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

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MAPLE- Developed in a Fourier Basis - Simplifying commands

After a succession of simplifying commands, I am trying to have a truncated serie Fourier of the expression T3. I get this kind of result T33 : [1] : ...
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2answers
26 views

Transforming linear combination of the cosine and sine function

In the proof of Transforming $a\cos(x)+b\sin(x)$ to $r\cos(\phi-x)$ ...
3
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0answers
39 views

Develop intuition in trigonometry problems?

I'm going to write the JEE-Advanced exams, that has problems on trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple ...
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1answer
28 views

Finding the angle value given 1 point and the centre of a circle

I got the coordinates of the center of a circle $(a,b)$ as well as one other point $(x, y)$. From those I can derive the radius by applying square root to the result of following formula. $$ (x-a)^2 ...
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4answers
82 views

$\sin 4x +\sqrt{3} \sin 3 x + \sin 2 x=0$

This question is from a 2012 VMK entrance exam I was trying to solve it first by expanding $\sin 4 x = 2 \sin 2 x \cos 2x$, then by noticing that if divided by 2, one can get, e.g. $ ...
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3answers
46 views

How to solve this inequality? $2\cos(x+1)>0$ [on hold]

Please help me answer this question. How can I solve the following inequality? Solve the following inequality: $2\cos(x+1)>0$. Thank you.
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1answer
46 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{2b!}$, Then prove that the value of ...
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2answers
151 views

Finding $\lim_{x\to 0}\frac{\sin(x+x^3/6)-x}{x^5}$

I'm trying to find the limit of this expression: $$\lim_{x\to0}\frac{\sin\left(x+x^3/6\right)-x}{x^5}$$ My solution is as follows: $$ \begin{align} ...
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3answers
68 views

Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!)

I'm doing a test about limits and continuity and got these two wrong. $\mathbf{Q1}$: The function $f(x) = \tan x$: $\hspace{1em}\mathtt{a)}$ is continuous $\hspace{1em}\mathtt{b)}$ is ...
3
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3answers
87 views

Equation of a tangent to the graph of a function parallel to a line [on hold]

Please help me find the answer to this question. Thanks. What is the equation of a tangent to the graph of a function $y=x-\frac{1}{x^2}$ which is parallel to the line $y=3x$?
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3answers
50 views

Need help solving $\;\arcsin(\sqrt3\sin x)=1$

I need help solving $$\arcsin\left(\sqrt3\sin x\right)=1$$ I've tried substituting various x's in, but not exactly sure what it means to find x fitting to the arcsin.
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1answer
42 views

Exact value of $\frac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$

Let $\alpha\in\left(0,\dfrac\pi2\right)$. What is the exact value of $$\dfrac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$$ Firstly, I tried to simplify $1-2\tan^2\alpha$ and got ...
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2answers
67 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
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3answers
50 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
5
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2answers
77 views

Trigonometric equation, find $\sin \theta $

Find $\sin \theta $ if $a$ and $c$ are constants $$ 1-\left(c-a\tan\theta\right)^2=\frac{\sin^2\theta\cos^4\theta }{a^2-\cos^4\theta } $$
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3answers
115 views

Angle in a triangle within a circle.

A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC $\perp OB$. AC = 12 cm. BC = 5 cm. Calculate the size of $\angle AOB$, marked $\theta$ on the ...
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2answers
27 views

$x$-intercepts of secant function

I have tried setting $f(x) = 0$ and solving for $x$ by undoing the operations, and what I end up with is $x= -\pi/6$. The book gives the answer as B, however, and I haven't been able to obtain those ...
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2answers
32 views

The exact value of csc -420 degrees (Find the exact value of each trigonometric funtion)

I'am very confused, I have looked all over google and I can not find out how too do this problem. I have the answer its number 14 since our teacher gives us the answer but we need to show work. I ...
2
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2answers
52 views

Determinant of a 4x4 matrix with trigonometric functions

I am stuck with my homework from math. I should calcutate the determinant of a matrix: $$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ...
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3answers
97 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
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1answer
15 views

Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...
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0answers
8 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
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2answers
77 views

Computing $ \lim_{x \to 0} \left( \frac {1}{x} - \frac {1}{\sin x } \right) $

How to calculate this limit: $$\lim_{x\rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$$ All I know is: $$\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$$ $$\lim_{x\rightarrow 0} \, x = 0$$ ...
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1answer
64 views

What does “versin” mean?

$$\newcommand{\versin}{\operatorname{versin}}2\versin A+\cos ^2 A= 1+\versin ^2 A$$ I don't understand the word 'ver' in this equation. What does it mean?
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1answer
14 views

How to find initial direction and angle of collision of a ball with a vertical wall?

I have a problem in my game. I have a wall where a ball hit to a wall from anywhere. I need to give it to the direction according to the collision law. Let suppose if a ball thrown from $(0, 0)$ and ...
1
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1answer
53 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
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1answer
28 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
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3answers
71 views

Find $ \int \frac {\mathrm{d}x}{(4x^2-1)^{3/2}}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
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2answers
49 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
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3answers
105 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
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4answers
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Equivalence of equations

$ \sin ^2 \alpha = \frac{\tan ^2 \alpha}{1+\tan^2 \alpha} $ $ 1+\tan^2 \alpha = \frac{\tan ^2 \alpha}{\sin ^2 \alpha} $ It is said that these two equations are equivalent. How can that be? I know ...
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6answers
105 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
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4answers
52 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
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0answers
30 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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1answer
20 views

The vertical projection of a chord of a circle?

I was wondering if anyone could help me with the problem below (finding x): So we are given t_i (the initial tangent angle to the circle), t_o (the exiting angle of the tangent of the circle), the ...
3
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1answer
20 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
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3answers
45 views

What is the required radius of the smaller circles around a larger circle so they touch?

I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box) I would like to be able to adjust the number of ...
3
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3answers
49 views

Using trig substitution to solve for integration?

So I used a trig sub for this problem: $$\int \frac{1}{x^2\sqrt{9-x^2}}dx.$$ ${x=3\sin\theta}$ ${dx=3\cos\theta\ d\theta}$ ${\sqrt{9-x^2}= 3\cos\theta}$ I ended up with $$\frac19 \int \frac{ ...
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2answers
71 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
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1answer
43 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
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1answer
32 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
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0answers
24 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
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63 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
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2answers
51 views

How to solve $\sin(\arctan((\frac{1}{2}))$ [closed]

Can you solve $\sin(\arctan((\frac{1}{2}))$? It says I have to use a right triangle
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4answers
57 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
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4answers
104 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
3
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2answers
99 views

Simple Equation Does my proof work?

Its the inequality equation $|a+b| \leq |a|+|b | $ I managed this by cases. Let $c = a$ and $d=b$ if $a>b $ let $c = b$ and $d = a$ if $b>a $ if $a=b$ let $a=c$ Hence we have $|c+d| \leq ...
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0answers
104 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-2mx+N = ...
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2answers
94 views

Prove that $\lim_{x\to\frac{2}{\pi}}\big\lfloor\sin\frac{1}{x}\big\rfloor=0$ [closed]

Prove that $$\lim_{\large x\to \frac{2}{\pi}} \left\lfloor\sin\left(\frac{1}{x}\right)\right\rfloor=0$$ using the $\varepsilon$-$\delta$ definition of limits. Note that $\lfloor 0.1\rfloor = 0,\; ...
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2answers
52 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...