Tagged Questions
0
votes
1answer
14 views
approximation of law sines from spherical case to planar case
we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
3
votes
1answer
262 views
Approximating $\arctan x$ for large $|x|$
I would like to know if there is reasonably fast converging method for computing large arguments of arctan.
Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
2
votes
3answers
126 views
Simple test if point is above or below sine curve
Is there any simple formula or algorithm for determining if a point lies above or below the sine curve? For instance, if I have a point $(x, y)$, how can I test whether or not $y > \sin(x)$? ...
3
votes
1answer
215 views
Can the trigonometric functions be expressed, explained, or proven in terms of arithmetic?
I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not ...
1
vote
0answers
77 views
Solving or approximating an equation with radicals and arctan function
I have solved a differential equation recently, which left me with this whopper of inverse function to figure out. I know what $c$ is, I just haven't calculated its exact value based on the initial ...
4
votes
1answer
131 views
Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$
I read on Wikipedia that
$$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$
to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
0
votes
0answers
68 views
Approximating a function with a sine function: transform into constant amplitude?
I have a smooth function, it is stationary. So I tried approximating my function with regression by fitting a sine function that changes period, phase & frequency every observation to get the ...
6
votes
3answers
2k views
How to justify small angle approximation for cosine
Everyone knows the picture that explains instantly the small angle approximation to the sine function (as defined by the parametrisation of the unit circle): "what's the length of that arc?" "See how ...
14
votes
7answers
753 views
Rapid approximation of $\tanh(x)$
This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
10
votes
3answers
556 views
Sine Approximation of Bhaskara
An Indian mathematician, Bhaskara I, gave the following amazing of the sine (I checked the graph and some values, and the approximation is truly impressive.)
$$\sin x \approx \frac{{16x\left( {\pi - ...
5
votes
5answers
613 views
Numerically Efficient Approximation of cos(s)
I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application.
I don't need every last ...
3
votes
4answers
376 views
Find approximation to $\sin(x)$
How to find the approximation to $\sin(1.58)$ ? By using the Newton's method
$$x_{n+1} = x_{n} - \frac{f(x)}{f'(x)}$$
You always will get $0$. Using this method:
$f(x+\Delta x) \approx f(x) + ...
2
votes
2answers
104 views
Numerical Approximation Involving Trig
I have a graphics problem that reduces to this:
(Computer equation)
alpha = arctan(X / ((Y / (Z * cos(alpha) - k)) * Z * cos(alpha)))
(LaTeX)
$$\alpha = ...
2
votes
2answers
118 views
Calculate other tangents which are related
I am using a small microcontroller, which has limited processing resources. I need to calculate the three tangents:
...
2
votes
2answers
353 views
Approximating a cosine
Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a ...
