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

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10
votes
4answers
131 views

Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$

How to evaluate the following integral $$\int_{0}^{\pi/2}\frac{x\sin x\cos x}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}dx$$ For integrating I took $\cos^{2}x$ outside and applied integration by parts. ...
0
votes
1answer
34 views

Basic Geometry Question. [duplicate]

What do you think can be the value of angle $x$ in the image below and how. You may use any concept related to triangles.
1
vote
0answers
4 views

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] : ...
1
vote
2answers
27 views

Transforming linear combination of the cosine and sine function

In the proof of Transforming $a\cos(x)+b\sin(x)$ to $r\cos(\phi-x)$ ...
13
votes
2answers
342 views
+250

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
19
votes
0answers
466 views
+100

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
2
votes
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 ...
3
votes
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 ...
0
votes
1answer
30 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 ...
3
votes
4answers
9k views

How to calculate reflected light angle?

On a two dimensional plane, line $X$ is at an angle of $x$ radians and an incoming light travels at an angle of $y$ radians. How can I calculate the angle of the outgoing light reflected off of the ...
4
votes
2answers
153 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} ...
0
votes
4answers
44 views

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 ...
7
votes
3answers
718 views

Integration substitution: How does Wolfram Alpha come up with this step?

I have to integrate $$ \int \frac{1}{(\sin x) (\cos x)} \, dx $$ I looked at the Wolfram Alpha step by step solution to figure out how to do it. First, it rewrites the integral as: $$ \int (\csc ...
9
votes
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. $ ...
-6
votes
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.
3
votes
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 ...
1
vote
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.
3
votes
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 $y=3x$?
0
votes
0answers
21 views
+50

Given three vectors involving trigonometric functions, how many $\theta$ satisfy a particular box product relation?

If $$\vec a =(1+\sin \theta )\hat i+\cos \theta \hat{ j}+\sin2\theta\hat k\\ \vec b =(\sin( \theta +2\pi/3))\hat i+\cos ( \theta +2\pi/3) \hat{ j}+\sin( 2\theta +4\pi/3)\hat k\\ \vec c =(\sin ( \theta ...
1
vote
2answers
42 views

Find angle and hypotenuse of right angled triangle

Find the missing side and the hypotenuse of a right triangle that has a side length of 5 cm and a perimeter of 30 cm. I'm confused. Can somebody please explain to me how to do this step by step? Not ...
1
vote
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 $(4(2x+1)^2)$?
3
votes
5answers
95 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
1
vote
2answers
473 views

Formula to calculate the height of a Satellite Image in degrees

Do you know of a formula or function that can calculate the real world height of a satellite image in degrees? The image is from Google Static Maps and the image uses Mercator projection which makes ...
5
votes
1answer
43 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 ...
2
votes
1answer
302 views

Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder?

I have a solid that is a combined shape of a cylinder and a concentric cone (a round sharpened pencil would be a good example) Know values are: Total Volume = 46,000 Height to Base Ratio = 2/1 ...
1
vote
4answers
71 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
0
votes
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 ...
1
vote
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? $$ 2\sin\sqrt{2x-1}+1 $$ ...
3
votes
1answer
449 views

Fitting a sine function to data

I have a sequence of $n$ points $(x_i,y_i)$, for $i=1,\dots,n$. I would like to find the function, of the form $y=V\sin(x+\phi)$, which best fits the points. Which numerical method could I use? I have ...
5
votes
2answers
77 views

Trigonometric equation, find $\sin \theta $

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

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
21
votes
12answers
4k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
0
votes
1answer
15 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 ...
3
votes
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 ...
2
votes
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: $$\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$$ $$\lim_{x\rightarrow 0} \, x = 0$$ ...
0
votes
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 ...
0
votes
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 ...
2
votes
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) & ...
0
votes
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 ...
3
votes
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?
0
votes
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 ...
5
votes
3answers
218 views

Solving trigonometric equation with unknown and restricted domain

Given that $ \tan^2(\fracθ3) = 1$ and $θ\in [0, 4\pi]$ find θ. I'm not sure how to progress with the restricted domain. Here's what I've got so far: Solving for the domain $[0, 4\pi]$. $$ ...
0
votes
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 ...
82
votes
13answers
8k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
2
votes
1answer
97 views

Not understanding arc midpoint computation

I'm trying to find the midpoint of an arc, so I found this page wherein Gregory V. Akulov and Oleksandr G. Akulov describe the midpoint formula. I pasted the formula & description from the site ...
2
votes
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. ...
1
vote
1answer
61 views

Confusion regard Fourier Series formulae

EDIT: I need help today, please. It's very important for my homework. I need to understand this concept. Thank you! I have a doubt regarding the Fourier series formula. In one of my notes, it is ...
9
votes
0answers
64 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| ...