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

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Help with Trigonometry homework - prove an identity

I need to prove the following identity: $\sin^2 2\alpha-\sin^2 \alpha = \sin 3\alpha \sin \alpha$ What I have tried, is to work on each side of the identity. I have started with the left side: ...
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tangents of infinite sums — reference request

This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this: $$ ...
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Equation of sine wave around a circle

Consider a sine wave having $4$ cycles wrapped around a circle of radius 1 unit (its center needs not be the origin of a Cartesian coordinate system). Assume that the length of axis of the sine wave ...
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110 views

$\left(1+2\cos\theta\right)\left(1-\cos\theta\right)=1+\cos\theta-2\cos^2\theta$

I am having trouble trying to show this. I have reams of paper on the floor and I am tired, I need some help. I know that $1+\cos\theta-2\cos^2\theta$ equals $\cos\theta + ...
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Prove $\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$

How can I prove the fact $$\frac{1}{4-\sec^{2}\frac{2\pi}{7}} + \frac{1}{4-\sec^{2}\frac{4\pi}{7}} + \frac{1}{4-\sec^{2}\frac{6\pi}{7}} = 1.$$ When asked somebody told me to use the ideas of ...
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As shown in the figure: Prove that $a^2+b^2=c^2$

Geometry: Buildings in the triangle Other triangles with the same property: $1.$ 12 18 6 12 30 102 $2.$ 15 30 15 15 15 90 $3.$ 24 30 54 24 6 42 $4.$ 30 10 40 30 20 50 ...
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2answers
138 views

Proving a sum using induction

I am having a problem with this question. I need to prove by induction that: $$\sum_{k=1}^n \sin(kx)=\frac{\sin(\frac{n+1}{2}x)\sin(\frac{n}{2}x)}{\sin(\frac{x}{2})}$$ The relation is obvious for ...
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80 views

Is this conjecture about integration of sinusoids on a specific interval correct?

I haven't formally learned integrals, but I was trying to apply what I do know. Is $$\int_{a}^{b}m\sin(k(x+j)) dx =0$$ as long as $b-a\equiv 0 \pmod {\frac{2\pi}{k}-j}$ and $m, k, j \in ...
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1answer
117 views

sines and cosines law

Can we apply the sines and cosines law on the external angles of triangle ?
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141 views

Find the value of the given trig function

Find the value of the trigonometric function $\,\sec(v-u)\,$ given that $$\sin u=−\frac{20}{29}\;\;,\;\; \cos v =−\frac{3}{5}$$ (Both u and v are in Quadrant III.) Thanks!
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287 views

Find the exact value of the trigonometric function

Find the exact value of $$\cot(v-u)$$ given that $\sin u=−3/5$ and $\cos v = − 7/25$ (Both $u$ and $v$ are in Quadrant III.)
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299 views

Prove the Hyberbolic identity

(Note: The ? is where I have to fill in something) Prove: $$ \begin{align} \sinh2x & = 2\sinh x\cosh x\\ \sinh2x & = \sinh (x + ?)\\ & = \sinh x(?) + \cosh x \sinh x\\ & =\space?\\ ...
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Why $\cos^2 (2x) = \frac{1}{2}(1+\cos (4x))$?

Why: $$\cos ^2(2x) = \frac{1}{2}(1+\cos (4x))$$ I don't understand this, how I must to multiply two trigonometric functions? Thanks a lot.
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5answers
561 views

Derivate of $(\sin \theta)^{n-1}$

Please, explain me why the derivative of $$(\sin \theta)^{n-1} = (n-1)\cos\theta \cdot (\sin \theta)^{n-2}$$ Thanks so much!
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2answers
303 views

How to relate $2\sin(3\pi/8)-2\sin(7\pi/8)$ and $\csc(3\pi/8)$?

