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

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1answer
44 views

Trig equation, solve for x

I couldn't solve the question : $3\cos^2 x-2$ for $x\in(180,270)$, evaluate $2\sin(x/2+30)$ My answers were $x=35.26+360n$ $x=324.74+360n$
1
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0answers
265 views

Definite integral involving exponential, powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} e^{-qx^{ak}(x^2+d^2+2 \, dx \cos[t])^{-a/2}} dt $$ I am not able to find any related formula. Note that this integral follows ...
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0answers
235 views

Definite integral involving powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} {m \over m + x^{ak}\left[\,x^{2} + d^{2} + 2dx\cos\left(t\right)\,\right]^{-a/2}} \,{\rm d}t $$ I am not able to find any ...
0
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1answer
151 views

Ptolemy Trig/Geometry Help Please

For this problem, I need to find a bunch of values for this semicircle, and use the methods of Ptolemy to find a value for sin 18. This is for a low level math history class, and I'm bot good with ...
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1answer
197 views

Power of tangent is odd and positive- integration

So I am currently studying trig substitution and am curious about the process involved in finding an answer. $$ \int \frac {\tan^{3}(x)}{\sqrt {\sec(x)}} dx$$ $$=\int \sec(x)^{-1/2}{\tan} ...
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7answers
843 views

Fundamental Theorem of Trigonometry

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
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1answer
46 views

Precalculus Trig Problem

I have no clue where to start this problem, some help would be awesome and much appreciated! $$\text{If }\sin(x)-\cos(x)=-0.3, \text{ evaluate }\sin(2x).$$
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1answer
366 views

Finding the Third Side of a Triangle Composed of Vectors

Imagine a beam of light emanating from a point in space and striking a surface. Then, for any other point on that surface, I want to know the distance from that point to the light vector. What I ...
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5answers
347 views

Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?

We have the followings: $\sin(\frac{\pi}{1})=\frac{\sqrt{0}}{\sqrt{1}}$ $\sin(\frac{\pi}{2})=\frac{\sqrt{1}}{\sqrt{1}}$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{\sqrt{4}}$ ...
0
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1answer
168 views

How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
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2answers
53 views

Troubles when evaluating some limits with trig functions

I know that $\displaystyle\lim_{x \to 0}{\frac{\sin x}{x}} = 1$, but I cannot use L'Hopital Rule as we haven't studied it yet. Here are the limits: $\displaystyle\lim_{x \to 1}{\frac{\sin(1 - ...
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3answers
108 views

Proof of a trigonometric expression

Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$. This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.
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2answers
560 views

Hard contest type trigonometry proof

Suppose that real numbers $x, y, z$ satisfy: $$\frac{\cos x + \cos y + \cos z}{\cos(x + y + z)} = \frac{\sin x + \sin y + \sin z}{\sin (x + y + z )} = p$$ Then prove that: $$\cos (x + y) + \cos (y + ...
4
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1answer
158 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
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1answer
65 views

Complex exponential sumation: $\sum_{-M}^M e^{-2\pi ikf}$

Does someone know how to prove that $\displaystyle {\sum_{k = -M}^M e^{-2\pi ikf} = \frac{sin((2M + 1)\pi f)}{sin(2\pi f)}}$ where $f$ is a constant such that $-\frac{1}{2} < f < \frac{1}{2}$ ...
0
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1answer
40 views

Trigonometric Prοof

How can one show that for any angle $\theta$ such that $0<\theta<2\pi/5$ the following equation is true? $\sqrt{1-\cos\theta} ...
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2answers
314 views

How can I prove this two identities? $\cos^2 x=\frac{1+\cos(2x)}{2}$ and $\sin^2 x=\frac{1-\cos(2x)}{2}$

How can I simply prove the two following equations? $$\cos^2x=\frac{1+\cos(2x)}{2} \,\,\,\,\,\,\,\,\,\,\text{ and }\,\,\,\,\,\,\,\,\,\, \sin^2 x=\frac{1-\cos(2x)}{2}$$ I already proven them using ...
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4answers
90 views

Does $-\tan(x)=\tan(-x)$ for all $x$?

Just to clarify, does $-\tan(x)=\tan(-x)$ for all $x$?
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1answer
613 views

Why tan(1/z) has a non-isolated singularity at z=0?

