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
130 views

Tait-Bryan to Rotation matrix to translating from global to local space

Re-writing my entire question to be more math-oriented and hopefully make more sense. I have two objects, each at a position defined by P1 and P2 (XYZ). Each has a heading based on yaw/pitch/roll, ...
0
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1answer
104 views

equation for the radius of a circle that is tangent to two lines and passing through a specific point on one of the lines?

I'm interested in finding the equation for the radius (and optionally the center point) for a circle that is tangent to two lines and passing through a specific point on one of the lines. So far, I've ...
1
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1answer
59 views

Trying to find an $\arctan(x/y)$ identity.

I have this equation : $$\theta = \arctan\left(\tfrac xd\right) + \arctan\left(\tfrac yd\right).$$ $\theta$ is an angle and I am trying to express $d$ as a function of $\theta$. So is there a way ...
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3answers
40 views

Implicit differentiation with trig function

I have the following expression which I need to implicitly differentiate: $$ xy^2 + x^2 + y + \sin(x^2y) = 0 $$ I'm a little confused as I'm not entirely sure what to do with the trig function. ...
0
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1answer
199 views

Calculating the points of a annular sector type shape.

The problem involves a circle inside a square sharing a common center point. The circle is always smaller than the square so that their edges never intersect. Then an annular sector (see cyan shape in ...
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1answer
58 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
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2answers
68 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
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2answers
27 views

Definite integral of trig function

I'm looking for some assistance on the following problem: Let $$ T(x) = \int_{4r^3}^{4} tsin(t^3)dt $$ Find $$T'(r)$$ I'm struggling to find the antiderivative of the sine function, particularly as ...
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6answers
174 views

Find the first derivative $y=\sqrt\frac{1+\cosθ}{1-\cosθ}$

$$y=\sqrt\frac{1+\cosθ}{1-\cosθ}$$ my professor said that the answer is $$y'=\frac{1}{\cosθ-1}$$ she said use half angle formula but I just end up with $\frac{(-2\sinθ)\sqrt{(1-\cosθ)(1+\cosθ)}}{2(1-\...
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3answers
291 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
34
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2answers
810 views

Is Pythagoras the only relation to hold between $\cos$ and $\sin$?

Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$. (Vague) Question. Is this the only relationship between the functions $\cos$ and $\sin$? More precisely: Let $\...
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2answers
31 views

Slight help with inverse trigonometry question

I apologize for the lack of LaTeX, i will try to learn LaTeX and update this question as soon as possible. I am having some trouble with an inverse trigonometry question and was hoping that someone ...
1
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1answer
59 views

Evaluate an integral involving tangent and secant: $\int \tan^2x\sec^2x\,dx$

Evaluate $\displaystyle \int \tan^2x\sec^2x\,dx$ I tried several methods: First method was I changed $\tan^2x = \sec^2x-1$, and then substitute $\sec x$ to $t$, but it doesn't work. Second ...
2
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1answer
64 views

Equilateral Triangle Problem With Trig

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle P. The distance $|AP|=3 cm, |BP|=4 cm, |CP|=5 cm.$ What is the area of the ...
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1answer
117 views

Constructing triangle using side length-median relationship

$$\begin{align} m^2_a&=\frac{2b^2+2c^2−a^2}4\\[4pt] m^2_b&=\frac{2c^2+2a^2−b^2}4\\[4pt] m^2_c&=\frac{2a^2+2b^2−c^2}4 \end{align}$$ Solving for $a$, $b$, $c$ in terms of $m^2_a$, $m^2_b$, $...
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1answer
349 views

Desmos.com simulating spinning orbital object

https://www.youtube.com/watch?v=U_VsPV1WJbg As shown in the video, the face with the eyes and mouth are orbiting an unplotted circle with radius = 4, but also spinning (rotating) in a circular motion ...
3
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1answer
97 views

Question of trigonometry

If $\cos^2 A=\dfrac{a^2-1}{3}$ and $\tan^2\left(\dfrac{A}{2}\right)=\tan^{2/3} B$. Then find $\cos^{2/3}B+\sin^{2/3}B $. I tried componendo and dividendo to write the second statement as cos A but ...
2
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1answer
146 views

Proving $\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi }{2-x}\right)$

How to solve the following question? If $n$ is an integer, show that \begin{eqnarray} \left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(...
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1answer
417 views

Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that $$f'...
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votes
2answers
107 views

Trig Equation - 2 years out of math & lost [closed]

$$\cos^2(2x) + \sin^4(x) = 2$$ So lost on how to solve these things and it's already midnight. 3 days I've spent reviewing and doing practice, but I can't find any proper information on how to go ...
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1answer
155 views

What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$? [duplicate]

How to compute $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$ I tried to rewrite it in terms of $\sin$ $$ \csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}...
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3answers
1k views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is simple:...
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1answer
73 views

If $\alpha$, $\beta$ are two values of $\theta$ satisfying the equation $\cos\theta/a+\sin\theta/b=1/c$, prove that $\cot ((\alpha+\beta)/2) = b/a$

What I did was $$b\ \cos (\theta) + a \sin (\theta) = \dfrac{ab}{c} \\ b\ \cos (\theta) = \frac{ab}{c} - a\ \sin (\theta) $$ Square both sides and using sum of roots and product of roots as $$\...
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2answers
99 views

If xsinθ = ysin(θ + 2π/3) = zsin(θ + 4π/3) then prove that Σxy = 0?

