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

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93 views

Differentiating w. r. t. $x$

Differentiate $$ \text{arccot} \frac{1-x}{1+x} $$ with respect to $x$ After putting $x= \cos \theta$, I got $$\text{arccot} \left(\tan^2 \frac{\theta}{2}\right)$$ Then how do I reach the answer? ...
1
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3answers
280 views

Exact value of trig functions

Can we find the exact value(not numerical/approximation) of sin 1? I tried to do so by solving a cubic equation using Cardano formula but I ended up with complex nested radicals): I was told to use ...
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1answer
59 views

Trigonometry Compund Angles Problem

I am studying for my trigonometry examination. But I cannot figure out how to this sum from the chapter "Compound and Multiple Angles". I am in the eleventh standard. The sum goes like this : ...
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2answers
225 views

Trig identity problem from Gelfand's Trigonometry

Having a problem with an exercise from Gelfand and Saul's Trigonometry, in the section dealing with the half-angle formulae. The exercise (7.a. on p.151) asks the reader to show that: ...
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3answers
329 views

Find the domain and range of $y=\cos^{-1} \sqrt{1-x}$

Find the domain and range of $y=\cos^{-1}\sqrt{1-x}$. Can someone please help me with question above, as to how it's done? Thanks. I am unfamiliar with what you do when there is a square root.
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2answers
1k views

$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$

I was playing around with trigonometric functions when I stumbled across this $$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$$ Immediately I checked it to see if it was flawed so I devised a ...
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4answers
1k views

Solving for $\tan \theta$ given $\sin \theta/2$

QUESTION: I'm having a hard time figuring this problem out. I've looked through my lectures and cannot find a problem that relates to this one. I do have my identities pulled up in front of me. I'm ...
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1answer
1k views

Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula. If we know the lat,lng of two points we can calculate their distance. I found the ...
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1answer
45 views

Simplifying the trigonometric equation

$$\sin(x)\cos(x)\tan(x) + \frac{2\sin(x)\cos^3(x)}{\sin(2x)}$$ Can someone please show me step by step working of how I may be able to solve this?
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1answer
60 views

Block selection in tile-based game

I know that this problem can probably be solved with some basic trig, but for some reason, I can't figure out what exactly to do for the life of me. I'm making a game that's not unlike Terraria (it's ...
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4answers
168 views

How can I prove that $\operatorname{arctg}(x) + \operatorname{arctg}(\frac{1}{x}) = \frac{\pi}{2}$, given that $x > 0$?

Which would be the easier way to prove that $\operatorname{arctg}(x) + \operatorname{arctg}(\frac{1}{x}) = \frac{\pi}{2}$ in cases where $x > 0$? I don't need explicit solutions, rather keywords ...
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5answers
452 views

Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$

How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$ Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't ...
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1answer
87 views

Maximum of a trigonometric Polynomial

Given $$x+y+z=\pi$$ $$3\sin(x)+4\sin(y)+18 \sin(z)=A$$ Question:find maximum of $A$. I spend so many time on this question. answer is $ 35\sqrt{7} /4$, but why?
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1answer
104 views

Can someone verify my trig proof

Give a convincing argument that $\sin (2\cdot 3\pi/5) \neq 2\cdot\sin(3\pi/5)$ This is my proof, can someone verify it. $$ \sin 2x = \sin x$$ if $$ 2 \sin x \cos x = \sin x$$ if $$ 2 ...
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1answer
257 views

Show that there exists unique real solution

Show that there exists unique real numbers $a$ and $b$ satisfying $$3\sin a-2\cos b=6a-12,$$ $$\cos a + 3\sin b=6b+6.$$ Thank you!
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1answer
98 views

Period of Trigonometric Functions

I have always been taught that in the scenario of a Sine,Tan,Cos function of $f(x) = a\sin b(x+c) +d$, the period of the sine and cos functions $= \dfrac{2\pi}{b}$, and the period for the tan ...
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2answers
243 views

Rewriting exponential function using Euler

I have the following function that I want to express using trigonometric functions: $$f(x) = \frac{1}{2}\left(\frac{2i+2}{2i+1}e^{ix} + \frac{-2i+2}{-2i+1}e^{-ix}\right)$$ I have come so far as: ...
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1answer
62 views

