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

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3
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1answer
55 views

Is $\arctan2$ irrational?

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
1
vote
1answer
34 views

the maxima of given function

What's the maxima of $$2^{\sin(x)}+2^{\cos(x)}$$ I found max by taking logs and then differentiating and equating to $0$ at $x=45°$ so the answer is $2^{\frac{\sqrt{2}+1}{\sqrt{2}}}$ am I right or I ...
2
votes
2answers
34 views

Find the value of $ sin(2\theta)$ when $cot(\theta) + tan(\theta) = 2.5 $

I have an homework question that goes like: $cot(\theta) + tan(\theta) = 2.5 $ is valid on some angles $\theta$ at section $0 < \theta < \pi/2$. Find the value of $sin(2\theta)$. (There is no ...
0
votes
1answer
10 views

Having trouble drawing an arc using atan2 and specified range of arc.

I'm writing code. My arc invariant is: $$ \theta_0, \theta_1 \in [-2\pi, 2\pi], \\ \theta_1 - \theta_0 \leq 2\pi $$ where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle. ...
2
votes
1answer
25 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
0
votes
0answers
6 views

Find largest value of the constants that makes the equality true

$$sin(h^2) = a + O(h^b)$$ Find the largest value of a and b such that the equality holds. I tried to use the truncated Taylor series expansion of $sin(h^2)$, but the derivatives of the function are to ...
1
vote
0answers
27 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
0
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0answers
25 views

plotting circles of variable size on circumference of a circle

I am writing a layout algorithm for graph (i.e. vertices and edges) data and want to implement a slight twist on a hive plot. I would like to be able to draw each node as a circle (though different ...
0
votes
2answers
29 views

trigo substitution and identites?

When I use trigo substitution to solve an integral I get an expression like that: $$\frac{1}{4}\tan\left(\arcsin\left(\frac{x-2}{2}\right)\right)+C$$ How can I simplify it?
0
votes
1answer
52 views

Challenge in trignometry and integration [on hold]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
0
votes
2answers
35 views

The obtuse angle $B$ is such that $\tan B = -\frac{5}{12}$. Find the exact value of $\sin B$.

The obtuse angle $B$ is such that $\tan B = -\frac{5}{12}$. Find the exact value of $\sin B$. Is this possible to do without a calculator? If so, how?
0
votes
5answers
72 views

Solving Trigonometric Equation.

Solve for $\theta$ $[0°<\theta<180°]$ $$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$ My solution is here: $$\sin2\theta + \sin4\theta=\cos\theta + \cos3\theta.$$ After using the ...
0
votes
3answers
39 views

Trigonometric Idntities under the condition $A+B+C=π$

If $$A+B+C=\pi$$, prove that $$\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}=4\cos\frac{\pi-A}{4}.\cos\frac{\pi-B}{4}.\cos\frac{\pi-C}{4}$$. My solution: Here $$A+B+C=\pi$$ $$A+B=\pi-C$$ Taking ...
1
vote
1answer
16 views

Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer. Show that $$T_2 (x) = 2x^2 – 1.$$ My attempt: $T_2 (x) = \cos (2 \arccos x)$ ...
3
votes
1answer
97 views

Do any mathematican still reserach about trigonometry?

Do any mathematican still reserach about applied trigonometry? If so, what are the subject area called in the PhD level except fourier analysis? In many area, you could see a lot of trig and ...
1
vote
1answer
33 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A\cos(\omega t + \phi)$ $A > 0$ $\phi\in(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes $0.123$ ...
-1
votes
4answers
80 views

If $A+B+C=π$, prove that

If $A+B+C=π$, prove that $$\cos^2A+\cos^2B-\cos^2C=-2\cos A\cdot\cos B\cdot\cos C.$$ ATTEMPT: Given $$A+B+C=π,$$ $$A+B=π-C$$ Taking "cos" on both sides $$\cos(A+B)=-\cos C.$$ Now, ...
2
votes
1answer
39 views

How would one evaluate $\sin(72\pi/11)$?

How would one evaluate $\sin(\frac {72\pi} {11})$?. The prime number in the bottom is getting me stuck. I couldn't see how to use it using the sum of two angles trig identity.
0
votes
2answers
24 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
2
votes
0answers
39 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
0
votes
3answers
45 views

If $\frac{m+1}{m-1}=\frac{cos(\alpha-\beta)}{sin(\alpha+\beta)}$„ then

If $$\frac{m+1}{m-1}=\frac{\cos(\alpha-\beta)}{\sin(\alpha+\beta)}$$, prove that : $$m=\tan(π/4 +\alpha).\tan(π/4 +\beta)$$. My attempts/ Here .. ...
1
vote
3answers
25 views

Prove that $m\tan (\theta-30°)=n\tan (\theta+120°)$

If $m\tan (\theta-30°)=n\tan (\theta+120°)$ then prove that : $$\cos 2\theta=\frac{m+n}{2(m-n)}$$ My attempt\ Here, $$m\tan (\theta-30°)=n\tan (\theta+120)$$ $$\frac{\tan (\theta-30°)}{\tan ...
2
votes
4answers
58 views

