# Defining only in terms of a unit circle

• Dec 23rd 2010, 01:07 AM
kamykazee
Defining only in terms of a unit circle
We ofcourse know that given a unit circle, certain values of trigonometric functions apply ($\displaystyle \displaystyle sin\frac {\pi}{4}=\frac {\sqrt2}{2}$, for example).

However, if we were to take a circle with a bigger radius, that would automatically lead the values of such functions to change.

Whenever i used sin(pi/4) in a problem i always took its value as the one mentioned in terms of a unit circle, when in fact it varies according to the lengths of the triangle.

So my question would be: How can we take $\displaystyle \displaystyle sin\frac {\pi}{4}$ as having a standard value of $\displaystyle \frac {\sqrt2}{2}$ when in fact it varies. What am i missing here?
• Dec 23rd 2010, 01:32 AM
Prove It
Because when you scale the radius by a factor of $\displaystyle \displaystyle r$, the perpendicular (sine value) is also scaled by a factor of $\displaystyle \displaystyle r$.

So to find the sine value, you have to divide by the radius.
• Dec 23rd 2010, 01:51 AM
kamykazee
Sorry for my mistake, i was used to the unit circle where the radius was 1, thus the sine was actually the perpendicular itself. When i went to a bigger circle i forgot to divide it by the hypothenuse of the drawn triangle and i was still taking the value of the sine as being equal to the perpendicular itself. Also since sine is a ratio between the sides of a triangle it would have been illogical to exceed one. I'm sorry for my stupid question.