Please help! I don't know how to solve this question. I tried putting the whole thing equal to "k" and then calculating values of x,y and z in terms of k and putting there. But it messes up the ...
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1answer
75 views

Why do both trig functions have the same Macluarin series?

Both the degree version and the radian version of the trig functions have the same Maclaurin series, yet they are different. How is this possible? How can two different functions have the same ...
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5answers
178 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
1
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1answer
92 views

Roots of $f(x)=a_0+a_1\cos x+a_2\cos 2x+\dots+a_n\cos nx$

If $a_i$'s are nonzero real numbers such that $a_n > {\sum^{n-1}_{i=0}}|a_i|$ prove that the number of roots of $f(x)=a_0+a_1\cos x + a_2\cos 2x+\dots+a_n\cos nx$ is at least 2n.
0
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2answers
97 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos 2nx}{...
2
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1answer
82 views

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$, compute $b^2-c^2$

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$. Then $(b^2-c^2) = $ $\bf{Options::}$ $(a)\;\; 4ac\;\;\;\;\;\;(b)\;\; a^2\;\;\;\;\;\;(c)\;\; 4bc\;\;\;\;\;\;(d)\;\; 4ab$ ...
2
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2answers
119 views

What is $\frac{2x}{1-x^2}$ when $x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$?

If $$x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$$ Find $$\frac{2x}{1-x^2}$$ I got till here by simplification by taking the previous value of x, ie, $$x={\frac{\sqrt{1-\cos\theta}}{1+\cos\theta}}$$...
0
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1answer
55 views

Trigonometric Inequalities

If $$a\sin A+b\sin B+c\sin C=k$$ then the minimum value of $$\sin^2A + \sin^2B + \sin^2C =?$$ ATTEMPT- I tried to use A.M-R.M.S inequality but it didn't help
2
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1answer
87 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
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2answers
41 views

Trigonometric idnetity

IF $\sin \alpha = 3 \sin (\alpha+2\beta)$, then the value of $\tan (\alpha+\beta)+2 \tan \beta=$? ATTEMPT: $\sin \alpha = 3 (\sin (\alpha+\beta) \cos \beta + \cos (\alpha+\beta) \sin \beta)$ ...
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1answer
731 views

how to prove that $\cos n\pi=(-1)^n$?

I'm asked to prove that $$\cos n\pi=(-1)^n\qquad n\in\mathbb {Z}$$ I'm not sure how to approach the problem, I want to know if there is a different way to use induction
0
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1answer
49 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
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1answer
63 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
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3answers
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Ambiguity of notation: $\sin(x)^2$

Several people have told me that $\sin(x)^2 = \sin(x^2)$. However, on several computing platforms, such as the TI-84 and Wolfram|Alpha, $\sin(x)^2 = \sin^2(x)$. Can I safely conclude that the notation ...
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1answer
1k views

Geometry Find the Radius of a circumcircle given the area of the triangle

Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle. I also know that to find the radius I ...
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1answer
39 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi \...
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3answers
407 views

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg.

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg. I have tried adding 2x + x for the two legs, however ...
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1answer
260 views

Angle of Elevation and slope

Standing on top of a gentle 5degree slope I see the top of a tall building at an angle of elevation of 35degree 15’. I am 160 cm tall and it is 12 m from where I am standing to the foot of the ...
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0answers
50 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
2
votes
1answer
52 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
2
votes
1answer
717 views

Real life math to explore/solve [closed]

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
0
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0answers
38 views

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form?

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form such that when you draw 4 equilateral triangles the foot of the perpendicular of the equilateral ...
3
votes
3answers
130 views

Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...
3
votes
1answer
100 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
3
votes
2answers
70 views

$\tan \left(\sec ^{-1}(x)\right)$

$$\tan \left(\sec ^{-1}(x)\right)$$ I know that sec(?)=$\frac{x}{1}$ and that sec=hyp/adj, therefore I conclude that hyp=x and adj=1 and that op=$\sqrt{x^2-1}$ Since Tan = opp/adj I thought the ...
0
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1answer
67 views

Using complex analysis to convert $b\cos \theta +a \sin \theta$ to a single trigonometric function

Using product $(a+bi)(\cos \theta+i \sin \theta) $ show that $$b\cos \theta +a \sin \theta=\sqrt{a^2 + b^2}\sin(\theta+\arctan(b/a))$$ and using this result show by induction that $$ \frac{\...
1
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4answers
46 views

Range of inverse trigonometric function

Find the range of $y$. $$y=\tan^{-1}\left(\frac{2x}{1+x^2}\right)$$ I used the following approach: Let $$x=\tan\theta$$ $$\therefore \theta=\tan^{-1}x$$ Since the principal solution of $\tan^{-1}$ ...