Can someone please explain me this concept. Any sort of help will be highly appreciated.
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2answers
114 views

Evaluating $\int e^{-x}\tan(x) \ \mathrm dx$ .

I was told years ago by a visiting professor that this integral:$$\int e^{-x}\tan(x)dx$$ has an elementary form, but I have never been able to find it. Any suggestions? I don't think it's possible ...
4
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0answers
166 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
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2answers
61 views

Proving a trigonometric identity.

I really need some help with this question. I need to prove this identity: $$\frac{2\sin^{3}x}{1-\cos x} = 2\sin x + \sin 2x.$$
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4answers
630 views

Why is $\sin(x) = \sin(180^{\circ}-x)$

I cannot seem to understand why this is true. Same for $\cos(x) = -\cos(180^{\circ}-x)$ and $\tan(x) = -\tan(180^{\circ}-x)$. Without the use of the compound angle formulas. Thanks
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1answer
202 views

An Unexpected Circle…

I played around with $$z=\frac{-1+e^{it}}{\phantom{-}2+e^{it}}$$ and found that, when I draw the real against the imaginary of $z$, it pretty much looks like a circle. But neither ${\frak{R}} z ...
0
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1answer
252 views

How to treat system of linear first order differential equations with trigonometric function coefficients?

I'm having trouble solving the following IVP: $$x_1^\prime = -x_1\tan t + 3\cos^2t$$ $$x_2^\prime = x_1 + x_2\tan t + 2\sin t$$ where $x_1(0) = 4$ and $x_2(0) = 0$. I'm not sure what to do when the ...
2
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1answer
74 views

Contract expression of circle segment area contingent on height

I want to determine a function for the area of the segment's height. I have made it this far, but I would like to contract the equation further - sadly, I do not know how to do this while still ...
2
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1answer
69 views

solving $\frac{x}{3}+{[\frac{x}{3}]} = \sin(x) + [\sin(x)]$ for real x , in an efficient manner

$$\frac{x}{3}+{\left[\frac{x}{3}\right]}=\sin(x)+[\sin(x)]$$ I know the answer and the solution. $$-1 \le \sin(x) \le 1\Rightarrow-2\le\sin(x)+[\sin(x)]\le 2 \Rightarrow -2 \le ...
2
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1answer
374 views

Proving that $\sin(x)$ is continuous at $0$

Given: $|\sin x| < |x|$, valid for $0<|x|<\frac12\pi$ (EDIT: $\frac12 \pi$ not $\frac1{2\pi}$) Conclusion: $\lim_{x\to 0}\sin(x) = 0$, also expressed as $|\sin(x)| < \epsilon$ if ...
3
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1answer
278 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
2
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2answers
71 views

Round 2: Derive this trig identity from the common ones? $\;\cos^2(3x) = \frac{1+\cos(6x)}{2}$

$$\cos^2(3x) = \frac{1+\cos(6x)}{2}$$ Just came across another wacky identity today. Is this an easy derivation from the more popular identities, or is this one you just take it at face value and ...
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1answer
20 views

Trigonometric equation tangent

I want to solve trigonometric equation $\tan x=\pi-x$. one of these solutions in interval [0,$\pi$] is $x=\pi$. Find another root please.
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6answers
538 views

How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
0
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1answer
876 views

Find the area of a circle that is inscribed in a circular sector with a radius $R$ and an angle $2x$.

The circle within the sector touches the radii R and the arc. So what is the area of the inscribed circle? The answer is actually $$S = \pi R^2\frac{\sin^2x}{(1+\sin x)^2}$$ How can I derive this?
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1answer
79 views

Representing a function as a real part of a complex variable?

To represent a simple sinusoidally varying function $V(t)$ let us use $V(t)=Re(\hat V e^{i\omega t})$ where $\hat V$ can be a complex constant.Let $\hat V =-iV_o$ Therefore, $V(t)=V_o \sin(\omega t)$. ...
2
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2answers
90 views

Why can't I use trigonometric functions here?

I tried solving this answer by using trig functions to get the answer (E). First I calculated angle FBC which was 30 and then i used sin 30 to get the length of FC. then I got the length of line BF ...
0
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1answer
234 views

Is $\cos(90 - 2x) = \sin (2x)$?

