8
votes
1answer
80 views

Hardy's approximation for the cosine

I was reading about the Hardy's approximation for the cosine function (here and also in Mathworld): for 0<x<1 What I would like to know is, how was this approximation derived? What other uses ...
2
votes
3answers
45 views

How was this approximation of transcendent equation solution found?

I have an equation for $\xi$: $$\xi\gamma=\cos\xi,$$ where $\gamma\gg1$. I've tried solving it assuming that $\xi\approx0$ and approximating $\cos$ by Taylor's second order formula: ...
2
votes
0answers
59 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
1
vote
1answer
81 views

Why is $\tan 54^\circ\approx \frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

This question was asked as an equality on MSE and I am quite surprised to find that its strictly false However I would like to see why is their difference of the order $10^{-15}$? $$\tan ...
1
vote
1answer
41 views

Approximation for $\sin(\beta\sin(x))$

Can someone explain why, assuming $\beta\ll 1$, we have $$\cos(\beta \sin(2\pi f_mt))\approx 1$$ and $$\sin(\beta \sin(2\pi f_mt))\approx \beta \sin(2\pi f_mt) $$ the equations are part of a FM ...
0
votes
1answer
40 views

Exact calulation of trigonometric functions

My question is a bit related to computer science and I am not quit sure if this place is a good one for it. Let me know if I should move it. So as you know values of functions like sin or cosine can ...
1
vote
1answer
97 views

Rational approximation of $\tanh\,(\sqrt[4]{s}$)

I'd like to find a rational representation of $$f(s) = \frac{\tanh\,\sqrt[4]{s}}{\sqrt[4]{s}}= \frac{a_0 + a_1 s + a_2 s^2 + ... + a_n s^n}{b_0 + b_1 s + b_2 s^2 + ... + b_m s^m} $$ For the case ...
3
votes
1answer
78 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
24
votes
4answers
621 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
1
vote
4answers
123 views

Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$ d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)} $$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
1
vote
1answer
86 views

Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
4
votes
4answers
135 views

$\sin x$ approximates $x$ for small angles

In physics, particularly in waves, we make use of the fact that for small angles (less than $\pi/12$-ish), the sine function value of an angle is pretty close to the value of the angle itself (in ...
4
votes
1answer
111 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
4
votes
2answers
134 views

Ratio between trigonometric sums: $\sum_{n=1}^{44} \cos n^\circ/\sum_{n=1}^{44} \sin n^\circ$

What is the value of this trigonometric sum ratio: $$\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ} = \quad ?$$ The answer is given as ...
1
vote
1answer
38 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 ...
5
votes
1answer
1k 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
269 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 $y > \sin(x)$? ...
4
votes
1answer
390 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
148 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 ...
5
votes
1answer
161 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
86 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 ...
7
votes
3answers
5k 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 ...
18
votes
8answers
2k 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
941 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
2k 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
640 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) + ...
3
votes
2answers
140 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
124 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
380 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 ...