Trying to simplify $2\sin(3\pi/8)-2\sin(7\pi/8)$ down to $\csc(3\pi/8)$. The two expressions have equal decimal approximations but I'm literally at my wit's end trying to relate them based on ...
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338 views

Range of a trignonmetric function

I came across this in an Engineering entrance book, What is the range of this: $a^2 \sin^2 x + b \sin x \cos x + c \cos^2 x$ What is the method to find it? I tried the graph approach but didn't know ...
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146 views

Solve $x \arccos(x)+x/2=\cos(2x)$

$$x \arccos(x)+x/2=\cos(2x)$$ I dont know how to solve this one. It looks relatively easy but it is not, not for me at least. Anybody to help?
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1answer
74 views

How zeros of second derivative map to equilibrium value

This question arose from plotting some functions in MATLAB and GeoGebra. Assume we have a function of the form $$f(x)=A \sin(P_n(x))+d \qquad \text{or}\qquad f(x)=A \cos(P_n(x))+d$$ where $$P_n(x) ...
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1answer
204 views

Looking for good visual aids for sin cos tan .

Looking for good pictures ( or videos ) with explanations for visualizing sin cos and tan. Any relation to 3D Graphics is a bonus.
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Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
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1answer
1k views

How to simultaneously solving trig equations?

I was doing some math work, and have to solve two trig equations simultaneously, but have no idea how to approach this, can anyone help, just need to be pointed in the right direction. I have to ...
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1answer
215 views

Expressing $\int \tan^n x\,dx$ with a sum

I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that $$\begin{align} \int \tan^n x\,dx ...
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1answer
529 views

Show that $2θ + 2\sinθ - 1 = \pi/3.$

Need help with this question: The diagram shows that the cross section ABCD of a glass prism. AD = BC = 4cm and both are at right angles to DC. AB is the arc of a circle, centre O and radius 6cm. ...
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283 views

Product of Sine: $\prod_{i=1}^n\sin x_i=k$

From the article Products of Sines, we have $\sin 15^\circ\sin75^\circ=\sin 18^\circ\sin54^\circ=\frac{1}{4}$. We can rewrite this as $\sin \frac{\pi}{12}\sin\frac{5\pi}{12}=\sin ...
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655 views

$\cos^n x-\sin^n x=1$

For $0 < x < 2\pi$ and positive even $n$, the only solution for $\cos^n x-\sin^n x=1$ is $\pi$. The argument is simple as $0\le\cos^n x, \sin^n x\le1$ and hence $\cos^n x-\sin^n x=1$ iff $\cos^n ...
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134 views

Lost in Trig Identities

I've been working on this problem on and off for a couple hours and have not been able to find out how to go about it even after looking through the other questions and some google searching. Prove ...
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153 views

Derivatives of trig functions

How can I prove that $\frac{d}{dx} (\csc x)= -\csc x \cot x$? Specifically, how does one see the step $\cos x/\sin x = \cot x$?
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92 views

Why $\sqrt{\sin^2 x}<0.5$ can be transformed in $|\sin x|<0.5$?

Why $\sqrt{\sin^2 x}<0.5$ can be transformed in $|\sin x|<0.5$. Then $|\sin x|<0.5$ can be transformed in $-0.5<\sin x<0.5$? What is the proof of the inequality?
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142 views

A curious identity on sums of secants

I was working on proving a variant of Markov's inequality, and in doing so I managed to come across an interesting (conjectured) identity for any $n\in\mathbb{N}$: $$\sum_{m=0}^{n-1} ...
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914 views

How many distinct real root does the equation $x^{4}-x^{3}\cdot\sin(x)-x^{2}\cdot\cos(x)=0 $ have?

How many distinct real root does the equation $x^{4}-x^{3} \cdot \sin(x)-x^{2} \cdot \cos(x)=0 $ have? Is there any quick solution(less than 2 minutes)?
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Prove the trigonometric identity $(35)$

Prove that \begin{equation} \prod_{k=1}^{\lfloor (n-1)/2 \rfloor}\tan \left(\frac{k \pi}{n}\right)= \left\{ \begin{aligned} \sqrt{n} \space \space \text{for $n$ odd}\\ \\ ...
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2answers
371 views

What is the limit of this function as $n$ tends to infinity?

