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Questions tagged [trigonometry]

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

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Multiple angle formulas for tangent

There are direct multiple angle formulas for sine and cosine, in terms of Chebyshev polynomials. For example, the cosine of $n\theta$ is $$ \cos{n\theta}=T_n(\cos\theta) $$ Where $T_n$ is the n-th ...
Francesco Sollazzi's user avatar
0 votes
0 answers
31 views

I need help with mathematical induction on trigonometry [closed]

I need to prove that $\cos x \cos 2x \cos 4x \ldots \cos 2^{n}x= \sin 2^{n+1}x/2^{n+1}\sin x$ while $x \in \mathbb{R}$.
milica's user avatar
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1 vote
1 answer
65 views

How to see the graph of this? $n=\sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$ [closed]

$$n \text{ (constant)} = \sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$$ Code is this, but I can't emulate this. Why? Even WolframAlpha can't make graph too. ...
Myeongjun Chae's user avatar
0 votes
0 answers
56 views

Inverse trigonometry , find x [closed]

What is the value of x ? There are many different ways to do . Is my method correct? If so can anyone help me to go further. After this part , i have no idea to go with !!
Master's user avatar
  • 13
1 vote
1 answer
25 views

In concentric circles Triangle formed from intersection of a line making 45 degree with x axis where inner circle meets x axis to outer circle

I have two concentric circles one of radius 5 cm and outer one of 10 cm, their centers being 0,0 I want to calculate P B and H of the triangle formed by intersection of a line on outer circle making ...
Ken Kaneki's user avatar
-2 votes
0 answers
50 views

$\sin A+\cos A+\sec A+\operatorname{cosec}A+\tan A+\cot A=7$, find the value of $\sin(2A)$. [closed]

How can I find the value of $\sin(2A)$, when $$\sin A+\cos A+\sec A+\operatorname{cosec}A+\tan A+\cot A=7?$$
Sayam's user avatar
  • 11
1 vote
0 answers
60 views

A solution to an equation on trigonometric functions

Consider the equation $$ 2 \sin 2x + \cot \frac{(n-1)x}{2} = 0 $$ on $x$ for fixed positive integer $n$. Is there an explicit solution or an approximate solution for $x$ in each interval $(\frac{2k\pi}...
user1150713's user avatar
-2 votes
0 answers
26 views

Let $\prod_{i=1}^np_i^{a_i\cos^2(a_i)} = 2019$, then [closed]

Let $$\prod_{i=1}^np_i^{a_i\cos^2(a_i)} = 2019$$ where $p_i$ denotes the $i$-th prime number ($p_1=2 , p_2=3, p_3=5$, etc. up to $p_n=2017$), and $a_i\in\Bbb R$ How can we prove the four following ...
Hitesh's user avatar
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3 votes
1 answer
98 views

Is $f(x) = \sin x$ the unique function satisfying all five: $f(0)=0;\ f'(0)=1;\ f(\pi/2)=1;\ f'(\pi/2)=0;\ -1\leq f''(x)\leq 0$ for $x\in [0,\pi/2] ?$

I would like to prove or find a counter-example to the following proposition (which I came up with), please. Suppose $f:[0,\pi/2]\to [0,1]$ is twice differentiable in the interval $[0,\pi/2]$. ...
Adam Rubinson's user avatar
-3 votes
0 answers
26 views

Proving Trigonometric identities, show [closed]

enter image description here I am having trouble with this trigonometric identity, which shows that one side equals the other side. I would appreciate help since my math knowledge is limited.
SQL absorber's user avatar
-3 votes
1 answer
103 views

I need the solution for the equation $\sin x−\sin(2x)−\cos(3x)=0$ [closed]

I need the solution for the equation $\sin x−\sin(2x)−\cos(3x)=0.$ It has to be solved using trigonometric identities (not by graph). I have tried using the identity $\sin x-\sin y=2\sin\left[\frac{1}{...
Xabcd's user avatar
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0 answers
92 views

Solving the equation $\sin x-\sin(2x)-\cos(3x)=0$

I need the solution for the equation $\sin x-\sin(2x)-\cos(3x)=0$. It has to be solved using trigonometric identities (not by graph). I have tried using the identity $\sin x-\sin y=2\sin\left(\frac{x-...
Xabcd's user avatar
  • 21
1 vote
1 answer
45 views

