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

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5
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3answers
70 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
0
votes
4answers
34 views

Equivalence of equations

$ \sin ^2 \alpha = \frac{\tan ^2 \alpha}{1+\tan^2 \alpha} $ $ 1+\tan^2 \alpha = \frac{\tan ^2 \alpha}{\sin ^2 \alpha} $ It is said that these two equations are equivalent. How can that be? I know ...
3
votes
6answers
90 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
0
votes
4answers
46 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
1
vote
0answers
25 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
0
votes
1answer
19 views

The vertical projection of a chord of a circle?

I was wondering if anyone could help me with the problem below (finding x): So we are given t_i (the initial tangent angle to the circle), t_o (the exiting angle of the tangent of the circle), the ...
3
votes
1answer
16 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
1
vote
3answers
42 views

What is the required radius of the smaller circles around a larger circle so they touch?

I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box) I would like to be able to adjust the number of ...
3
votes
3answers
48 views

Using trig substitution to solve for integration?

So I used a trig sub for this problem: $$\int \frac{1}{x^2\sqrt{9-x^2}}dx.$$ ${x=3\sin\theta}$ ${dx=3\cos\theta\ d\theta}$ ${\sqrt{9-x^2}= 3\cos\theta}$ I ended up with $$\frac19 \int \frac{ ...
1
vote
2answers
64 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
3
votes
1answer
40 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
0
votes
1answer
30 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
2
votes
0answers
23 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
1
vote
2answers
49 views

How to solve $\sin(\arctan((\frac{1}{2}))$ [on hold]

Can you solve $\sin(\arctan((\frac{1}{2}))$? It says I have to use a right triangle
4
votes
4answers
55 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
3
votes
4answers
103 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
3
votes
2answers
97 views

Simple Equation Does my proof work?

Its the inequality equation $|a+b| \leq |a|+|b | $ I managed this by cases. Let $c = a$ and $d=b$ if $a>b $ let $c = b$ and $d = a$ if $b>a $ if $a=b$ let $a=c$ Hence we have $|c+d| \leq ...
7
votes
0answers
85 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-2mx+N = ...
-2
votes
2answers
92 views

Prove that $\lim_{x\to\frac{2}{\pi}}\big\lfloor\sin\frac{1}{x}\big\rfloor=0$ [on hold]

Prove that $$\lim_{\large x\to \frac{2}{\pi}} \left\lfloor\sin\left(\frac{1}{x}\right)\right\rfloor=0$$ using the $\varepsilon$-$\delta$ definition of limits. Note that $\lfloor 0.1\rfloor = 0,\; ...
1
vote
2answers
50 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...
0
votes
1answer
24 views

Isosceles has maximum vertex angle between triangles of equal area

I'm trying to prove the following that in the image below (E1 & E2 are parallel, AB=AC) no matter where I move the vertex point A on line E1 (keeping BC as is), the vertex angle A is going to ...
2
votes
2answers
54 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
1
vote
0answers
26 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
0
votes
2answers
38 views

Sine on a Circle

I'm walking a quarter mile circular walking track. The width of the track is 8 feet across. If I walk from one side of the track to the other, walking a sine wave that has a 20 foot period, how much ...
0
votes
0answers
31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
6
votes
0answers
85 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
2
votes
2answers
52 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
0
votes
0answers
19 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
1
vote
3answers
180 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
0
votes
1answer
42 views

How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?

Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$ Now, ...
2
votes
4answers
77 views

prove that $\sqrt{2} \sin10^\circ+ \sqrt{3} \cos35^\circ= \sin55^\circ+ 2\cos65^\circ$

Question: Prove that: $\sqrt{2} \sin10^\circ + \sqrt{3} \cos35^\circ = \sin55^\circ + 2\cos65^\circ$ My Efforts: $$2[\frac{1}{\sqrt{2}}\sin10] + 2[\frac{\sqrt{3}}{2}\cos35]$$ $$= 2[\cos45 \sin10] ...
0
votes
3answers
35 views

trigonometric identity of sin squared in terms of tan squared.

Why is $\sin^2(x)=\frac{\tan^2(x)}{1+\tan^2(x)}$? And why is $\sin^2(x)=\frac{1}{\cot^2(x)}$? I've tried starting from $\tan^2(x)=\frac{\sin^2(x)}{1-\sin^2(x)}$ but that wasn't really working out ...
0
votes
1answer
25 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
0
votes
0answers
39 views

can somebody help me with this? [on hold]

Just gonna ask you guys if it's possible to prove that $\sec A\tan A - \sin A\sec A = 1 - \tan A$
2
votes
1answer
36 views

Finding the third side of a triangle, given ratio of two sides and difference of two angles [on hold]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
-1
votes
1answer
34 views
2
votes
2answers
59 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
3
votes
3answers
86 views

Simplify a quick sum of sines [duplicate]

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$ I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.
1
vote
4answers
77 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
2
votes
1answer
22 views

sine and cosine and difference of angles

I'm doing a math review, and I am getting a different answer than the guide, and I need some guidance. Here is the problem: Suppose $\cos(x)=\frac{1}{2}$ and $\sin(y)=\frac{1}{2}$, where $x$ ...
0
votes
3answers
66 views

Find $\int\sin^4(x)\cos^2(x)\,dx$

Find $$\int\sin^4(x)\cos^2(x)\,dx$$ My Attempt: $$\int\sin^4(x)\cos^2(x)\,dx = \frac18 \int ((1-\cos(2x)-\cos^2(2x)+\cos^3(2x))\, dx$$ How to proceed from here?
1
vote
2answers
29 views

Finding the minimum point looks easy with a graph but hard with a formula

My research has lead me to the following function: $$ \frac{\sin(x) [\sin^2(x)\cdot F+ \cos^2(x)/F ]} { 1 - \cos(x) } $$ $F$ is a parameter, and I would like to find the minimum value of this ...
0
votes
2answers
38 views

Question about sines of angles in an acute triangle

Let ABC be a triangle such that each angle is less than 90 degrees. I want to prove that sinA + sinB + sinC > 2. Here is what I have done: Since A+B+C=180 and 0 < A,B,C < 90, at least two of ...
5
votes
3answers
61 views

Proof that $\cos(\pi/4)=\frac{\sqrt2}{2}$

Normally I just look up or remember that $\cos(\pi/4)=\frac{\sqrt2}{2}$, or type "$\cos(\pi/4)$" into WolframAlpha to check the answer. But what about the first time someone wanted to know what ...
2
votes
3answers
41 views

Triangle of maximum perimeter for a given area

What type of triangle has the maximum perimeter for a certain area? Suppose I start with a rectangle of that area (axb=Z). I can stretch one dimension of the rectangle until infinity, reducing the ...
3
votes
4answers
52 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
0
votes
2answers
29 views

In a triangle ABC , $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to…

In a triangle ABC, prove that $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to $\frac{abc}{R^2}$, where $a$, $b$, and $c$ are sides of the triangle and $R$ is the circumradius. My work:- By ...
12
votes
1answer
203 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
4
votes
3answers
43 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
2
votes
2answers
45 views

Equivalence of trigonometric identity

Is writing $$ \cot{2\theta}=\frac{a-c}{2b} $$ equivalent to $$ \cot{\theta}=\frac{a}{b},\tan{\theta}=\frac{c}{b} $$ becuase of the trigonometric identity $$ ...