Trigonometry question- triangles

In the triangle ABC, the side AB has length 5 units and the angle BAC = 28$^\circ$ For what range of values of the length of BC will two distinct triangles ABC be possible?
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2answers
213 views

How to prove $\frac{\cos\theta \cdot \theta}{\sin\theta} = \frac{\sin\theta}{\theta}$

I am working on a problem that is looking to prove $\lim_{x\to0}\frac{\sin\theta}{\theta} = 1$. At the particular point I am working on, I have to prove $1 < \frac{\theta}{\sin\theta} < ...
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4answers
147 views

What is $\sin 2v $ if $\sin v= 12/13$

I'm having an exam tomorrow and stumbled over an old exam question that says: $$\sin v = \frac{12}{13},\qquad \pi/2 < v <\pi, $$ What is $\sin 2v$? Answer exactly! I've been sitting with ...
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2answers
99 views

A question about Indefinite trigonometric integration

I am practising a unit on Integration. I am going through some past year papers, and there are some types of questions that I could not solve. So if anyone could help me in this, I'd really ...
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0answers
152 views

Proving $\int_0^1\frac{\log (1-x)}{x}\mathrm dx=-\frac{\pi^2}6$ [duplicate]

It is a well known fact that $\displaystyle\sum_{k=1}^{\infty}\frac1{k^2}=\frac{\pi^2}6$. I wanted to prove this using elementary techniques. By doing some easy algebra, I found it was sufficient to ...
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2answers
85 views

How do I improve my approach to solving integrals to get this and similar ones in the future correct?

$$\int \sqrt{ 8 (\cos t \sin t)^2 } dt = \sqrt{2} \int 2\sin t\cos t dt = \sqrt{2} (\sin t)^2 + C$$ Which seems correct to me, but if I take the definite integral from $0$ to $\pi$, then: $$\sqrt{2} ...
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2answers
91 views

Show that $\sin^2 x- 6\sin x-5=0$ has more than one real solution

Without a calculator, prove $\sin^2 x- 6\sin x-5=0$ has more than one real solution. I have repeatedly solved this but I have only got one solution. Can someone help me out? I want to learn so ...
3
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2answers
141 views

Finding all $x$ such that $|\tan x | \leq 2\sin x$

I need to find all real numbers $x$ that satisfy: $$|\tan x | \leq 2\sin x \text{ and } x \in [ -\pi, \pi]$$ in terms of unions of intervals. I know it's equivalent to: $-2\sin x \leq \tan x \leq 2 ...
3
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4answers
394 views

Find the value of $\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right)$

What is the value of $$\textrm{cosec}^2\left(\frac\pi7\right) +\textrm{cosec}^2\left(\frac{2\pi}7\right)+\textrm{cosec}^2\left(\frac{4\pi}7\right) \qquad\qquad ? $$ I tried to write ...
3
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4answers
177 views

$4^\text{th}$ power of a $2\times 2$ matrix

$$A = \left(\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right)$$ is given as a matrix. What is the result of $$ad + bc \text{ if } ...
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0answers
122 views

Prove That $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ When $n$ is Odd

The original problem is: Prove that $1/\sin(i \pi/n)+1/\sin(j \pi/n) \ne 2$ when $n$ is odd. I tried, and found that although everything looks similar, I actually know nothing about this kind of ...
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1answer
77 views

how to prove an identity related to $\int_0^\infty\sin(x^{1+a})dx$?

i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$ and the computer give me the following result ...
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1answer
3k views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...
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3answers
218 views

Solving $5 \sin (2x)=1$ witouth a calculator?

My friend asked me if I could help her with a few problems, and they are mostly quadratics of trig functions and easy manipulations. But I was surprised to find equations such as $5 \sin (2x)=1$ and ...
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5answers
4k views

How can I solve $\cos^2 x + \sin x +1 = 0$?

The solution set of the equation $$\cos^2 x + \sin x +1 = 0$$ is? I haven't studied trigonometry, I'm kinda lost on this issue ...
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3answers
469 views

If $\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1$ then find the value of $\sin(c)$

If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the ...
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1answer
43 views

How to properly sort a set of axis-aligned boxes so they are drawn correctly under this projection?

