the useful material faster.
It's also thought that the triangle approach is easier to understand than the unit circle approach, which is perhaps why it comes first in the curriculum.