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
56 views

Evaluate the sum of $\log\tan k$, $k=1\dots87$

Evaluate the sum $$\sum_{k=1}^{87}\ln(\tan k)$$ I first of all wrote them as ratios of $sine$ and and $cosine$ using the fact that $tan(x)$=$sin(x)$/$cos(x)$ Then I later use the co- function ...
8
votes
3answers
153 views

Differential Equation of the Form $\frac{dy}{dx}=\sin(x+y)$ [duplicate]

I have been attempting to solve the above differential equation for some time now, and I remain stuck on one step. After substituting $u=x+y$, separating the variables, and integrating both sides, I ...
0
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2answers
78 views

Restricting intervals for parameter equations

What interval $I$ should we restrict the parameter t, when the graph of parametric equations $\begin{align*} x=t+1, y=2t^2 +1 \end{align*}$ for $t\in I$ is identical to the graph of the parametric ...
0
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3answers
268 views

Find the smallest interval for parametric equations

$\mathcal{G}$ is the graph of parametric equations $\begin{align*} x = \cos(4t), y = \sin(6t). \end{align*}$. Find the length of the smallest interval $I$ such that the graph of the parametric ...
0
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1answer
56 views

Integral of ArcCos with Difficult Argument

I have $$\int_{d-1}^{1}2 x \arccos\left(\frac{x^2+d^2-1}{2 x d}\right)\,\textrm{d} x$$ but can't find the right substitution. I have little experience integrating $\arccos$ with anything but trivial ...
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2answers
39 views

How to find the remaining segment of this triangle without the Law of Cosines?

I am given a triangle $T$ with vertices $A, B,$ and $C$ and the the following info about $T:$ $\overline{AC} = 6$ Km $\overline{BC} = 9$ Km The angle formed by these two segments is $120^{\circ}$ ...
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1answer
79 views

Need help using Atan2 instead Arctan for transforming cartesian to polar coordinates.

I need some help to use Atan2 function instead of arctan to the following equation. This is used to find the inverse of lon, lat angles based on a lon lat reference point projected onto an x,y plane ...
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1answer
45 views

Lattice points on parametric equations

The definition of a lattice point is a point that has integer coordinates. Find the number of lattice points $(x,y)$, where $-100\le x\le 100$ and $-100\le y\le 100$ are on the graph of the ...
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1answer
264 views

Find the intersection of two parametric equations at one point

Find all $k$ such that the graph of parametric equations $\begin{align*} x &= 2+ 4\cos s, y= k-4\sin s, \end{align*}$ intersects the graph of the parametric equations $\begin{align*} x&=1+\cos ...
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1answer
57 views

How to convert $r = 2s\cos (\theta+t)$ into Cartesian coordinates?

How would I convert $r = 2s\cos (\theta+t)$ into Cartesian coordinates? I believe $r$ currently is in polar coordinates. However, polar coordinates are in form $(r,\theta)$ and the equation given is ...
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3answers
41 views

How to find the point of intersection of four parametric equations

I am trying to find the TWO points of intersection of the parametric equations: $x = \cos t, y = \sin t$ and $x = 2+4 \cos s, y = 3+4 \sin s$. Would I set $\cos t = 2+4 \cos s$ and $\sin t = 3+4 \...
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1answer
52 views

Area of an isosceles triangle where the tangents of some angles are in geometric progression

In $\triangle ABC$, $AB=BC$ and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $\overline{BE}=10$. The values of $\tan CBE$, $\tan DBE$, and $\tan ABE$ form ...
3
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0answers
69 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
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1answer
61 views

How to solve a triangle knowing just two legs?

Given a right triangle, if I know two of its legs are $a = 3,5$ and $b = 5,5$, is it possible to solve the triangle with this info? This is, can I determine all of its sides and all of its angles with ...
2
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2answers
43 views

Minimum value of trigonometry function

How can I get the minimum value of function $$f(x) = (2 + \sin x)(5 - \sin x)$$ I have used the differential ways but the answer was not match with the key answer. By the way the key answer is $6$.
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1answer
92 views

Finding the length of the sides, the measure of the angles and area of spherical triangles?

