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
58 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
votes
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
114 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
338 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
95 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
votes
1answer
131 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 ...
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1answer
405 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 ...
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2answers
106 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
154 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 $\sin$ $$ ...
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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 ...
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1answer
59 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
84 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
74 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
90 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
94 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 ...
2
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1answer
80 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)\;\; ...
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, ...
0
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1answer
54 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
85 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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votes
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
712 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 ...
0
votes
1answer
61 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
1k views

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 ...
0
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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 ...
1
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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
395 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 ...
0
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1answer
255 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
49 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
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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 ...
3
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1answer
648 views

Real life math to explore/solve

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 ...
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0answers
37 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
128 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
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2answers
67 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 ...
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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 $$ ...
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4answers
45 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 $\tan^{-1}$ ...
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2answers
164 views

Solve the equation $a+b+c=abc$ for $a,b,c\in\mathbb{Z}$

Solve for $a,b,c$ (where $a$, $b$, and $c$ are integers) the equation $$a+b+c=abc.$$ I would prefer a solution using trigonometry and I think that it might use the formula $\tan A + \tan B + \tan ...
0
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2answers
110 views

Alternative of finding theta when sin $\theta$ and cos $\theta$ are given

For example, we're given a problem in which sin $\theta = \sqrt3/2$ and cos $\theta = -1/2$. To find out the angle $\theta$, I look at the unit circle and I get the answer. However, I was just curious ...
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2answers
275 views

Determinant of a matrix with trigonometry functions.

Prove that the matrix is invertible for any value of $\beta$. I've done several exercises of this type. But I'm not sure with this one: $$\begin{bmatrix}\cos \beta & \sin \beta & 0\\ ...
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3answers
1k views

Find out $\theta$ when sin $\theta$'s and cos $\theta$'s value are given

Given: $\sin \theta = \frac12$, $\cos \theta = \frac{\sqrt{3}}{2}$. What I have tried: It is very easy looking at the angles' table and figuring out the value when the values of cos $\theta$ and sin ...
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4answers
116 views

Minimizing $\tan^2 x+\frac{\tan^2 y}{4}+\frac{\tan^2 z}{9}$

Given that $\tan x+2\tan y+3\tan z=40 , \ \ \ x,y,z \in \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right),$ We need to find the minimum value of $ \tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9}$ ...
0
votes
1answer
119 views

determine shortest distance between circle intersections

I have three circles positioned shown in the fig. Each of them has the same radius. I know the distance between each of them (A-B, B-C, A-C). My goal is to find the shortest path between B and C. The ...
3
votes
1answer
101 views

Show that $\max(\mathrm{Re} (\exp(it)\cdot z) = |z| $

I need to show that $\max(\mathrm{Re} (\exp(it)z) = |z| $, with $t\in \mathbb{R}$ and $z\in \mathbb{C}$. Therefore I have calculated $\exp(it) = \cos(t) + i \sin(t)$. If we write $z= a+bi$, then $$ ...
1
vote
2answers
60 views

Trignometric problem (using De Movier's Theorem)

Ok so this question, I started out writing tan as sin and cos in the right side of the equation, simplified as much as possible and ended up with a very (sort of) fascinating equation which is ...
0
votes
0answers
66 views

period of cubic trigonometric functions

Can anybody explain how you would find the period of cubic trigonometric function. so I need to find the period of $f(x)=\sin^2\left(\frac{x}{3}\right)$. So I have began the question by finding the ...
1
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1answer
6k views

How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
2
votes
5answers
301 views

Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$

Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$ No idea how to even approach the ...
0
votes
5answers
75 views

Find the value of $\sec x$ using knowing that $9\sin x + 40\cos x = 41$.

I am trying to find the value of $\sec x$ using equation $9\sin x + 40\cos x = 41$. I have tried to solve but I failed.