Proving $\tan A=\frac{1-\cos B}{\sin B} \;\implies\; \tan 2A=\tan B$

If $\tan A=\dfrac{1-\cos B}{\sin B}$, prove that $\tan 2A=\tan B$. My effort: Here $$\tan A=\frac{1-\cos B}{\sin B}$$ Now $$\begin{align}\text{L.H.S.} &=\tan 2A \\[4pt] &=\frac{2\tan ...
2
votes
0answers
52 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
0
votes
0answers
18 views

Proving that If $A+B+C=π$ then, [duplicate]

If $A+B+C=π$ then prove that : $\sin(B+2C)+\sin(C+2A)+\sin(A+2B)=4\sin\frac{B-C}{2}.\sin\frac{C-A}{2}.\sin\frac{A-B}{2}$. My attempts: Here $A+B+C=π$ Now, $$\begin{align} ...
1
vote
1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
1
vote
1answer
33 views

Showing that $\alpha$ satisfies the equation $\sin 2x=x$

This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme. Question: Here's what I have done: $$A(OBA) = \frac 12r^2α$$ [basic ...
0
votes
2answers
33 views

What is this procedure called for angle radians?

So, my lecturer says that $-\cos(\frac{\pi}{8}) = \cos(\frac{9\pi}{8})$. What did he do to get that? Please recommend a source where I can brush up on my knowledge of angles.
0
votes
3answers
42 views

How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
0
votes
2answers
54 views

How does $\frac{\sin\theta}{\cos\theta}$ become $\frac{y}{x}$

I ended up in the wrong math class (trigonometry) for my level but am trying to survive by catching up on some more basic principles. I'm wondering if the same principle (and if so, what is it) is ...
0
votes
0answers
32 views

let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \frac{1}{3}$, prove $\alpha $ is irrational [duplicate]

Let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \dfrac{1}{3}$, prove $\alpha$ is irrational. (Proof by contradiction) If we consider $\cos \left(\dfrac{m\pi}{n} \right)=\cos \left(\dfrac{ m\pi ...
0
votes
2answers
54 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
3
votes
5answers
140 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
0
votes
1answer
26 views

Trigonometry markup

Imagine we have the following problem; $$\cos(x) = \cos(a) \Rightarrow x=a+k\times 2\pi\\ or \\x=-a+k\times 2\pi$$ And we have the following answers.. : $$a=\frac{\pi}{3} \\or \\a=-\frac{\pi}{3}$$ ...
0
votes
1answer
25 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
0
votes
2answers
37 views

Solving $\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$ [on hold]

Can you solve for $α$ in degrees/radians and tell me exactly how to do so? $$\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$$
0
votes
0answers
37 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
2
votes
3answers
30 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
1
vote
1answer
39 views

Trigonometric solution for $\int_{0}^{2 \pi} \sin^n (x) \cos^m (x) dx $?

At home I came across the exercise and had to compute: $\int_{0}^{2\pi} \sin^n (x) \cos^m (x) dx $ with $m$, $n \in \mathbb{N} $ My current set of tools for solving problems of that kind is rather ...
0
votes
1answer
30 views

Conditional Proof in Trigonometry

If $\sin\theta + \sin\alpha=m$ and $\cos\theta + \cos\alpha=n$, prove that: $$\frac{\sec(\theta+\alpha)}{2}=\frac{\sqrt{m^2+n^2}}{2}$$ My attempt\ given: $$\sin\theta+\sin\alpha=m$$ $$2 ...
0
votes
1answer
20 views

Trigonometry Proving

If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$ Attempts: Here $\sin\theta + \sin\alpha=x$ ...
4
votes
2answers
51 views

How do I show that $n=2$ is the only integer satisfy :$\cos^n\theta+ \sin^n\theta=1$ for all $\theta$ real or complex?

It is well known that :$\cos²\theta+ \sin²\theta=1$ for all $\theta$ real or complex ,I would like to ask about the general equality :$\cos^n\theta+ \sin^n\theta=1$ if there is others values of the ...
2
votes
4answers
91 views

Show if $A^TA = I$ and $\det A = 1$ , then $A$ is a rotational matrix

Show if $A^TA = I$ and $\det A = 1$ where $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then $A =\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & ...
3
votes
0answers
70 views
+100

Find elevator height given rope length?

This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online. ...
2
votes
4answers
44 views

Prove that $16\cos^5A-20\cos^3A+5\cos A=\cos5A$

Prove that $$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$ My solution begins here; $$ \begin{align} \text{RHS} & =\cos5A \\ & =\cos(A+4A) \\ & =\cos A\cos4A-\sin A\sin4A \\ & =\cos ...
7
votes
5answers
454 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
0
votes
0answers
17 views

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
0
votes
3answers
38 views

Why dividing by trigonometric functions gives wrong answer when solving trigonometric equations?

Hello I have a problem with solving Trigonometric equations. Why this is not true for $0\le\theta\le360$ $$2\sin\theta\cos\theta=\sin\theta$$ $$2\cos\theta=1$$ Set of solutions $\theta=60,360$ and ...
2
votes
2answers
28 views

Find Length of line which has rotating object.

I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point. For easy understanding, let's consider ...
2
votes
1answer
38 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...