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

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-1
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0answers
32 views

Point in the circle [closed]

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
95 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
44 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
112 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
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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
37 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 [closed]

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
13 views

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

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
62 views

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

I have used jensen's inequality but couldn't move on.
1
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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
4answers
122 views
+100

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 $xy+yz+zx=1$? [closed]

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$ ?
33
votes
16answers
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
18 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 ...
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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 ...
0
votes
1answer
20 views

How to determine if a node has a positive y axis?

I am trying to determine if a node on this diagram has a positive y axis: Things I know: The blue node in the centre will always be at 0,0. second blue node will always have a positive x axis. ...
3
votes
4answers
49 views

Substituting a value of sine function in a trigonometric equation

I am trying to really understand trigonometric equations and I've stumbled upon a rather confusing example. Solve the following equation: $\sin x= 2|\sin x|+ {\sqrt{3}}\cos x$ First step is to ...
23
votes
3answers
445 views

Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

Let $\varphi=\frac{1+\sqrt5}2$ (the golden ratio). How can I simplify the following expression? $$7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$$
-4
votes
1answer
27 views

Sin theta equation [closed]

Good afternoon, I just started this new class, and am having a slight issue. I understand the $s= r \times θ$ formula to find the length of a circular arc, but I don't understand how to plug in the ...
2
votes
2answers
84 views

Prove or disprove $\frac{\sqrt{1+\tan x}}{\cot x} = \frac{1+\sin x}{\cos x}$

I have tried to prove the identity \begin{equation} \frac{\sqrt{1+\tan x}}{\cot x} = \frac{1+\sin x}{\cos x} \end{equation} by $t$-substitution but seem not to work. Please don't solve(don`t post the ...
6
votes
3answers
652 views

Two different trigonometric identities giving two different solutions

Using two different sum-difference trigonometric identities gives two different results in a task where the choice of identity seemed unimportant. The task goes as following: Given $\cos 2x ...
0
votes
3answers
52 views

What is this question asking? Geometry

Rhombus $ABCD$, with side length 6, is rolled to form a cylinder of volume 6 by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin\left(\angle ABC\right)$? Is it asking for the sin of the angle ...
0
votes
8answers
85 views

Trigonometric Property

How can I show that the following property holds? $2\cos(4a)+2\cos(2a)+1=\displaystyle\frac{\sin(5a)}{\sin(a)}$ I've been trying to derive it to no avail. What would be a way to approach similar ...
1
vote
3answers
25 views

Trigonometry on circle as function of distances to (-r,0 ) and (r,0)

I have two point A and B on a circle centered at the origin $ O = (0,0)$ with radius r And I am only told: A and B are both on the upper half plane ($ y \ge 0 $ ) the distance $a_1$ from A to the ...
1
vote
5answers
66 views

Proving $1+\cot^2(-\theta)=\csc^2(\theta)$

I'm stuck on this one proof that I just can't get for some reason. It seems really simple too, and I've tried just about everything I can think of, but I just keep going in circles. ...
3
votes
2answers
100 views

Trigonometric Limit without L'Hopital [closed]

I am having problems solving this limit without L'Hopital or series. $$ \lim_{ x\to 0 } \frac{x\cos(x) - \sin(x)}{2 x^3} $$ I tried some trigonometric manipulations without success. I tried ...
1
vote
6answers
123 views

Best way find $\lim_{x\to 0}( \frac {\sin x}{x})^{\frac 1x}$

$\lim_{x\to 0}( \frac {\sin(x)}{x})^{\frac 1x}$ $$$$ I can use Tailor to get to $\lim_{x\to 0}(1+\epsilon(x))^\frac 1x$ $$$$ $(\epsilon(x)\underset{x\to\infty}\to 0) $ $$$$ but does that mean that ...
0
votes
0answers
40 views

Simplify trigonometric equation

$$ \alpha sin\theta + \beta sin\phi + \gamma sin(\theta+\phi) = 0 $$ where $\alpha, \beta, \gamma$ are constants. I want to simplify this into a linear relation between $\theta, \phi$ I wonder if ...
47
votes
9answers
821 views

Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
1
vote
3answers
72 views

$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$

For $0<\theta<360$ $$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$$ $$\Longrightarrow \sin(2\theta + 34)= \sin (82-\theta)$$ since sine is an odd function $$2\theta + ...
-2
votes
2answers
57 views

What is the minimum value $P$ can have in $\cos(P\sin x) = \sin(P\cos x)$ [closed]

What is the minimum value $P$ can have in $\cos(P \sin x) = \sin(P \cos x)$, if there is a solution to the above equation in $x \in [0, 2\pi] $?
0
votes
4answers
53 views

Solving Trig Equation $\cos(2x)=-\sin(2x)$

Proceeding as follows: $$\cos(2x)=-\sin(2x)\Rightarrow \cos \left(2x\right)=-\cos \left(\frac{\pi }{2}-2x\right)$$ How to proceed further? Can I remove the $cos$ from both sides and proceed or no?
2
votes
2answers
45 views

Equal sign or approximation sign?

I want to solve the equation $2\enspace \sin^2\enspace \theta+2\enspace \sin\enspace \theta-1=0$, $0\leq\theta<2\pi$. Using the quadratic formula, I found one of the solution to be ...
1
vote
1answer
22 views

Chain fixed at two points, how far does it drop down?

Not too sure whether this should be in maths or physics, but oh well. If you have a metal chain of length h metres and you have 2 points, the distance between them being x metres, If h is less than ...
1
vote
1answer
32 views

Maximum value of trigonometric expression?

If $ f(x) = \cos x ( \sin x + \sqrt [2] {\sin^2 x + \sin^2 \theta} )$, where $\theta $ is a given constant, then maximum value of $f (x) $ is?The answer is in terms of $\sin \theta$ or $\cos \theta ...
1
vote
2answers
45 views

When is $a \space \sin^2(x) + b \space \cos^2(x) \le 1$?

When is the above expression less than or equal to $1$, meaning for what values of $a$ and $b$ will the above expression be less than or equal to $1$?