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
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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 : $$m=\tan(\frac\pi4 +\alpha)\cdot \tan(\frac\pi4 +\beta)$$ Attmept As given, ...
1
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3answers
15 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
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4answers
40 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 ...
1
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0answers
27 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.
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0answers
17 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
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1answer
23 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
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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
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2answers
29 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.
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2answers
32 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
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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
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0answers
28 views

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

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
49 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 ...
2
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5answers
83 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
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1answer
23 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
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1answer
20 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
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2answers
33 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
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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 $m\rightarrow2^m$. ...
2
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3answers
27 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
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1answer
37 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
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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
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1answer
19 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 $\sin\theta + \sin\alpha=x$ ...
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2answers
44 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
87 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 & ...
2
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0answers
34 views

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
42 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 ...
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5answers
445 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
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0answers
13 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
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3answers
37 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
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2answers
27 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 ...
0
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1answer
40 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 $\lambda \cosh(x) + \sinh(x)$, ...
2
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1answer
63 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 $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
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0answers
28 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 ...
6
votes
3answers
78 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 ...
1
vote
1answer
45 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 ...
0
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0answers
23 views

Find the third angle

Three planes are orthogonal to each other. I have found the rotation about the 2 axis (x and y). Is there a way to find the third angle around z provided the angles around x(70 degrees) and y(-1 ...
2
votes
2answers
90 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
0
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5answers
63 views

If $\tan^2(\theta)+2\sec^2(\theta)=5$. Find the value of $\sin^2(\theta)$

I have a trig problem which i can't really understand where to start. It says If $$\tan^2(θ)+2\sec^2(θ)=5.$$ Find the value of $$\sin^2(θ).$$ I think it has something with to do with Pythagorean ...
3
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4answers
42 views

Identity of $8\sin^2(t)\cos^2(t)$

I know this probably has a simple answer, but I am having trouble understanding the steps to find the identity for this problem. This is the answer I was provided: $$8\sin^2(x)\cos^2(x) = ...
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0answers
25 views

parametr with trigonometric equations

can you help me with this equation: $$ \frac{4p+3}{6} - \frac{\sin{8x}}{2} - (p+\frac{2}{3})\sin(4x-\frac{\pi}{4})=0 $$ for which values of $p$ equation has has 3 distinct solutions in range ...
6
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1answer
152 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} ...
7
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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 ...
24
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1answer
724 views

Moriarty's calculator: some bizarre and deceptive graphical anomalies

Background: This is a problem I first came across a few years ago in a calculus textbook (a James Stewart one), where it addressed some of the pitfalls of using graphing calculators. The original ...
8
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3answers
147 views

Differential Equation of the Form $\frac{dy}{dx}=\sin(x+y)$ [duplicate]

I have been attempting to solve the above differential equation for some time now, and I remain stuck on one step. After substituting $u=x+y$, separating the variables, and integrating both sides, I ...
3
votes
0answers
58 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
12
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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| ...
122
votes
3answers
7k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
31
votes
2answers
650 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 ...
7
votes
3answers
498 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 ...
25
votes
2answers
396 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
8
votes
4answers
131 views

Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$

If $0\le\alpha\le\frac{\pi}{2}$, then which of the following is true? A) $\sin(\cos(\alpha))<\cos(\sin(\alpha))$ B) $\sin(\cos(\alpha))\le \cos(\sin(\alpha))$ and equality holds for some ...