Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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6
votes
4answers
106 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
0
votes
0answers
8 views

What does 3D gaze direction contains? And how to convert it to yaw and pitch?

I am trying to use a dataset. But I am facing two problems or confusions in understanding it. Can anbody guide me what 3D gaze direction stands for or means (angles, (x, y, z) coordinates or what)? ...
2
votes
3answers
56 views

How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?

How do I solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x $? Note : I have tried using trigonometric transformation but it seems very complicated to get the result .. may ...
2
votes
3answers
45 views

Find the general values of $x$ satisfying the trigonometric equation

Find the general values of $x$ satisfying $$ \frac{\tan^2 x \sin^2 x}{1-\sin^2 x \cos2x}+\frac{\cot^2 x \cos^2 x}{1-\cos^2 x \cos2x}+\frac{2\sin^2 x}{\tan^2 x+\cot^2 x}=\frac{3}{2} $$ It ...
2
votes
1answer
29 views

How to use Chebyshev Polynomials to approximate Sine and Cosine? [on hold]

How to use Chebyshev Polynomials to approximate Sine and Cosine? Optional: Compare the methods of using Chebyshev Polynomials and using the Taylor Series to approximate Sine and Cosine. Possible ...
0
votes
0answers
22 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
1
vote
3answers
33 views

Trigonometry equation maximum

Given the equation: $\cos x + \sqrt3 \sin x = a^2$ find the maximum value for $a$ for which the equation has solutions and for this case solve the equation, $a \in \mathbb{R}$. I'm guessing I need ...
3
votes
1answer
61 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
1
vote
0answers
30 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
0
votes
1answer
38 views

When solving trigonometric irrational equations does the condition of existence of the radicand under an even root matter?

Hi everyone I would like to ask a thing about the following equation: $$\cos(x) + \sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = 0$$ It is trigonometric and irrational, the root's index is 4 (even ...
0
votes
1answer
29 views

linear or bilinear interpolation

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to ...
-1
votes
2answers
23 views

Euler's formula for off-center circle [on hold]

A circle with radius $R$ and center at $(a,b)$ is given by the formula $(x-a)^2 +(y-b)^2 = R^2$. A circle with radius $R$ whose center is at the origin is given by Euler's formula: $R e^{i \theta}$. ...
0
votes
1answer
41 views

Derivative of a trigonometric function

What is the derivative of $$\cos^2 a (\tan a - \tan b)$$ Please anyone explain in detail. The differentiation is with respect to $a$. I tried to obtain the answer using chain rule, but didn't get it. ...
2
votes
1answer
62 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
1
vote
0answers
40 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
1
vote
2answers
33 views

Find the least positive angle satisfying the trigonometric equation

$\sin^3 x+\sin^3 2x+\sin^3 3x=(\sin x+\sin 2x+\sin 3x)^3$. I did solve the question, but my method is highly tedious. I combined the sin and then opened the cubic.... Is there some trick? Something I ...
2
votes
1answer
47 views

Solving $\sin x=2\sin(2\pi/3-x)$

How can I solve the equation: $$\sin x=2\sin\left(\frac{2\pi}{3}-x\right)$$ Without using the formula: $$\sin(a-b)=\sin a \cos b-\sin b \cos a$$? Thanks.
0
votes
6answers
36 views

Trigonometrical inequation problem

Solve the inequation: $\sin^4x+\cos^4x \geq 1/2$. I did this: $(1-\cos^2x)^2+\cos^4x \geq 1/2$ $-2\cos^2x+2\cos^4x \geq -1/2$ $-2(\cos^2x-\cos^4x) \geq -1/2$ $\cos^2x(1-\cos^2x) \leq 1/4$ ...
0
votes
3answers
34 views

Find the roots of equation involving $\arctan x$

I try to find the roots of the equation: $$y=x-2\arctan\left(x\right)$$ I know that one of them is $(0,0)$ but there are two others that should solve $$\dfrac{x}{2}=\arctan\left(x\right).$$ Is ...
0
votes
2answers
73 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
1
vote
2answers
86 views

Evaluate the indefinite integral $\int \frac{\cos \theta}{ \sqrt{2 - 9 \sin^2 \!\theta}} \mathrm{d}\theta$

I want to evaluate $$\int \dfrac{\cos \theta \, \mathrm{d}\theta}{ \sqrt{2 - 9 \sin^2 \theta}}$$ but I can't seem to get the answer, my working is as below:
-1
votes
0answers
32 views

Point in the circle [on hold]

I have a circle with center $(x, y, z)$. Circle lies in plane which normal also known $(n_1, n_2, n_3)$. How calculate points on circle for given angle?
5
votes
2answers
91 views

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does ...
6
votes
2answers
49 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges ...
0
votes
2answers
46 views

Integrating inverse trigonometric functions

I want to find the integral of $$\frac {\sin^{-1}(\ln x)}{x}$$ I know the best way to find th integration of trigonometric shirt substitutions is to substitute to eliminate the inverse trigonometric ...
0
votes
2answers
43 views

Is this always true that if the angle in degrees is negative, its radian counterpart will also be negative and vice-versa?

