0

When a sine wave (or a cosine wave) is graphed, it repeats itself at a fixed period. However, the value $\sin(\frac{\pi}{2})$ is geometrically undefined because you can't have a triangle with two right angles. The same argument is true for all obtuse angles. So why do we still graph these and use these values as valid inputs to trignometric functions?

u8y7541
  • 718
  • Google "unit circle" –  Apr 14 '17 at 03:44
  • That depends on how you define the sine function in the first place. It is useful to us to define sine from an analytical point of view as $\sin(z):=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\dots=\sum\limits_{k=0}^\infty (-1)^{k}\frac{z^{2k+1}}{(2k+1)!}$. From that perspective, any input makes sense to talk about, even complex inputs. – JMoravitz Apr 14 '17 at 03:45
  • @ZacharySelk Yes, I do know that sine and cosine can be defined as the y and x coordinates as you go through points on a circle. However, it still doesn't make sense using the original definition. – u8y7541 Apr 14 '17 at 03:45
  • 2
    So your question isn't really about repeating, as much as it is "Why did we switch from triangles to the unit circle?" – pjs36 Apr 14 '17 at 03:47
  • @u8y7541 The original definition can be seen as a being special case of the unit circle definition. Since the original definition agrees with the unit circle definition for angles in the range $(0, \pi/2)$, it is natural to use the unit circle to extend the definition of the trigonometric functions. – Dustan Levenstein Apr 14 '17 at 03:47
  • It looks like this question (was located on the related bar to the right) should contain a lot of the information you are looking for. – JMoravitz Apr 14 '17 at 03:53
  • @JMoravitz Thanks, reading it was helpful. – u8y7541 Apr 14 '17 at 03:56
  • Why do you think triangles is the "original" definition? I was taught with circles first and it always made more sense to me. Anyway "original" definition is irrelevent. It wasn't good enough because you can't have two right angles. So we made it better. The physical actual side of triangle and the right angle have nothing to do with anything, so we shouldn't let them trip us up. – fleablood Apr 14 '17 at 05:08

1 Answers1

1

If you choose to define sin and cos using the unit circle, then you can have a function with more values which can still be used to make inferences about right angled triangles. Any theorems proven about sin will then be applicable to the specific cases where the angles involved are less than 90 degrees.

A function $f$ which agrees with another function $g$ on a smaller domain is called an extension of $f$. Finding extensions of functions with nice properties to larger domains (where they still retain some/all nice properties of the 'original' function) is a common theme in mathematics. Extending sin and cos to the unit circle is an example of this.

Bernard W
  • 2,141