Solving a trigonometric equation originating from law of sines

Often when using trigonometry, especially the sine law to solve geometry problems, I end up with the following equation sin (x) sin(a) - sin (x+b) sin(c) = 0 However, I am often stuck as it is common ...
Jelly Qwerty's user avatar
1 vote
2 answers
70 views

Find $x$ in this concyclic quadrilateral

$ABCD$ is a concyclic quadrilateral, with $\angle A = 60^\circ$, and $ AB = 10, BC = x , CD = x+2 , DA = x+4 $. Find $x$. My attempt: Using the vector method, we can express the horizontal and ...
c'est pas normale's user avatar
1 vote
0 answers
30 views

What is an angle? [duplicate]

I am a high school student and recently I started trigonometry and one question that comes to my mind every time is that "What is an angle?" I mean when we say that angle between two sides ...
Himanshu Singh Nirwan's user avatar
0 votes
1 answer
84 views

Calculating circle offset for each angle when reference point is not circle center

I try to calculate offset change at outer rim of a circle for each angle with respect to C1 but can not figure out. its is easy on the paper with geometry but very difficult to formulate it. The ...
ebbac44's user avatar
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3 votes
3 answers
118 views

Find $x$ in this quadrilateral

A quadrialteral $ABCD$ has $AB = 10$, $\angle A = 50^\circ, \angle B = 120^\circ$, $ BC = x , CD = x + 2 , AD = x + 4 $. Find $x$. My attempt: Applying the law of cosines to $\triangle DAC$ and $\...
c'est pas normale's user avatar
1 vote
1 answer
65 views

Convert pair of parametric trig equations to $y=f(x)$ form

My apologies if this already has an answer, I've spent some time looking but haven't found anything that (to me) looked directly applicable. I have a set of parametric equations describing a periodic ...
SteveP's user avatar
  • 11
1 vote
1 answer
33 views

Identify and describe the surface described by the equation: φ = c where π/2 < c < π.

In this context, φ refers to the polar angle between the z-axis and the radial line ρ as described by the spherical coordinate system with the following equations: x=ρcos(θ)sin(φ) y=ρsin(θ)sin(φ) z=...
largecoconutballs06's user avatar
3 votes
3 answers
128 views

Find the exact value of $\DeclareMathOperator{\cosec}{cosec} \cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$ [duplicate]

Find the exact value of $\cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$. The equation can be written as $$ \cosec(x) + \cosec(60^\circ-x) - \cosec(60^\circ+x), $$ where $x = 10^\circ$. I ...
Hitesh's user avatar
  • 67
3 votes
2 answers
119 views

Find $X$ and $\theta$ in this quadrilateral

$ABCD$ is a convex quadrilateral, with $\angle A = 120^\circ$. $\angle B = \angle C =\theta$. $AB = 10$, $AD = BC = X $ and $ CD = 2 X $. Find $ X $ and $ \theta $. My attempt: From the law of ...
c'est pas normale's user avatar
1 vote
1 answer
34 views

angle of $\frac{\sum_i a_i e^{j\theta_i}}{\sum_i a_i}$

Let $\displaystyle a=\sum_{i=1}^N a_i\ne 0$ be a sum of $N$ complex numbers, $a_i$'s. Suppose we rotate each $a_i$ by a small angle $\theta_i$, i.e. $\vert \theta_i\vert \le \phi$ and, say, $\phi=\pi/...
syeh_106's user avatar
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-1 votes
0 answers
26 views

Trigonometric formulas derivation? [closed]

We know that how other formulas like quadratic formula, distance formula, section formula are derived but what about trigonometry formulas? We are just told to memorize their values. Pardon me for ...
Payal Payal's user avatar
1 vote
0 answers
78 views

On the divergence of $\{\cos(x\pi n!)\}$ for all $x\in \mathbb{R}-\{0\}$

It is well known that $\left\{\sin\left(n\right)\right\}$ and $\left\{\cos\left(n\right)\right\}$ are divergent sequences. Recently I encountered with a problem where I needed to prove that $\cos\left(...
Yathi's user avatar
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1 vote
0 answers
44 views

