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

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21 views

Prove that the sum of the cosines of a convex quadrilateral is zero only when it is cyclic or a trapezium.

This is question 3 from the BMO paper in 1990. ABCD is a quadrilateral and cos(A)+cos(B)+cos(C)+cos(D)=0, I have been unable to proof that ABCD must be cyclic or a trapezium.
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2answers
96 views

Why $\sin(x)+\sin(\pi x)$ is not periodic?

Why $\sin(x)+\sin(\pi x)$ is not periodic? There is an answer here which tries to explain it, but I somehow do not get it. If we assume that $T>0$ is a period of $\sin(x)+\sin(\pi x)$, then ...
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1answer
58 views

$\cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$?

I need to show that $$ \cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$$. For $x \notin \mathbb Z$ one can use $\cos(\pi x ) = \frac{\sin(2 \pi x )}{2 \sin(\pi x )}$ ...
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3answers
19 views

Sketching the graph of trigonometric functions involving absolute value function

How do I sketch the graph of $\tan |x|?$ I know that the modulus over $|x|$ can be thought of as the angle (in radians) will always be positive which implies that the angle is always measured ...
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3answers
60 views

limit of $ x/\sin(\pi x)$ as $x$ approaches zero?

I reorganized as $x\csc(\pi x)$ and input $0$, which would be $0$, but the answer is apparently wrong? It says the answer is $1/\pi$. Any help?
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0answers
23 views

Vector Bearing Force Application.

We have this question, for a simple exercice. A Force of $125N$ acts on an object. Another force of 85N acts on the same object an an angle of $72 ^\circ $from the first force find the magnitude of ...
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2answers
42 views

Does $f(10^n)$ begin every time by $n$ $9$ after the comma?

Trying to find a formula to give me the rank of people in a game, I found for small values that if $$f=\frac{2}{\pi} \arctan$$ then $f(10^n)$ always become by $n$ $9$ after the comma if $n \ge 1$. It ...
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2answers
26 views

Grouping Trig Terms

I'm confused as to how the $\cos^2\theta$$\cos^2\phi$ + $\cos^2\theta$$\sin^2\phi$ simplifies to $\cos^2\theta$ and likewise for the $phi^2 d/dx$ part also. Did I miss something in double angle calc? ...
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0answers
27 views

Area of a triangle on an Argand diagram

I am working on two problems: 1) Find three distinct roots of the equation $8z^3 + 27 = 0$ I solved this and ended up with \begin{align*}z_1 &= \frac32 \left( \cos (\pi/3) + ...
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0answers
26 views

Intuition behind negative radians in an interval [duplicate]

Say a function's domain is $[−\pi,\pi]$. How should I interpret this interval? It starts from where? To where? In what direction?
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1answer
184 views

How to prove this trigonometric integral?

$$ \displaystyle \int_{-\pi/4}^{\pi/4} {{\left(\dfrac{\cos x - \sin x}{\cos x + \sin x}\right)}^{\cos(2t)} \ dx} = \frac{\pi}{2 \sin(\pi \cos^2 t)}$$ I could simplify it to $\displaystyle \int_0^1 ...
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1answer
28 views

Value of a series in $\tan$ [duplicate]

Question: The value of: $$(1+\tan 1^o)(1+\tan 2^o)(1+\tan 3^o)...(1+\tan 44^o)$$ is: A) 2 B) A multiple of 22 C) A multiple of 4 D) Not an integer I'm not sure how I would ...
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3answers
107 views

Find all real numbers $a,b$ such that $|a|+|b|\geq\frac{2}{\sqrt{3}}$ and $|a\sin x+b\sin{2x}|\leq 1$ for all real $x$.

Find all real numbers $a,b$ such that $|a|+|b|\geqslant\frac{2}{\sqrt{3}}$ and $|a\sin(x)+b\sin(2x)|\leqslant 1$ for all real $x$. We could write the inequality as $$ ...
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2answers
41 views

How fast was the car driving?

According to this article, http://electrek.co/2016/05/06/tesla-model-s-crash-large-crumple-zone-gallery/ a car flew through the air and landed 25m away. Assuming an ideal ramp angle of 45 degrees, is ...
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1answer
24 views

Inequalities involving the Roots of unity

Let $\epsilon \ne 1$ be a nth root of unity. Then prove that, $\bullet \ |1-\epsilon|\ge\frac{2}{n-1}$ $\bullet |\sin\frac{k}{n}| \ge \frac{1}{n-1}$ Here are my solutions: $\bullet$ To prove ...
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1answer
104 views

Show that $\lim_{n\to \infty}\prod_{i=n}^{Bn}\frac{\arctan(i\phi)}{\arccos\left(\frac{\phi}{i}\right)}=B^{\frac{2}{\pi}}$

Inspired from Gosper's formula $$\lim_{n\to \infty}\prod_{i=n}^{2n}\frac{\pi}{2\arctan(i)}=4^{\frac{1}{\pi}}$$ (See Pi Formulas on MathWorld) Through mathematical experimental we found another ...
2
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1answer
26 views

Trigonometry, obtuse angles and a negative length?