What does $\cos(90 - 2x)$ equal? I know $\cos (90 - x) = \sin x$ but does this equal $\sin 2x$? Thanks.
8
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2answers
303 views

maximum of $f(x)=\sin{(\pi Ax)}\left(\,\csc{(\pi x)}+\csc{(\pi (\frac{1}{A}-x))}\,\right)$

Let a function $f(x)$ be $f(x)=\sin{(\pi Ax)}\left(\,\csc{(\pi x)}+\csc{(\pi (\frac{1}{A}-x))}\,\right)$ where $A\geq 2$ is a positive integer and $\csc{(x)}=\frac{1}{\sin{(x)}}$. I want to prove ...
1
vote
1answer
225 views

Solve $x - 2\arctan(x)= 0$

$x - 2\arctan(x)= 0$. I can find one root (0) from the equation $\tan(x/2) = x$ but there are two others, namely ($-2.3312, 2.3312$) that I don't know how to find. Looking for help! Thanks :) ...
0
votes
2answers
90 views

Simplify the trig function $\frac{2\cos{2x}}{\sin{2x}}$

How would I simplify the trig function $\frac{2\cos{2x}}{\sin{2x}}$? Is this a trig identity of can I simplify it down further?
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2answers
975 views

How to prove that $\lim_{x \to 0} \sin(x) = 0$ using the epsilon-delta definition?

How do I prove that $\lim_{x \to 0} \sin(x) = 0$ using the episilon-delta definition of a limit? Do I have to divide the domain of $x$ into 4 cases for each quadrant? Update: Based on the ...
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1answer
199 views

Prove that $\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is an increasing function

Could anybody show that $f(x)=\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is increasing on the interval $x\in[0, 0.25]$? Of course, $\csc{(x)}=\frac{1}{\sin{(x)}}$. Here is ...
3
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1answer
93 views

How find this $\sum_{i=1}^{5}\tan^4{\frac{i\pi}{11}}$

show that:$$\tan^4{\dfrac{\pi}{11}}+\tan^4{\dfrac{2\pi}{11}}+\tan^4{\dfrac{3\pi}{11}}+\tan^4{\dfrac{4\pi}{11}}+\tan^4{\dfrac{5\pi}{11}}=2365$$ my try: I think first we can find this ...
3
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2answers
244 views

Verifying trig identities

$$\sec(2x)=\frac{\sec^2x+\sec^4x}{2+\sec^2x-\sec^4x}$$ I have no idea how to verify this. I've tried changing it into cosine but it doesn't work.
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2answers
1k views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$$
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2answers
112 views

Assuming *only* that $\sin^2{\theta}$ + $\cos^2{\theta} = 1$, show that $\sin{\theta}\cos{\theta}\le1/2$

Assuming only that $\sin^2{\theta}$ + $\cos^2{\theta} = 1$, show that $\sin{\theta}\cos{\theta}\le1/2$. I only know how to do it using calculus!
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1answer
68 views

Derivative of trig function and limits

Find $$ \lim_{x\to \frac{\pi}{2}} \frac{tan(3x)}{tan(7x)} $$ I want to find it using l'hopital's rule My answer was : $$ \frac{\frac{sin(3x)}{cos(3x)}}{\frac{sin(7x)}{cos(7x)}} $$ $$ ...
0
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3answers
45 views

$3 \tan^{-1 }(x-2) +\pi =0$

Please help. I don't know where to start. Do I use the addition rule, or do I start with the double angle formula? I've never done one with the pie sign before!! It should be $3\tan^{-1} (x-2) + \pi ...
1
vote
1answer
84 views

Which icosahedral triangles projected onto sphere's surface contain points in P?

I am working on a Python script to: Compute the vertex coordinates of a geodesic sphere/icosahedron, Project the triangles onto a sphere, then Find which spherical triangle contains an arbitrary ...
0
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4answers
216 views

Derivative of trig function (l'hopital's rule)

Find: $$ \lim_{x\to 0} \frac{\sin^26x}{\sin^27x} $$ My answer was : $$ \frac{2\sin6x \cdot \cos6x}{2\sin7x \cdot \cos7x} $$ When I took the derivative of the numerator and denominator I got this $$ ...
2
votes
2answers
70 views

Use De Moivre's Theorem to determine $(-1 +i)^{184}$ in the form $x + iy$

Use De Moivre's Theorem to determine $(-1 +i)^{184}$ in the form $x + iy$ I first rewrite the equation in polar form. To do this I first determine $z$ $z = -1 + i$ I then solve $|z| = \sqrt{-1^2 + ...