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
votes
1answer
70 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 ...
0
votes
0answers
10 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A*cos(\Omega*t + \phi)$ $A > 0$ $\phi$ range is $(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes ...
-1
votes
2answers
54 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
28 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.
1
vote
3answers
24 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 ...
4
votes
4answers
395 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 ...
0
votes
2answers
23 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
4answers
180 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
2
votes
4answers
51 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 ...
0
votes
1answer
49 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
2
votes
0answers
34 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
2answers
77 views

Is there a sequence of $v_{n}$ such that $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}> 1$?

Let $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}$, where $v_{n}$ is an increasing, monotonic sequence of natural numbers: is it true that there is no sequence of $v_{n}\in\mathbb{N}$ such ...
8
votes
3answers
514 views

Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$

I am trying to prove: If $\cos(\pi\alpha) = \frac{1}{3}$ then $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ So far, I've tried making it into an exponential, since exponentials are easier to ...
0
votes
3answers
40 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 ...
4
votes
2answers
465 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
0
votes
3answers
38 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 .. ...
20
votes
4answers
917 views

Elementary Proof of Euler's Sine Expansion $\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$

I've been looking at proofs of Euler's sine expansion, that is $$\frac{\sin x}{x}=\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$$ All the proofs seem to rely on complex analysis and ...
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
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 ...
1
vote
0answers
40 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
47 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
3
votes
5answers
96 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 ...
7
votes
4answers
697 views

Two different trigonometric identities giving two different solutions

Using two different sum-difference trigonometric identities gives two different results in a task where the choice of identity seemed unimportant. The task goes as following: Given $\cos 2x ...
1
vote
1answer
31 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
30 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
2answers
52 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
2answers
52 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 ...
0
votes
1answer
24 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
21 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 (...
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 ...
0
votes
2answers
36 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? $$\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$$
2
votes
4answers
90 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 & ...
6
votes
6answers
584 views

Proving a trigonometric identity

How can one prove the validity of this trigonometric identity? \begin{equation} 2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1) \end{equation}
0
votes
0answers
32 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 $m\rightarrow2^m$. ...
6
votes
3answers
80 views

What is the integral value of $\frac{\tan 20^\circ+\tan40^\circ+\tan80^\circ-\tan60^\circ}{\sin40^\circ}$?

I have tried possibly all approaches. I first expressed $80$ as $60+20$ and $40$ as $60-20$ and then used trig identities.I later used conditional identities expressing $\tan ...
6
votes
1answer
156 views
+50

Formula for cos(k*x)

I need to prove that: \begin{align} c_k =&\; \cos(k\!\cdot\!x)\\ c_k :=&\; c_{k-1} +d_{k-1}\\ d_k :=&\; 2d_0\!\cdot\!c_k +d_{k−1}\\ d_0 :=&\; −2\!\cdot\!\sin^2{(x/2)}\\ \end{align} ...
2
votes
3answers
29 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 ...
12
votes
1answer
207 views

Inequality with summation of cosine terms $\left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| \leq 1 + 2\sum_{j=1}^k \cos (\frac{2\pi }{q}j)$

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
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
41 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
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
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 $\sin\theta + \sin\alpha=x$ ...
7
votes
5answers
447 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
4
votes
2answers
48 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 ...
0
votes
1answer
41 views

Plane rotation: range of angles to produce all posible x'y' planes

Given an $(x, y, z)$ system I create a new system $(x', y', z')$ by applying two rotations $\theta$ and $\phi$. In the new system the $(x',y')$ plane, i.e.: the $z'=0$ plane, can be written as: $$ ...
2
votes
1answer
64 views

Proof that there are no solutions this equation. (3 variables, Square root and Sine) [on hold]

Hypothesis: There do not exist three different positive integers $a,b,c$ such that $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
0
votes
1answer
24 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
5
votes
1answer
123 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!