I've been looking for an answer as to why when $\cos x<0$ and $\tan x<0$ the angle is obtuse. I found a few identical explanations online where, a right-angle triangle is formed in the second ...
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2answers
35 views

What is the relationship between the width, height, and radius of an arc?

What is the relationship between the width ($w$), height ($h$), and radius ($r$) of an arc? Specifically, the relationship in terms of $h$. I know this is a simple question - I'm a hobbyist ...
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1answer
73 views

Why this is not true : ${i}^{n}=0$ for every even positive integer $n$?

Let $i$ be a unit imaginary part , we have for $\theta=\frac{\pi}{2}$: $\left(\cos \theta + i \sin \theta \right)^n = (0+i\sin(\frac{\pi}{2} ))^n=i\sin(n\frac{\pi}{2})=0$ (Using Moiver formual) , ...
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0answers
44 views

Solving $\sin(\theta) + \cos(\theta) = -1$ using the T-Formula [duplicate]

In class we've learned 2 methods for solving trigonometric equations. T-Formula: If you're not familiar with the T-Formula look here; T-Formula Auxiliary Method: If you're not familiar look here ...
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2answers
58 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
0
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1answer
47 views

Can't find solution to trigonometric equation, need help to understand why!

I am struggling with the solution of an equation and I think as well with a lack of understanding when there is a solution for such a trigonometric problem and when there will be infinitely many ...
3
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2answers
38 views

Find range of $\alpha$ of $ \frac{4x^2+1}{64x^2 - 96x \sin \alpha +5} \leq \frac{1}{32}$ for all real x.

I simplified it to get $ \frac{64x^2 + 96 x \sin \alpha +27}{64x^2 - 96 x \sin \alpha +5} \leq 0$. I dont have any idea how to proceed further.
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0answers
57 views

Max value of $9\lambda^2 -2 \mu^2$

Suppose vector $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ such that: \begin{align} \lvert \mathbf{a} \rvert &= \mu \lvert \mathbf{b} \rvert \\ \lvert \mathbf{c} \rvert &= \lambda \lvert ...
0
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1answer
41 views

Given the sides of a triangle, find $\tan\left(\frac{A}{2}\right)$ and $\tan\left(\frac{B}{2}\right)$

In a triangle the sides are given as $a=25$ units, $b=52$ units and $c=63$ units. The problem is to find $\tan\left(\frac{A}{2}\right)$ and $\tan\left(\frac{B}{2}\right)$ where $A$ is the angle ...
0
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1answer
18 views

Given sides of a right triangle, calculate one of its angles

I need help with this problem for trig: Given a right triangle with $c = 17.9$ and $a = 6.4$, find the measure of angle $\alpha$ in degrees, rounded to one decimal place. Draw a picture and show ...
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20 views

Elements of The Excentral triangle

I am learning geometry and found the following question which I partially solved. I want to see the method of approach taken out here in StackExchange. The problem goes as follows: The triangle ...
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4answers
94 views

Is this procedure valid for finding the limit?

$$\begin{align*} \lim_{x\to \pi/4} (1-\tan x )^{\cos x -\sin x } &= \lim_{x\to \pi/4} \frac{(1-\tan x )^{\cos x}}{(1-\tan x )^{\sin x }} \\ &= \lim_{x \to \pi/4} \frac{\ln (1-\tan x )^{\cos x ...
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2answers
32 views

How do I calculate the height of an arc?

I'm a hobbyist engineer, having one of those moments where my mind goes blank. I know this is a simple problem, but I can't remember how to approach it. I have an arc defined by width and angle. ($w$ ...
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1answer
34 views

Completing the Square (Trigonometric Substitution)

I need help completing the square. The equation is $\sqrt{1-49x^2} \, dx$ I have done $$-x^2+12x+13+k-k$$ $$k-(x^2-12x-13+49)$$ $$49-(x-6)^2$$ $$u=x-6$$ $$du=dx$$ We then have $$\sqrt{49-u^2}du$$ ...
0
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1answer
26 views

Finding all possible values of $θ$ in radians

The question I'm given is: "if $\csc\theta = -17/15$ and $\theta$ is in the third quadrant, determine all possible values of the expression $\tan\theta + 3\sec \theta$. State all possible values of ...
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3answers
77 views

If $\alpha=\frac{2\pi}{7}$,prove that $\sin\alpha+\sin2\alpha+\sin4\alpha=\frac{\sqrt7}{2}$ [duplicate]

If $\alpha=\frac{2\pi}{7}$,prove that $\sin\alpha+\sin2\alpha+\sin4\alpha=\frac{\sqrt7}{2}$ We need to find $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ ...
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2answers
47 views

For $0<x<\frac{\pi}{4}$,prove that $\frac{\cos x}{\sin^2x(\cos x-\sin x)}>8$

For $0<x<\frac{\pi}{4}$,prove that $\frac{\cos x}{\sin^2x(\cos x-\sin x)}>8$. I have no idea how to solve this problem.Somewhat i tried. We need to prove that $\cos x>8\sin^2x(\cos ...
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0answers
36 views

Finding the area of the different portions of a rectangle that lie in a grid.

