Question

How can I approximate the sine or cosine value of a really smaller number (like 0.1) without just assuming it equals the value?

Answer 1

\[\cos(x)\approx1-\frac{x^2}{2}\]

Answer 2

for small values of x only

Answer 3

You could use a Taylor/Maclaurin series

Answer 4

@Bobo-i-bo How did you arrive at that? From what trig identity does it come from?

Answer 5

I think he got it from the Taylor series for cosx.

Answer 6

Yeah, you could justify it by the Taylor Series. Another justification is this if you know the small angle approximation for sin(x) (which is sin(x)~x for x small):
\[\cos(x)= \cos^2( \frac x 2)- \sin^2(\frac x 2)=1-2 \sin^2(\frac x 2)\]
\[\approx 1 - 2(\frac x 2)^2 = 1 - \frac{x^2} 2\]

Answer 7

Thank you for the explanation @Bobo-i-bo! I wish I could give a medal to you too agent0smith.