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
152 views

Sum and Difference of 2 cosine functions

This question has been troubling me for days, I really haven't got a clue how to handle it: $f(x) = -3+2\cos(x)$ $g(x) = \cos(x-\dfrac{1}{4}\pi)-2 $ Get the sum ($s(x)=f(x)+g(x)$) and difference ...
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4answers
5k views

Drawing sine and cosine waves

I like mathematics and pretty much every mathematical subject, but if there is one thing I thoroughly dislike, it is drawing (functions, waves, diagrams, etc.) We have this important trig test coming ...
2
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1answer
136 views

Trig question I don't really understand

$4\cos^2 \left( x + \dfrac{1}{4}\pi \right)$ = 3 My final answer: $ x = \frac{11}{12}\pi+k\pi $ and $x = \frac{7}{12}\pi + k\pi $ In the correction model it is $x = \frac{7}{12}\pi + k\pi $ and ...
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1answer
96 views

How to tackle these trig questions correctly

$ \sin(2x) \cdot \cos(2x) + \sin(2x) = 0 $ In the correction model I have something I don't understand is done in the first step: $ \sin 2x(\cos 2x + 1) = 0 $ Is this step correct? And can someone ...
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1answer
449 views

How to solve these types of trig equations

Lets use an example: $$ \sin^2 \left(\dfrac{\pi}4x\right) = 1 $$ I am at this point: $$ \frac{\pi}4 x=\frac{\pi}2 + k\cdot2\pi \quad\text{or}\quad \frac{\pi}4 x=-\frac{\pi}2 + k\cdot2\pi $$ But ...
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1answer
376 views

Application of Trigonometry 2

My question is- At the foot of a mountain the elevation of its summit is 45 degrees.After ascending one kilometer towards the mountain up an incline of 30 degrees,the elevation changes to 60 ...
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1k views

how to calculate the exact value of $\tan \frac{\pi}{10}$

I have an extra homework: to calculate the exact value of $ \tan \frac{\pi}{10}$. From WolframAlpha calculator I know that it's $\sqrt{1-\frac{2}{\sqrt{5}}} $, but i have no idea how to calculate ...
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2answers
333 views

Application of Trigonometry

My question is- From an aeroplane vertically over a straight road,the angles of depression of two consecutive kilometer-stones on the same side are 45 degrees and 60 degrees.Find the height of the ...
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1answer
313 views

How to solve a set of cosine equations?

suppose I have an equations of the following with two unknowns $A$ and $\theta$ $A\sin(x+\theta)=D$ I have two points $(E,F) (G,H)$ how do I go about solving this equation analytically. I can solve ...
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5answers
1k views

Prove that $\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(k-1)\pi/n)=-1$

How do you prove that $$\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(n-1)\pi/n)=-1,$$ where $n \geq 2$?
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1answer
85 views

triangles and trigonometry

A triangle has sides $a,b,c$ and angles $\alpha,\beta,\gamma$ such that: $$ a \,\cos\beta + b \, \cos\gamma+ c \, \cos\alpha = \frac{a+b+c}{2}$$ Prove that the triangle is isosceles. I tried writing ...
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2answers
205 views

Simplification of trigonometric expression regarding planetary orbits

I am trying to solve an orbital problem concerned with analytically calculating the solstice points of an orbit. I have managed to reach a point in the problem where I need to simplify the ...
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2answers
161 views

Finding $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$

I try to find $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$. Left side: $$\sin(4 \alpha + \frac{\pi}{6}) =$$ $$= \sin4\alpha \times \cos \frac{\pi}{6} + \cos 4\alpha ...
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1answer
157 views

evaluation of the integral $\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $

Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks. ...
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2answers
2k views

$\sin 4\alpha = 2\sin 2\alpha \times \cos 2\alpha $?

