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

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2
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1answer
47 views

How to get ellipse cross-section of an ellipsoid

I'm trying to get the major and minor radius of an ellipse which represents the cross-section of a given ellipsoid. This is particularly of interest in the field of RF propagation in terms of Fresnel ...
5
votes
1answer
138 views

A difficult trigonometry problem [closed]

How to prove that ...
0
votes
1answer
35 views

When doing 3D rotations my angle flips 180 degrees

I'm implementing 3D rotations for a set of 3D circles. To do that I'm using the parametric equation as described in http://demonstrations.wolfram.com/ParametricEquationOfACircleIn3D/. It works as ...
0
votes
3answers
36 views

How $1 + \cos(4 \pi t + \pi) = 1 - \cos 4 \pi t$

I know some trigonometry but I wonder how $1 + \cos(4 \pi t + \pi) = 1 - \cos 4 \pi t$ ?
1
vote
4answers
62 views

simple trigonometric functions

how would you solve this $$\cos^2 x - 2\sin x \cos x - \sin^2 x = 0$$ I tried to simplify it but I got $\cos 2x - \sin 2x$. I can't simplify that further.
2
votes
3answers
46 views

Trigonometric relation in sin, cos, tan

Show that if $x, y,$ and $z$ are consecutive terms of an arithmetic sequence, and $\tan y$ is defined, then $$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $$ I'm not sure ...
4
votes
1answer
57 views

estimation of $\pi$ and $e$ by using the Taylor series of $\cos x$

how can one show, that $3<\pi<3.2$, $2.7<e<3$ by just knowing, the estimation of $\cos(x)$, namely: $$1-x^2/2+x^4/24-x^6/720\le\cos(x)$$ ? If I substitute Pi/2 into this estimation, I ...
1
vote
1answer
44 views

Finding The Lim Of Expression With Trigonometry

Evaluate $$\lim_{n\rightarrow \infty}\left(\frac{\sin ^3n^2-5\cdot \sin ^2n^2+3}{\sqrt{\ln \left( \left| \dfrac{1}{\tan ^{19}e^ {- n }}+3\right| \right)+\sin \left( \dfrac{1}{\tan ^{24}e^ {- n }} ...
4
votes
3answers
179 views

Trigonometry. Finding the angle alpha

Refer the diagram below : What should be the angle alpha such that the variable x is between 7mm and 7.3mm.
0
votes
2answers
40 views

How to prove $\sqrt\frac{1 - \sin x}{1 + \sin x} = \frac{1}{\cos x} - \tan x$

Prove $\sqrt\frac{1 - \sin x}{1 + \sin x} = \frac{1}{\cos x} - \tan x$ I tried but I couldn't figure it out, give me a hint please.
0
votes
3answers
73 views

Prove the derivative of $\sin(1/x)$ exists

How do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...
2
votes
1answer
20 views

Eliminating theta problem from given equations and proving the identity

If $\cos\alpha=\cos\beta\cdot\cos\phi=\cos\gamma\cdot\cos\theta$ and $\sin\alpha=2\sin\dfrac{\phi}{2}\cdot\sin\dfrac{\theta}{2}$ then prove that ...
0
votes
3answers
44 views

Finding unique solutions

Show that f(x; k) ≡ cos x − kx = 0 has a unique solution, x0(k), in the interval [0,pi/2] for all k > 0 . I don't know where to start
0
votes
0answers
14 views

Can a function of a combination of low and high ordered trig terms be represented by a function of just higher ordered terms?

I have a function some variables $s, m$ and the RHS has been fixed to the form $A+Bcos^2+Ccos^4$. $A, B$ and $C$ emerge from various scenarios of $s$ and $m$ on the LHS. But I want to know if it is ...
0
votes
2answers
41 views

Solving $a \sin(bx + c) + d\sin(ex + f) = g$, where $a-g$ are constants

Could anyone tell me how I can solve an equation of this form: $a\sin(bx + c) + d\sin(ex + f) = g$ Where the variables $a$ to $g$ are constants. And does anyone know a place (i.e. a website) where I ...
13
votes
4answers
959 views

Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
0
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0answers
16 views

What does a notation such as $|v1/v2|^2$ mean?

I have phasors $V_1=V_a*e^{j\phi1}$ and $V_2=V_b*e^{j\phi2}$ Va and Vb are the amplitudes of AC-signals. The relationship between the amplitudes can be defines as: $a ≡ |(v1_a)/(v2_b)|^2$ But ...
5
votes
1answer
172 views

Closed-form of $\int_0^{\pi/2}\frac{\sin^2x\arctan\left(\cos^2x\right)}{\sin^4x+\cos^4x}\,dx$

I have just seen two active posts about integrals of inverse trigonometric function, $\arctan(x)$, here on MSE. So I decide to post this question. This integral comes from a friend of mine (it's not a ...
1
vote
2answers
42 views

Are all products of trigonometric functions integrable?