I'm trying to understand this problem in the textbook but I got lost in one part: Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the ...
5
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1answer
66 views

Existence and unicity of a constant $c$ such that $\lim_{x \rightarrow \pi} \frac {x + c}{\sin x}$ is finite

Supposing $c = - \pi$. Then, the quotient will assume the form $0/0$. Using L'Hospital $$\displaystyle \lim_{x \rightarrow \pi} \dfrac {x-\pi}{\sin x} = \lim_{x \rightarrow \pi} \dfrac {1}{\cos x} = -...
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3answers
75 views

Simple algebra involving trigonometry, but confusing

How do I get from $$\frac{\sqrt3}{2} + \frac12 \tan x = 2 \tan x \cdot \frac{\sqrt3}{2}$$ to $$\frac{\sqrt3}{2 \sqrt3 - 1} = \tan x$$ and then to $$11 \tan x = 6 + \sqrt3$$
3
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2answers
44 views

Why does this work in this type of function?

I was just thinking about this: In a function like y = 3x you can find the correct slope even if you were to add two coordinates together. Example: Find the slope by using x = 2 and x = 4 So, correct ...
2
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3answers
123 views

Finding the argument $\theta$ of a complex number

I want to find the Argument of $z = -\sqrt{2 - \sqrt{3}} + i\sqrt{2 + \sqrt{3}}$ where $z$ is a complex number of the form $z = a + bi$. I find that the modulus is $2$, but am having trouble ...
2
votes
1answer
62 views

Placing a cone in a circle

I am trying to make a neat charting app, yet I am having dificulty with some of the math. That's why I'm here! I have a "pie slice" at a specific angle. I don't want the slice to just be a simple ...
0
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1answer
24 views

Graphing and finding sine wave info with $\sin(x/4)$

I'd doing a chapter on graphing sine waves, and finding the amplitude, period, and so on. I know something like $y = 2 \sin(3x+ \pi) + 1$ can be turned into $y = 2 \sin[3(x + \pi/3)] + 1$ following ...
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1answer
45 views

Algebra with a little trig problem?.

$\frac{1}{r\tan \left(\frac{\alpha}{2} \right)} = a$ $\frac{1}{\tan \left(\frac{\alpha}{2} \right)} = b$ $\frac{f}{(f-n)} = c$ -$\frac{nf}{(f-n)} = d$ Solve what you can first: $\alpha = 2\arctan\...
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5answers
44 views

Solving $\cos x + 3^{1/2} \sin x = 1$ for $0\leq x \leq 360^{\circ}$

$\cos x + \sqrt3 \sin x = 1$ Not sure what the step would be to get an answer.
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1answer
36 views

stuck at simple trigonometric equations

I'm reading paper on inverse kinematic using simple trigonometric equations. In one part of the paper, the author skipped straight to final equation without any derivation. My trigonometric is not ...
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2answers
207 views

Relationship between trigonometric and hyperbolic sine

Why is the following identity true? $$ \sin(ix) = i\sinh(x)$$ When I do the calculation, I get this:$$\sin(ix) = \frac{{e^{i(ix)}}-e^{-i(ix)}}{2i}=\frac{e^{-x}-e^x}{2i}=-\frac{e^x-e^{-x}}{2i}=-\left(...
3
votes
1answer
69 views

Proving that $\sin^7\theta + \cos^7\theta <1$ using basic trigonometry and identities [closed]

How do I prove $\sin^7\theta + \cos^7\theta < 1$ for an angle between $(0,\pi/2)$?
2
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2answers
56 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
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3answers
7k views

Find the length of the chord given that the circle's diameter and the subtended angle

A chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...
0
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0answers
183 views

Find L for $r = \cos 3 \theta$.

Pictured above is the graph of $r = \cos 3 \theta$ for $0 \le \theta \le L$. Find the smallest value of $L$ that still produces the entire graph of $r = \cos 3 \theta$. I am having trouble starting ...
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1answer
39 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx \...
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2answers
72 views

What do we know about $\sin^{2} n$?

