Consider the case when $x = \pi/4$.
$\cos \pi/4 = 1 = \sin \pi/4$.
Now, if $a = 1$ and $b = -1$, $a\cos x + b\sin x = 0 $(for non-zero a and b).
Does this imply that $\cos x$ and $\sin x$ ARE NOT linearly independent?
Asked
Active
Viewed 353 times
1
-
http://math.stackexchange.com/questions/269668/linear-independence-of-sinx-and-cosx – Jessica B Aug 20 '14 at 11:17
-
Just a comment: $\cos \pi / 4 = \sqrt{2}/{2} = \sin \pi / 4$ not $1$ – DanZimm Aug 20 '14 at 11:27
1 Answers
6
The functions $\cos x$ and $\sin x$ are linearly independent. There are various ways to see this: I give a method which picks up on your own working.
First let $$a\cos x+b\sin x=0\ ,$$ and make sure you understand that this is an equality of functions: that is, it means that the LHS is zero for all values of $x$. Taking $x=\pi/4$, as you have done, shows that $a+b=0$; but taking say $x=0$ shows that $a=0$. Solving these two equations gives $a=0$, $b=0$ as the only possibility, and so $\cos x$ and $\sin x$ are linearly independent.
David
- 82,662