I have a feeling that this question has a really obvious answer, so forgive me if it turns out to be trivial. That being said, my question is whether all functions involving trigonometric functions ...
3
votes
1answer
51 views

Zeros of a function

Show that all zeros of $$f(z)=\sin z +z\cos z$$ are real. I tried to use zeros of $\sin z$ and $\cos z$ are real even though I couldn't get any ideal.
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0answers
9 views

Tidal torque on a ring mass

What is a tidal torque generated by a distant mass, say the Sun, on a simple circular ring inclined at some angle i? for example: ../ ./ the ring ----------------> sun. / the ring now lies in the ...
10
votes
7answers
366 views

How do I prove this seemingly simple trigonometric identity

$$a = \sin\theta+\sin\phi\\b=\tan\theta+\tan\phi\\c=\sec\theta+\sec\phi$$ Show that, $8bc=a[4b^2 + (b^2-c^2)^2]$ I tried to solve this for hours and have gotten no-where. Here's what I've got so ...
1
vote
2answers
38 views

Evaluting $\tan15°$ using difference formula

Evalute $\tan15°$ using difference formula Steps I took: $$\begin{align} \tan(45-30)&=\frac { \tan(45)-\tan(30) }{ 1+\tan(45)\tan(30) }\\ &=\frac { 1-\frac { \sqrt { 3 } }{ 3 } }{ ...
3
votes
3answers
136 views

What am I doing wrong? (Trigonometric Identity)

$$\frac { \cos\theta }{ 1-\sin\theta } =\frac { \sin\theta -\csc\theta }{ \cos\theta -\cot\theta } $$ Steps I took: $$\frac { \sin\theta -\frac { 1 }{ \sin\theta } }{ \cos\theta -\frac { ...
0
votes
3answers
57 views

Find exact value of trig function

How would I get the exact value for this equation? Question: Find $\arcsin\left[\sin\left(\dfrac{5\pi}{4}\right)\right]$. The answer they give me is: $-\dfrac{\pi}{4}$. I know how to get the ...
1
vote
1answer
39 views

Finding Exact Value $7\csc(x)\cot(x)-9\cot(x)=0$

The values for $x$ on $[0,2\pi)$ solving $7\csc(x)\cot(x)-9\cot(x)=0$ are? I think that $\dfrac{\pi}2$ is one but I can't find the others. what are the others?
1
vote
1answer
28 views

showing that cos(y)-cos(x)=2*sin((x+y)/2)*sin((x-y)/2)

how can one show that $\cos(y)-cos(x)=2sin(\frac{x-y}{2})sin(\frac{x+y}{2})$ by just using trigonometrical equalities like additional theorems? I was trying to show it by rewriting ...
0
votes
1answer
14 views

Maximum of a cosine graph

I am working on part b). I have determined, by graphing, that the maximum value is 4 (at x=0, y=4; the max point is 0,4) Now, to find an expression for part b, I was trying to find the period of ...
2
votes
0answers
32 views

Can I get $e^x=f(trigonometric function)$ without $i$ [closed]

I searched in many websites to get identity of $e^x$ with the trigonometric function but I didn't find. Can anyone help me to give me it if there is
2
votes
1answer
33 views

Finding the phase shift of a cosine function, given the graph

Here is what I have found so far: Vertical Displacement = 2 units down (-2) Amplitude = 4 Period = 2pi/3 I am now trying to find the phase shift. I moved the 'working x-axis' 2 units down, in ...
5
votes
3answers
110 views

Why does $\sin(\cos x)=\cos(\sin y)$ result in a lattice of circles?

Using Desmos graphing, I made an equation $\sin(\cos x)=\cos(\sin y)$ (here) which resulted in a strange lattice of symbols. I know that the trigonometric functions relate a triangle's sides to its ...
0
votes
1answer
13 views

What is the formula for the period of any trignometric function?