I want to know : if the angle in degrees is negative, will its radian counterpart will also be negative, or can it be anything(positive/negative)? I know that 180 degrees = pi radians. but, does ...
0
votes
0answers
24 views

initial height = 60“. There is 5 degree decline over 163.5”. What is ending height?

I'm building a roof for a structure and need to get the ending height correct. The initial height is 60". The adjacent length (the ground) is 163.5". The decline is 5 degrees. I have gotten the ...
3
votes
4answers
109 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
-1
votes
2answers
48 views

$A\sin X +B\cos X=c,\,B\sin X -A\cos X =d.$ Eliminate $X$

$A\sin X +B\cos X=c,\,B\sin X -A\cos X =d.$ Eliminate $X$ How can we eliminate the angle? I can't understand the question. How to try these type of questions?
5
votes
4answers
99 views

$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $

$$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $$ How can I solve this one, I mean I get something like this: $-3+\left(-1+2\cos ...
0
votes
2answers
29 views

Is the arctan of a negative number always negative?

Is the $\text{ arctan}$ of a negative number always negative and $\text{ arctan}$ of a positve number always positive ?
1
vote
2answers
80 views

Prove that $\sin^{2}{\theta} + \cos^{2}{\theta} = 1.$

I believe that I have been able to prove that Prove $\sin^{2}{\theta} + \cos^{2}{\theta} = 1, \forall \theta,$ but I would like to ask if my proof is correct / valid.
1
vote
3answers
36 views

Intersections of trigonometric functions and $x$

I was fiddling with my calculator and disovered something odd: $\sin x$ only intersects $x$ (as it seems) at $x=0$. Why is that? Furthermore, what is the significance of the intersection of $\cos x$ ...
0
votes
1answer
21 views

Mobius map problem [on hold]

In computer science, a neural network (NN) is a digital representation of a brain. It can have any number of numeric inputs, any number of numeric outputs, and can be trained to do pretty much ...
0
votes
0answers
12 views

Calculate pitch, yaw, and roll from mag, acc, and gyro data [on hold]

I've recently been a part of a project, in which we take a picture with a camera placed on an arduino board. The arduino board also contains a 9 degrees of freedom sensor, from which we should be able ...
1
vote
2answers
21 views

Finding the angle of inclination of a cone.

After my lecture on solving triple integrals with spherical coordinates, we defined $\phi$ as the angle of inclination from the positive z-axis such that $0\leq \phi \leq\pi$. What I don't understand ...
0
votes
0answers
11 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
1
vote
2answers
60 views

Minimum value of cosA+cosB+cosC in a triangle ABC

I have used jensen's inequality but couldn't move on.
1
vote
2answers
36 views

For given $t$ and $x$ and $y$, is there at least one $f$ such that $\cos ft = x, \sin ft =y$?

Suppose that $t$, $x$ and $y$ are given and are all in $\mathbb{R}$. Is there always at least one $f$ such that $\cos ft = x, \sin ft =y$? Edit: OK I forgot to add that given $x$ and $y$ are such ...
0
votes
0answers
23 views

Solve for 'y' for elipse rotated at an angle

How solve for y if we have set of x coordinates for elipse rotated at an angle 'A' ,has the origin at (h,k) and height as a and b
0
votes
1answer
26 views

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
1
vote
3answers
34 views

Solve for $x$ from an equation containing inverse trigonometric functions

How to solve the following for $x$? $$ \sin^{-1}\left(\frac{2a}{1+a^{2}}\right)+ \sin^{-1}\left(\frac{2b}{1+b^{2}}\right)= 2 \tan^{-1}(x ) $$ What conditions apply?
0
votes
4answers
60 views

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$? [on hold]

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$ ?
32
votes
14answers
3k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
3
votes
1answer
37 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
0
votes
1answer
15 views

Find coordinates of bounding box corners of rotated rectangle

I have a rotated rectangle inside a bounding box. It can be rotated to any angle. I know the coordinates of the "top left" corner of the inside rectangle (and I am able to work out the other 3 ...
1
vote
1answer
62 views

Prove that $\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$ has no rational roots.

Show that the following equation has no rational roots. $$\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$$ This is what I've tried: $$\left ( \frac{2\pi x}{x^2+x+1}=\frac{\pi}{6}+2k\pi ...
-1
votes
2answers
38 views

Is there a way of finding the general solution to this equation

I only know how to solve an equation $\sqrt{x}=\sin(3x)$ by newton raphson method of estimating the zeros of the equation. But I am looking for some other method of generalized solution to such an ...
1
vote
1answer
19 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; ...
2
votes
2answers
41 views

Help with indefinite integration

I am learning indefinite integration, yet am having problems understanding and recognizing where to substitute what. a good trick is to attempt convert algebraic expressions into trigonometric and ...