I am an undergraduate student working on a large research project and one part involves calculating the portions of a rectangle that lie in different parts of a cartesian grid. In the figure below, I ...
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0answers
14 views

“Opposite” point on ellipsis by axis (or vector)

I'm currently working on a little game and am stumped as to how I'd solve this math problem. What I'm trying to do is get the "rotation" needed for B, where B is always opposing A on the Y axis no ...
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0answers
87 views

Prove that $\frac{\cos\theta\cos\delta}{\cos^2\alpha}+\frac{\sin\theta\sin\delta}{\sin^2\alpha}+1=0$

If $$\frac{\cos\theta}{\cos\alpha}+\frac{\sin\theta}{\sin\alpha}=\frac{\cos\delta}{\cos\alpha}+\frac{\sin\delta}{\sin\alpha}=1,$$ where $\theta$ and $\delta$ do not differ by an even multiple of ...
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1answer
45 views

If for all $\displaystyle \theta \in [ 0,\frac{\pi}{2} ]$, we have $ | \sin \theta - p \cos \theta - q|\leq \frac{\sqrt{2}-1}{2}$. Then find $p+q$.

If for all $\displaystyle \theta \in [ 0,\frac{\pi}{2} ]$, we have $ | \sin \theta - p \cos \theta - q|\leq \frac{\sqrt{2}-1}{2}$. Find $p+q$. My Work: When $p=-1,q=\frac{\sqrt{2}+1}{2}$, we ...
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1answer
42 views
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13 views

How Cosh was used in this derivation?

Please see attached image. I'm really stuck on how to follow this proof for cut-off frequency. I'm not too sure how they've got to the third bulletpoint (9th line). Thanks
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3answers
23 views

Operation on inverse trigonometric functions

Prove that: $\tan^{-1}{6x-8x^3\over 1-12x^2} + \tan^{-1}{4x\over 1-4x^2} = \tan^{-1}2x$, $|2x|<{1\over \sqrt3}$
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1answer
29 views

Quaternion to Euler angles conversion

I have written the following MATLAB code for transforming Quaternion to Euler angles based on the mathematical formula from wikipedia: ...
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1answer
31 views

Express $\sec(2x)$ and $\tan(2x)$ in terms of $\tan x$

$\tan(2x)$ was easy, as I just had to use the identity: $$\tan(2x) = \frac{2\tan x}{1-\tan^2x}$$ I'm having a bit of trouble with $\sec(2x)$, however! I tried squaring it and substituting a few ...
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1answer
41 views

what trigonometrically satisfies $x^2+y^4=1$ belong too?

Plotting $x^2+y^4=1$ gives some 'squat cuircular type shape but spesically what satisfies these conditions in terms of sines and cosines ect? (An analogy would be circles $x^2+y^2=1$ that are ...
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3answers
160 views

solve $h = 36\sin(1.5t) - 15\cos(1.5t) + 65$ using $k\sin(1.5t - \alpha)$

The blades of a turbine are turning at a steady rate. The height $h$ metres, of the tip of one of the blades above the ground at time, $t$ seconds is given by the formula: $h = ...
2
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2answers
34 views

Show $3\cos 2x + 1 = 4\cos^2 x - 2\sin^2 x$

Show $3\cos 2x + 1 = 4\cos^2 x - 2\sin^2 x$ Using the formula $\cos 2x = \cos x - \sin^2 x$ I can say $3\cos 2x + 1 = 3(\cos^2 x - \sin^2 x) + 1$ $\Rightarrow 3\cos x^2 - 3\sin^2 x + 1$ But from ...
2
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1answer
72 views

Golden Ratio & Fibonacci - Charles de Gaulle 13-unit two-beamed cross problem.

Here is the question: The two-beamed cross, made popular by Charles de Gaulle, is formed from 13 unit squares as shown below. A straight line $BC$ drawn through point $A$ divides the cross in such ...
0
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2answers
31 views

How can you find two solutions to $\cos(3x - \frac{\pi}{2}) = 0$ by illustrating the situation on the unit circle?

How can you find two solutions to $\cos(3x - \frac{\pi}{2}) = 0$ by illustrating the situation on the unit circle? The solutions I got for this are $x = 0+2\pi k$ or $\pi/3 + 2\pi k$ where $k$ ...
0
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4answers
31 views

arc tangent to two lines and through a point

How can I calculate the radius of a curve that is tangential to two intersecting lines and also passes through a point that is not on either of the lines?
3
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3answers
112 views

A purely algebraic proof of $\vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta)$

I have seen a proof of the fact that $$ \vec{a}\cdot \vec{b} = \lVert\vec{a} \rVert\lVert\vec{b} \rVert\cos(\theta) $$ where $\vec{a}$ and $\vec{b}$ are two vectors. The proof relies on the Law of ...
0
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0answers
26 views

Relationship in a sequence of arcsin functions

Does any relationship exist in the series $(2/\pi) \arcsin \sqrt{1/2}$, $(2/\pi)\arcsin\sqrt{(1/2)^2}$, $(2/\pi)\arcsin\sqrt{(1/2)^3}$, $(2/\pi)\arcsin\sqrt{(1/2)^4}$ etc?