My question is pretty simple: Does the equation p=2pi/b work for ALL trig ratios? (cos, sin, tan, csc, sec, cot) I have the following questions, and am trying to find the period for them all. Must I ...
0
votes
1answer
27 views

Find the Exact Value of the following Cosine Function

Determine the exact value of cos(-5pi/4) I know: cosine theta = x/1 5pi/4 on the unit circle is 225 degrees Therefore, cos 225 degrees = x/1 In order to find the exact value, I need to find the ...
0
votes
1answer
37 views

Further Trigonometric Differentiation [closed]

I need to derive the maximum point of this equation for a modelling problem but am not too sure if my differentiated value is accurate. Would appreciate if someone could give it a shot! ...
0
votes
2answers
78 views

Show that the equation $ \cos(x) - kx = 0$ has a unique solution in $[0, \pi/2]$ for all $k>0$

Show that the equation $$f(x;k) \equiv \cos(x) - kx = 0$$ has a unique solution in $[0, \pi/2]$ for all $k>0$.
0
votes
0answers
22 views

finding angle of spirals along with length of it's line at a certain point

I'm tying to calculate the angles (the angle between each line segment and a horizontal ray to the right blue) of spirals at a certain point along with figuring out the length of the other lines. see ...
4
votes
7answers
113 views

Is it correct? $\tan x + \cot x \ge 2$ proof

The question is to prove $\tan x + \cot x \ge 2$ when $x$ is an acute angel. This is what I did $$\begin{align} \tan x + \cot x &\ge 2\\ \frac{1}{\sin x \cos x} &\ge 2\\ \left(\frac{1}{\sin ...
0
votes
4answers
32 views

Trig proving the identity

The question is : $\sin^2 \theta \sec^2 \theta = \sec^2 \theta -1$ I have tried many different methods but I cannot seem to figure out how to approach this so the both sides are equal.
1
vote
2answers
271 views

Lost solving a trigonometric identity

$$\frac { 2+\tan^2x }{ \sec^2x} -1$$ Steps I took: $$\begin{align} \frac { 2+\tan^2x }{ \sec^2x} -1 &=\frac { 2+\frac { \sin^{ 2 }x }{ \cos^{ 2 }x } }{ \frac { 1 }{ \cos^2x } } -1\\ ...
1
vote
1answer
30 views

Is there a simple way to explain GPS resolution?

I had always assumed that GPS satellites were geo-synchronous, and that the process of resolving your location was simple trigonometry. It turns out that GPS satellites are non-geo-synchronous - ...
1
vote
4answers
83 views

Geometry question involving triangles given with picture.

Here's the question: $\overset{\Delta}{ABC}$ is a triangle. $D$ is a point on $[BC]$. $|BD|=4$. $|AD|=|CD|$. $\text m(\widehat{CBA})=\alpha=30^\circ$. $\text ...
11
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4answers
2k views

Solving sin z = -z in reals

How do we prove that z=0 is the only real solution? I tried examining cases and quarters, but not sure how it is rigorously proven.
2
votes
1answer
16 views

Confusion with modeling a trigonometric function

I am studying trigonometry on Khan Academy and came across this problem: The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric ...
0
votes
1answer
45 views

Trig identities dividing fractions

The question is : $\dfrac{\csc x}{\sec x} = \cot x$ After solving a bit I get $$\frac{\frac1{\sin x}}{\frac1{\cos x}} = \frac{\cos x}{\sin x}$$ $$\frac{\cos x}{\sin x} = \frac{\cos x}{\sin x}$$ Is ...
1
vote
1answer
26 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
2
votes
4answers
69 views

Evaluate $\int_0 ^{\pi}\left (\frac{\pi}{2} - x\right)\sin\left(\frac{3x}{2}\right)\csc\left(\frac{x}{2}\right) dx$

How would you evaluate the integral $$\int_0 ^{\pi} \left(\frac{\pi}{2} - x\right)\sin\left(\frac{3x}{2}\right)\csc\left(\frac{x}{2}\right) dx$$ The answer from Wolfram is $0$. Would you use a ...
1
vote
3answers
21 views

Calculate uncertainty of sine function result

I have an angle given in degrees: $$\theta_{\min} = 63^{\circ} \pm 0.5^{\circ}$$ I need to calculate it's sine and still know the uncertainty of the value: $$n = 2\sin(\theta_{\min}) = ...
2
votes
1answer
62 views

Solving messy integral with modulus and trigonometry.

If $$a\in \mathbb R,\int_{a-\pi}^{3\pi+a}|x-a-\pi|\sin(x/2)dx=-16$$ then a can be? I tried substituting $x-a=u$ and then breaking into two integrals removing modulus then used $\int \sin x=-\cos ...
0
votes
0answers
11 views

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
0answers
44 views

Find all the triangles satisfying $\cos(A)\cos(B)+\sin(A)\sin(B)\sin(C)=1$ [duplicate]

I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$. In the case where the triangle has a right angle, ...