Table of Contents
What is the fixed point of Sinx?
Since sinx is periodic, we consider one period (-π, π]. As seen from the figure, in one period (-π, π], x∗ = 0 and x∗ = π are the two fixed points of ˙x = sinx.
What is the range of sin x and cos x?
Note that the domain of the function y=sin(x) ) is all real numbers (sine is defined for any angle measure), the range is −1≤y≤1 . The graph of the cosine function looks like this: The domain of the function y=cos(x) is all real numbers (cosine is defined for any angle measure), the range is −1≤y≤1 .
Is it ever possible for sin x cos x explain?
From the unit circle we see that sin x and cos x can only have the same value in two places, in x = /4 and x = 5 /4 (45° and 225°). The equation sin x = cos x can also be solved by dividing through by cos x. If we put k = 0 and k = 1 we get the solutions /4 (45°) and /4 + = 5 /4 (45°+ 180°= 225°).
How do you find fixed points?
Geometrically, the fixed points of a function y = g (x) are the points where the graphs of y = g (x) and y = x intersect. In theory, finding the fixed points of a function g is as easy as solving g (x) = x. The fixed points can also be found on figure 1, by looking at the intersection of y = x and y = x2 − 2.
How accurate is sin (and cos) for embedded systems?
Here is a simple fixed-point approximation to sin (and cos) appropriate for embedded systems without dedicated floating-point hardware. It is accurate to within ±1/4096 (0.01\% Full-Scale). No lookup-table is required, and portable C code is available.
Is there a fixed-point Sine Sine approximation for a 32×32 HW Mac?
With a full integer 32×32 HW MAC, but no floating point unit, fixed-point routines made the most sense for the math. If performance is required, this general approach could be optimized for your particular hardware. This post is inspired by, and a derivative of, the excellent Another fast fixed-point sine approximation.
How do you prove that a function has a fixed point?
Proof • If ��=�, or ��=�, then � has a fixed point at the endpoint. • Otherwise, ��>� and ��<�. • Define a new function ℎ�=��−� –ℎ�=��−�>0 and ℎ�=��−�<0 –ℎ is continuous • By intermediate value theorem, there exists �∈(�,�) for which ℎ�=�.
Why do we use cosine instead of Sine?
The cosine approximation had an average error of (2.82±2.05)E-6 and a maximum error of 12.6E-6. Given that the cosine design takes less hardware resources and is actually more accurate, it may make sense to implement cosine over sine. Sine itself can be obtained from cosine by shifting the input by 90°.