$\lim_{n\rightarrow\infty}\sqrt{n}\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$ I'm having a lot of trouble figuring it out. My first step is always to convert $\cos(\frac{\pi}{2}-\frac{1}{\sqrt{n}})$ to ...
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399 views

Clarification on Trigonometric Notation

Consider the following terms: $cos5x$ and $sin^2x$ Are these terms equivalent to: $5cosx$ and $(sinx)^2$ If not please explain. If so please confirm. Thanks
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Showing $\left\lvert \frac{\sin(nx)}{n\sin(x)} + \frac{\cos(nx)}{n\cos(x)} \right\rvert \le\left\lvert\frac{n+1}{n}\right\rvert $

It was shown in here that $$\left\lvert \frac{\sin(nx)}{n\sin(x)}\right\rvert \le1\,\,\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$$ iff $n$ is a non-zero integer. Using the similar argument in ...
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1answer
750 views

Coefficients in Trigonometric Functions

So, previously in my math classes I have been taught that The phase shift of $$\sin(ax+c)$$ is c. But in my new class they say that $$ \sin(ax+c) = \sin(a(x+c))$$ which I believe to be wrong. But I ...
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1answer
236 views

Generate a polynomial w/ integer coefficients whose roots are rational values of sine/cosine?

I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose ...
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4answers
88 views

How to verify this trigonometric identity?

Please help, $9\cos^2 B − 9\sin^2 B = 18\cos^2 B − 9$ Thanks
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1answer
201 views

Minimal set of trig identities to define all the trig functions

What are a minimal set of trig identities that can uniquely define the trig functions? I know that you can define, for example, $\sin(x)$ as the unique solution to the differential equation $f''(x) = ...
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2answers
230 views

Figure out a function expression from graph (sine and cosine)

I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions So far I am sure that the gray line is ...
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349 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
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$\pi$, Dedekind cuts, trigonometric functions, area of a circle

(I should say at the outset that this question is broad, and may need splitting up. Although I ask several questions, I present them as one because they are not independent of one another, and I am ...
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what's the range of $y=\frac{\sin x+a}{\cos x+b}$ .

what's the range of $y=\frac{\sin x+a}{\cos x+b}$ . It is a question I meet somewhere, I hope to find the most simple solution.
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1answer
291 views

dividing an offset circle into triangles

First of all - I am sorry if it is the wrong forum or if this is a very trivial question. I am not a mathematician nor a trigonometry genius - and therefor I would ask a simple answer that someone ...
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1answer
108 views

Is there an easy way to see that $\left(2 \tan^2x+2\right)^3 = 8\left(\tan^2x+1\right)^3$?

I'm reading this explanation of integrals with quadratics and the author pulled this out of nowhere. Is it obvious to everyone but me that this statement is true?
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512 views

Trigonometry without sine and cosine

Maybe an unusual (and too easy for you) question, but my younger brother is requested to calculate the height of the Eiffel Tower: Is this possible, given that he has not learned sine and cosine ...
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1answer
117 views

Request for proof of the following identity.

How do we establish the following identity? \begin{align} &\int_{0}^{\frac{\pi}{2}} \frac{1}{(\sin(\theta) + \cos(\theta)) \sqrt{\sin(\theta) \cos(\theta)}} \,d{\theta} \\ \stackrel{\text{def}}{=} ...
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170 views

Trig integral 2 ways: are both equivalent?

This problem came from some other website, where someone asked for help with the integral $$\int\frac {x \, dx}{1-\cos x}$$ After adding my suggestion of integration by parts to an existing ...
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Easy ways to remember trigonometric identities

Are there any easy ways or mnemonics to memorize the trigonometric identities like for example $$ \sin(3x) = 3\sin(x) - 4\sin^3(x) $$ I find them quite difficult to come up with, I almost always need ...
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509 views

How to prove $ \sin x=…(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})…$? [duplicate]

Possible Duplicate: infinite product of sine function Here is an other one which is more or less what Euler did in one of his proofs. The function sinx where x∈R is zero exactly at ...