Given a set S of axis-aligned, non-overlapping boxes {x,y,z,w,h,l}, where x,y,z are their center-positions and w,h,l their width, height and lengths, and given the following orthographic projection: ...
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2answers
502 views

Trigonometric Identities Like $A \sin(x) + B \cos(y) = \cdots$

Are there any identities for trigonometric equations of the form: $$A\sin(x) + B\sin(y) = \cdots$$ $$A\sin(x) + B\cos(y) = \cdots$$ $$A\cos(x) + B\cos(y) = \cdots$$ I can't find any mention of them ...
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1answer
567 views

find the rate of change of the area of triangle pulled by three people from its sides

this is the problem of my curious mind(I am it's designer!) . three people each having the rope attached by the end of the 3 sides of triangle ABC , pull the triangle with speed U in the direction ...
2
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1answer
172 views

Fourier expansion of sine of cosine function

What is the Fourier expansion of $$\sin\left(A\cos(\omega t)\right),\qquad 0<A<1,$$ in the frequency $\omega$ domain?
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0answers
127 views

Volume of spheroidal cap

A spheroid is bisected into two spheroidal caps by a plane, such that the shape of the area of the plane inside the spheroid is elliptical. The alignment of the plane is defined by two angles theta1 ...
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2answers
878 views

Calculate Triangle Ground using Height and Top Angle

Is it possible to calculate the ground of a triangle only using the height and top angle. Click here to see a poorly draw sketch of what I'm trying to calculate. So is it possible and how, to ...
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1answer
127 views

Function for cosine transformation around $\pi/2$

Given the cosine of an angle $x$ relatively close to $\pi/2$, is there a function $f$ such as: $f(cos(x))=cos((x+\pi/2)/2)$ ?
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167 views

Taylor expansions of $\text{atan}(\tan(x))$ and $\text{asin}(\sin(x))$

Do they actually exist? At least in a form that doesn't degenerate into a mantissa function or into repeated ranges of f(x)=x.
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0answers
48 views

Coefficients making expression independent of variables NOTE: updated

I have the following expression: $f_S(\lambda,\tau,\beta)=\displaystyle\sum_{n=1,\ \text{odd}}^{2S-1}\ ...
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1answer
513 views

finding maximum perimeter of a triangle

So, here we are given task to find maximum perimeter of a triangle with a given base 'a' and given vertical angle 'x' , now how should I proceed in given problem its confusing me Now supposing ...
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1answer
217 views

Solve the equation $\sin(5x) - \sin(3x) = \sqrt2/2$

I'm stuck on this problem and can't get any clue to solve this. Please help me, thanks. Solve the equation: $\sin(5x) - \sin(3x) = \sqrt2/2$ Thanks, I really appreciate if some one can ...
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3answers
121 views

Trigonometry, angles of depression.

A news helicopter hovers at a height of 500m. The angles of depression of a fire moving in the direction of the helicopter are first 10(deg) and then 15(deg), How far has the fire moved between thee ...
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2answers
71 views

This is a trigonometry question the I am having trouble with

$$\tan θ = 1$$ $$\sin θ = ?$$ Please if anyone could help with this it would be much appreciated.
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371 views

Elevation and depression, trigonometry.

A flagpole is mounted on top of a tall building. At a distance of $250$m from the base of the building, the angles of elevation of the bottom and the top of the flagpole are $38^\circ$ and $40^\circ$ ...
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1answer
55 views

some values of sine function

how to show that $$\sin \frac{\pi}{10} = \frac{\sqrt5 - 1}{4}$$ $$\sin \frac{\pi}{60} = \dfrac{\sqrt{30} + \sqrt{10} + \sqrt{20 + 4 \sqrt5} - \sqrt6 - \sqrt2 - \sqrt{60 + 12 \sqrt5}}{16}\,\!$$ ...
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1answer
229 views

Triangle with integral side lengths

$ABC$ is a triangle with integral side lengths. Given that $\angle A=3\angle B$, find the minimum possible perimeter of $ABC$. I got this problem from an old book (which did not provide even a hint). ...
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6answers
1k views

How to solve $\sin x \cdot\sin 2x\cdot\sin 3x + \cos x\cdot\cos 2x\cdot\cos 3x =1$

How to solve $\sin x \cdot\sin 2x\cdot\sin 3x + \cos x\cdot\cos 2x\cdot\cos 3x =1$ I don't know the solution for this. Help me! Thank all!