We all know that $-1 < \sin(n) < 1$. What about $\sin^2(n)$? What can we say about it? The main question is find the limit of $$\lim_{n\to\infty }\frac{\sin^2 n}{2^n}.$$
2
votes
2answers
184 views

Polar Plots and square roots

When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to ...
5
votes
1answer
108 views

Why is $\mathrm{arctan}(0)$ not infinity?

$\arctan x$ is defined as: $$\arctan x = \frac{1}{\tan(x)} = \frac{1}{\frac{\sin(x)}{\cos(x)}}$$ if I now have $x = 0$ I should get: $$\frac{1}{\frac{\sin(0)}{\cos(0)}} = \frac{1}{\frac{0}{1}} = \...
4
votes
4answers
777 views

Partial fractions and trig functions

A long time ago I wrote down a silly problem. It starts with Attempt to write $$\frac{1}{\sin(x)\cos(x)}$$ using partial fractions. and then goes on to prove a trig identity. I was wondering if ...
2
votes
5answers
107 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
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2answers
69 views

Euler formula, trigonometry.

Prove with Euler formula that $$ \cos(x-y) = \cos(x)\cos(y) - \sin(x)\sin(y). $$ I know how to find $\cos(x+y)$, but as for $\cos(x-y)$, I'm clueless. Thanks.
4
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1answer
124 views

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$ [duplicate]

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$? The period of the first term is $\pi$ and that of the second is $4\pi$. Does that mean that the period of the whole is $4\pi$?
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3answers
66 views

Rewrite $\sin(\omega t)$ in terms of exponentials

Could someone please give me a pointer or two. I am trying to rewrite $\sin(\omega t)$ and it should be something similar to $\dfrac{e^{2j\omega t}-e^{-2j\omega t}}{2j}$ but I can't quite seem to get ...
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2answers
58 views

How to understand sinus?

In $\Delta PQR$ we have $\angle PQR=60^\circ$, $QR=4$ and $PR=a$. For which values of $a$ are there 0, 1 and 2 triangles matching the description? I think I'm supposed to use the law of sines, but ...
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1answer
74 views

Generalized angle sum identity for $\arctan$?

The angle sum identity for arctan is: $$\arctan (\alpha)+\arctan(\beta)=\arctan\left(\frac{\alpha+\beta}{1-\alpha\beta}\right)$$ I was wondering if there exists a relationship for any linear ...
0
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5answers
80 views

What is the integral of $\frac{\sqrt{x^2 +4}}{x}dx$

I use trig substitution then get to this step but then I get stuck: $\int \frac{2\sec ^3\theta}{\tan \theta}d\theta$ anything I do seems to further complicate it. Thanks in advance.
0
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1answer
192 views

Finding third vertexes of any triangle where 2 vertex known and all sides length known

I am working with a CAD engine in the head but i working on code only. I have a rectangular tube that need to be put at an angle. I so have the diagonal of the tube where it has to start and stop but ...
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votes
1answer
97 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + b^2 \...
2
votes
1answer
38 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
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0answers
175 views

Calculate vertical lines intersection hexagon at regular interval

I would like to calculate the total size of vertical lines that dissect an hexagon regular, like the one on the image. I would like to know the internal size of the blue lines inside the hexagon, ...
0
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2answers
149 views

Evaluating trig functions for a point that passes through…

I have the question "Evaluate the trig functions for angle a in standard position whose terminal side passes through (3, 4): Sec a, csc a, and cot a. For cot a the answer given is 3/4, which makes ...
5
votes
2answers
333 views

For which $q\in\mathbb Q$ is $\sin(\frac\pi2q)$ rational?

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q\;?$$ Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < ...
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2answers
165 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
0
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1answer
44 views

Help With Solving Trigonometric equations

$(\sin x)^2 - 5\sin x \cos x=0$ What would be the first atep to solve this. I normally get the equation into a quadratic one but I cannot seem to spot the first step here. What I mean by $(\sin x)^2$...