Determining concavity of a certain function

Let $t\in(0,1)$ be a fixed number. Consider the function $f(x)=2\cos(t)\arccos(\cot(t)(\csc(x)-\cot(x)))+\arccos(\csc^2(t)\cos(x)-\cot^2(t))$. I believe that this function is concave down for $x>0$,...
user3816's user avatar
  • 301
1 vote
1 answer
57 views

Solving a trigonometric equation from a geometry problem

Question statement: In triangle $ABC$, angle $A = 54^\circ$ and angle $C = 24^\circ$. $P$ is a point on $AC$ such that $AP = BC$. Find angle $CBP$. (HK IMO Prelim 2015) I am looking into using ...
Jelly Qwerty's user avatar
0 votes
0 answers
50 views

Looking for similar trigonometric functions [closed]

Has anyone seen such a function before? I am not looking for the exact formula, but rather the type or structure of such a function. I.e. I would be happy to see the explicit formula of a function ...
mangolassi's user avatar
1 vote
1 answer
94 views

Evaluate $ x^2 = \sin(2x), x≠0 $ [closed]

There should be another solution ($x \approx0.967$), but can it be done without doing graphs? I tried drawing a triangle to solve but it failed miserably.
Assorted69's user avatar
2 votes
2 answers
71 views

Proving Perpendicularity and Equal Lengths in Rotated Isosceles Right Triangles

Given two isosceles right triangles $( \triangle ABC )$ and $( \triangle BDE )$ with $( \angle BAC = \angle BDE = 90^\circ )$, triangle $( \triangle BDE )$ is rotated counterclockwise by an angle $( \...
Oth S's user avatar
  • 345
1 vote
1 answer
53 views

In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$.

I was trying to solve this problem: In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$. After a long time of thinking ...
pie's user avatar
  • 5,605
1 vote
2 answers
88 views

Find $f(x)$ assuming that $f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$

If $f(x)$ is a real valued function such that $$f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$$ Find $f(x)$. I did $x\to\arcsin x$ and then $x\to \arccos x$ and I obtained $2\arcsin x=2\arccos x$ or $x=\frac{...
MathStackexchangeIsNotSoBad's user avatar
3 votes
2 answers
70 views

Integrating $\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$ two ways gives very different-looking answers. How to show algebraically they differ by a constant?

From Apostol's Calculus Volume 1 2nd ed., 6.22 #46, the task is to integrate $$\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$$ Method 1: The provided hint is to use the substitution $x-a=(b-a)\sin^2(u)$. Thus $b-...
newmacuser's user avatar
-2 votes
0 answers
24 views

prove the addition identity cos(x+y) using the law of cosines [closed]

I’ve drawn a triangle with sides a b and c and written down two equations for side c using the law of cosines with angle x+y and using the sine of angle x and sine of angle y. I don’t know how to ...
Ilove Math's user avatar
3 votes
2 answers
83 views

$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$

$$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$$ My attempt: Rewriting the numerator: We can use the double angle identity for sine: $$\sin(A)\sin(B) = \frac12\left(\cos(A - B) - \cos(A + B)\...
Nsnansn Jwjwj's user avatar
0 votes
3 answers
85 views

When solving $\sin{x}+\cos{x}=\sin{2x}+\cos{2x}$, where does the extra solution come from?

Background This question is from the 1907 Victorian Universities and Schools Examination Board Trigonometry (Senior) Examination. My solution is given below, but apparently there is an extra solution ...
Red Five's user avatar
  • 2,671
-2 votes
2 answers
89 views

Is there a way to simplify $\cos^{-1}\left(\frac{\cos x}{A}\right)$?