A trigonometry rule says that $\sin 2\alpha = 2\sin \alpha \times \cos \alpha$. Does this also apply to $\sin 2x$ when $x = n \times \alpha$? For example: $$\sin 4\alpha = 2\sin 2\alpha \times \cos ...
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3answers
764 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
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1answer
244 views

calculated reflected point within circle

The problem to solve is this. Imagine a circle. We know two points on the circumference, anchor A and anchor B, they could be anywhere on the circumference of the circle. Draw a line between these ...
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2answers
69 views

Calculate $\cos(v+\frac{\pi}{6})$ when $\cos v = -\frac{2}{3}$

I am to calculate $\cos(v+\frac{\pi}{6})$ when $\cos v = -\frac{2}{3}$ and $0 \lt v \lt \pi$. I know I can change $\cos(v+\frac{\pi}{6})$ into $$\cos v \times \cos \frac{\pi}{6} - \sin v \times \sin ...
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1answer
92 views

A simple question about angles on a circumference

Given two points on a circumference of radius $R$, $P_0$ and $P_1$ subtended by an angle $\theta$ at the center of the circumference, what is the angle at which a generic point $P_m$ inside the circle ...
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1answer
508 views

Auxiliary trigonometric identities [Identidades Trigonométricas Auxiliares]

I need to show two auxiliary trigonometric identities: 1) $\sec^2x = \tan ^2x + 1 (\cos x \neq 0)$ 2) $\csc^2x = \cot^2x +1 (\sin x \neq 0)$ How could I do it? [Original Portuguese] Identidades ...
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3answers
567 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
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2answers
399 views

Starting point of a Sine wave

We learned this at school, the function: $$y= a + b\sin (c(x-d))$$ has a starting point of $(d,a)$. But when I had to draw this function: $$g(x) = -2-\cos(x-1/2π)$$ I thought the starting point ...
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1answer
666 views

Triangle two angles and one length [closed]

In a triangle ABC, the angle at A is 41 degrees, the angle at B is 71 degrees, and the length of side AB is 8. To 2 decimal places, what is the length of side BC? I got this answer 5.66. is that ...
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1answer
86 views

Triangle one angle and two lengths [closed]

In a triangle ABC, the angle at B is 108 degrees, the length of side BC is 16, and the length of side AB is 12. To 2 decimal places, what is the length of side AC? So i worked out and out and got ...
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2answers
250 views

calculating limit without using direct formulae

In case of $\delta$-$\epsilon$ definition of limit, books usually show that some $L$ is the limit of some function and then they prove it by reducing $L$ from the function $f(x)- L$ and showing ...
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1answer
504 views

How to find the limit of $\sin(f(x))$, given the graph of $f(x)$

The full question is uploaded here: http://imgur.com/EZekb Basically, given the graph shown in the image above, I thought that the limit of $\sin(f(x))$ would be $\sin(2)$, since the limit of just ...
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3answers
615 views

Expressing in the form $A \sin(x + c)$

Express in the form $A\sin(x+c)$ a) $\sin x+\sqrt3\cos x$; b) $\sin x-\cos x$ sol: a) $A=\sqrt{1+3}=2$, $\tan c=\frac{\sqrt 3}1$, $c=\frac\pi3$. So $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$ ...
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3answers
114 views

Sine and Cosine equation ( diophantine )

$\cos(\frac{1}{ab} \pi) = \sin(\frac{a}{b} \pi)$ Let $a$ and $b$ be positive integers. What is the full set of solutions? An example is $a = 2$ and $b = 5$. I assume the best method is to take ...
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1answer
586 views

Bisect an angle

I have a simple question, and I've looked over the internet and can't find a numerical way of doing it. I only found how to do this using a ruler and a compass, which I can't use, since I'm doing a ...
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votes
2answers
255 views

Prove that $ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $

Prove that $$ \frac{1}{2}\cot^{-1}\frac{2\sqrt[3]{4}+1}{\sqrt{3}}+\frac{1}{3}\tan^{-1}\frac{\sqrt[3]{4}+1}{\sqrt{3}}=\dfrac{\pi}{6}. $$
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2answers
2k views

Explicitly finding the sum of $\arctan(1/(n^2+n+1))$

This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right)$$.
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3answers
685 views