Is there a way to simplify the following: $$\cos^{-1}\left(\frac{\cos x}{A}\right)$$ I'm thinking something along the line of pulling $A$ out and then getting $\cos^{-1}(\cos x)$ which would be $x$ ...
rdemo's user avatar
  • 329
7 votes
2 answers
97 views

Solving the integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx$

My textbook has the following problem: $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx$$ Using the trigonometric identity: $$\sin 2x = 2 \sin x \cos x$$ the ...
archthegreat's user avatar
0 votes
0 answers
40 views

Using the Maclaurin series of sin(x) to find Si(x)

The Maclaurin series for $\sin\left(x\right)$ is given by: $$\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}x^{2n+1}}{\left(2n+1\right)!}$$ So the series for $\frac{\sin\left(x\right)}{x}$ would be $$\...
Gabriel Turner's user avatar
0 votes
2 answers
186 views

summation of $\cos{x} + \cos{2x} + \cos{3x}$ and so on is $-\frac{1}{2}$

$$\cos{x} + \cos{2x} + \cos{3x} \ldots = y$$ $$ 2\cos{x} + 2\cos{2x} + 2\cos{3x} \ldots = 2y $$ By grouping every alternate term by $\cos{A} + \cos{B} = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$ $$ \...
dhruvk's user avatar
  • 11
1 vote
3 answers
82 views

Solving the integral $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x}$ [duplicate]

My textbook has the following problem: $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x}$. This website solves the problem in a very convoluted way and I was wondering if there's an ...
archthegreat's user avatar
1 vote
3 answers
43 views

3D trigonometry bearing question - finding the bearing of a point to a tower

I'm currently struggling with 3D trigonometry, particularly with drawing a proper diagram and interpreting bearings. The question is below: The angle of elevation of a tower QR of height 100m at a ...
Rita's user avatar
  • 11
-1 votes
0 answers
23 views

Find the most efficient path, time wise, to get from points S to Q at varying speeds [closed]

Both me and my friend are a bit stumped on this question and keep getting different answers. I think the sailor should land 4km up shore (1km from market) but am not entirely sure. Q: A fisherman can ...
Brian Desmond's user avatar
2 votes
2 answers
55 views

Alternative proof $g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$

The given inequality $$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$, comes out from an answer given to this other recent question. The ...
user's user avatar
  • 1
0 votes
1 answer
25 views

Range of values of θ that satisy these values.

The question:The value of θ for which $$x+y(\sin θ)=1, x(\sin θ)+4y=2$$ satisfy $$x>=\frac{4}{5}, y>=\frac{1}{3}$$ Note here θ must belong from $$(-\frac{\pi}{2},\frac{\pi}{2})$$ This is a ...
Mahit Chopra's user avatar
0 votes
1 answer
59 views

Maximum length of a boat able to turn at a junction of two orthogonal canals [closed]

Let's say you are making a boat, and the goal is to navigate down a canal of width $a$. But not only that, you want to make a turn into another canal that is perpendicular to the one you navigate of ...
MiguelCG's user avatar
  • 339
0 votes
1 answer
86 views

Prove that the average of the numbers $n\sin(n^{\circ}),n=2^{\circ},4^{\circ},6^{\circ},\ldots,180^{\circ}$ is $\cot(1)$.

Prove that the average of the numbers $n\sin(n^{\circ}),n=2,4,6,\ldots,180^{\circ}$ is $\cot(1^{\circ})$. I am trying to use complex numbers to solve the given problem. We can write $$\sin(n^{\circ})=...
Sillyasker's user avatar
1 vote
0 answers
68 views

Middle School Narrative Math Textbook

A couple decades ago, I remember teaching myself trig and calculus from some textbooks that I got from the library. They were set in a fantasy kingdom and concepts were introduced to solve various ...
hbacovci's user avatar
0 votes
1 answer
81 views

Is it possible to calculate the distance in red? [closed]

In the context of a physics experiment preparation I am trying to figure out if I can calculate theoretically the distance in red on the below schema in order to determine the waist of a laser beam. ...
Chris Ze Third's user avatar
0 votes
0 answers
20 views

Calculate offset from corner coordinate [closed]

Illustration Given 4 corner coordinates. Compared to leftBottom or leftTop, I want to determine the displacement of a point P in the area bounded by the 4 coordinates. The picture is looking north. ...
Erik Orosz's user avatar
0 votes
2 answers
107 views

Solving the system $\cos(x)-\cos(x+y)=0$ and $\cos(y)-\cos(x+y)=0$

At some point when dealing with a problem I had to solve the following system $$\begin{cases} \cos(x)-\cos(x+y)=0\\ \cos(y)-\cos(x+y)=0 \end{cases}$$ I seem to have problems with doing so, or maybe ...
Math Student's user avatar
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