Solve sum for theta

Is there any way to solve the following sum of trigonometric functions for theta without using a solver? $$25\sin(\theta)-1.5\cos(\theta)=20$$
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1answer
246 views

How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate $$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$ for $y\in\mathbb{R}$?
2
votes
2answers
430 views

Evaluation of a trigonometric partial sum

I just wanted to evaluate $$ \sum_{k=0}^n \cos k\theta $$ and I know that it should give $$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$ ...
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2answers
232 views

Calculating circumference from 2d coords

I'm trying to calculate the circumference of a circle given say three reference points from a 2d coordinates that would form an arc. The problem is the reference points may be slightly inaccurate so ...
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2answers
447 views

Possible to evaluate definite integral of inverse trigonometric function as function of $Y$?

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. Now I have the ...
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1answer
84 views

Evaluating $\int\frac{\cos^{n-1}\frac{x+a}{2}}{\sin^{n+1}\frac{x-a}{2}}\;dx$

Please help me to evaluate the following integral: $$\int\dfrac{\cos^{n-1}\dfrac{x+a}{2}}{\sin^{n+1}\dfrac{x-a}{2}}\;dx$$
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1answer
89 views

problem with some inverse trigo function graphing - need some help! [duplicate]

Possible Duplicate: Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same? I faced this equation while dealing with some inverse trigo functions. $\arcsin(\sin(x)) ...
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0answers
82 views

Definition of Inverse Cotangent

I would like to derive the following expression for inverse cotangent: $\cot^{-1} (z) = \frac{i}{2}[\ln(\frac{z-i}{z}) - \ln(\frac{z+i}{z})]$ But I don't want to take it as "definition" as this page ...
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2answers
4k views

“Show” that the direction cosines of a vector satisfies…

"Show" that the direction cosines of a vector satisfies $$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$ I am stumped on these things: "SHOW" that the direction cosines corresponds to a given ...
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1answer
389 views

Trilateration to know relative distance to a point.

I've been trying to find the distance $d$ in this system where I only can know the distances $a0, a1, b0, b1, c0, c1$ but not the position of $A, B, C, P0, P1$. I tried 2D trilateration using ...
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1answer
328 views

Relative side lengths of dual dodecahedron and icosahedron

If the side length of a dodecahedron=1, then what is the side length of its dual icosahedron whose vertices occupy the same space as the mid-points of the faces of the dodecahedron. I've read that ...
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1answer
83 views

How does this fraction simplify?

how does $$1+\cos2\theta = (2-\sqrt2)/2$$ give you $-\sqrt2/2$? i get that you subtract 1 from the left side, but how does doing so on the right give you $-\sqrt2/2$?
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1answer
247 views

Trigonometry identity help

My math book has a trigonometry identity $\frac{2\cos2x}{\cos x}$ and they simplify it to $2\cos^2x-1$ but do not show the steps. I have tried many time to simplify but can never get the same answer. ...
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2answers
204 views

Triple “Pythagorean identity”

It's not hard to find multiple trigonometric functions of period $2\pi$ that added to self shifted by some constant offset result in a constant. In classic pythagorean identity, you have ...
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3answers
304 views

Proof of the tangent addition theorem for complex numbers

How can the tangent addition theorem for complex numbers $$\tan(z+w) = \frac{\tan z + \tan w }{1 - \tan z \tan w}$$ be proved? For real numbers, the wikipedia page says one can use Euler's formula ...
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3answers
6k views

Is $\tan(\pi/2)$ undefined or infinity?

The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself. If we also observe a fraction $F=a/b$ ...
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3answers
159 views

Rigorous way to find the limit of this difference?

This is a question from an old released exam. By the triangle inequality, $s-r<1$, so I eliminate answers D and E. Intuitively, since the lower angle between $1$ and $r$ is fixed at $110^\circ$, ...
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5answers
1k views

Why do hyperbolic “trig” functions seem to be encountered rarely?

Hyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarely. ...