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

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0answers
51 views

Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
2
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2answers
85 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
1
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1answer
20 views

Detect when two edges make a “inner” angle or an “outer” angle

So, given three points, a direction of movement and the side of the movement, find out the "external" or "internal" angle value. In the left pic, I'm above the red line, moving from edge 1 to edge ...
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2answers
34 views

Determine all numbers $x$ such that $\sin x = \sin a$

Let $a$ be a given number. Determine all numbers $x$ such that $\sin x = \sin a$. You may suppose that $0 \le a \lt 2\pi$, and distinguish the cases $a = \frac\pi2$, $a = \frac{-\pi}2$ and $a ...
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2answers
112 views

Is this true that $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$? [on hold]

Assume that $A,B,C$ are positive real numbers and $A,B,C \in (0,\frac{\pi}{2}]$ and we have $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 $$ prove or disprove that $$A+B+C=\pi$$
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0answers
23 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
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0answers
48 views

Proving a limit of a trigonometric function: $\lim_{x \to 2/\pi}\lfloor \sin \frac{1}{x} \rfloor=0$

$$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$ 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 ...
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2answers
66 views

Prove that $\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$ in the $\epsilon$-$\delta$ way [duplicate]

Given: $$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$ How to prove this limit using the $\epsilon$-$\delta$ way? (the biggest problem is to find $\delta$)
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5answers
74 views

Solving $12-\sin(\theta)=\cos(2\theta)$

$12-\sin(\theta)=\cos(2\theta)$ What's the correct answer on the interval $[0, 2 \pi]$. Please help, I'm rather lost. I started with: $$12-\sin(\theta)=1-2\sin^2(\theta)$$ and then I cant get ...
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3answers
44 views

Is there another way to solve this Trigo in series? [duplicate]

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+...... +\cos^2(\theta+179^{\circ})$$ Attempt, $$\cos x=-\cos(180^\circ-x),\sin x=\cos(90^\circ-x),\cos ...
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1answer
58 views

Evaluating $ \int_0^\theta \cosh(a\sin x) dx$

The integral below seems quite simple, but I couldn't find anywhere the result. $$ I = \int_0^\theta \cosh(a\sin x) dx$$ I tried to expand it into Taylor expansion series and successfully evaluate the ...
2
votes
3answers
87 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...
5
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3answers
65 views

Given two points, how to find a circle through them that's also tangent to the $x$-axis?

A seemingly simple geometry problem that is surprisingly difficult. I want to find the radius of a circle that is tangent to the $x$-axis, but also must contain two given points. I understand there ...
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2answers
95 views

Resolved! Thank you! [on hold]

Calculator says: $π/12±2πn,11π/12±2πn,−π/12±2πn,13π/12±2πn$
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0answers
32 views

Trigonometric identity reduction

I want to be able to reduce some trigonometric expressions that have powers of sine and cosine. For example, for arbitrary real numbers $a$, $b$, and $c$, we can reduce the expression $$ a\cos^2\theta ...
3
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1answer
54 views

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
3
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2answers
105 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
0
votes
3answers
32 views

how to parameterize the ellipse $x^2 + xy + 3y^2 = 1$ with $\sin \theta$ and $\cos \theta$

I am trying draw the ellipse $x^2 + xy + 3y^2 = 1$ so I can draw it. Starting from the matrix: $$ \left[ \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 3 \end{array}\right]$$ I ...
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2answers
37 views

Integrating $\sin^3(x)/(2+\cos(x))$

I could use some help solving the following integral: $$\int \frac{\sin^3(x)}{2+\cos(x)} dx$$ So far I tried using the equality: $$\sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4}\sin(3x)$$ which ...
1
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1answer
28 views

Finding value (Trigo Series) [duplicate]

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+......+\cos^2(\theta+179^{\circ})$$ Can anyone teach me where to start with? I've no idea.
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4answers
78 views

Exact value of sin (θ/2) if cos θ = 3/5

Exact value of $\sin\frac{\theta}{2}$ if $cos θ = \frac{3}{5}$ and $360° < θ < 450°$: Okay, so I put this into the half argument property and got: $$\sin\frac{\theta}{2} = \pm ...
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3answers
53 views

How to solve the equations of the type $\sin a + \sin b = \sin x$?

I came across a question in my book that's like this: $$\sin20 + \sin40 = \sin x $$ I don't know if the values of the $a$ and $b$ make a difference (or in this case, the fact that $b = 2a$) but I'd ...
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0answers
20 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
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1answer
55 views

Eyebrow calculation [on hold]

Given a width of 71 and a height of 35, what are the following dimensions: left side, right side, radius, and base?
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0answers
53 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
4
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3answers
57 views

Trying to prove a trigonometric identity

I've been trying to solve it for quite some time but I still don't get it why it is true. The original equation is: \begin{equation*} 1-\frac{\sin{^2}\theta}{1-\cos\theta}=-\cos\theta. ...
8
votes
7answers
682 views

Squaring a trigonometric inequality

A very, very basic question. We know $$-1 \leq \cos x \leq 1$$ However, if we square all sides we obtain $$1 \leq \cos^2(x) \leq 1$$ which is only true for some $x$. The result desired is $$0 \leq ...
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2answers
33 views

Find the value without using calculator (Trigo)

$\sin ^210+\cos ^240+\sin10\cos40$ How to find the value without using calculator and without the formula $\sin^2A-\sin^2B=\sin(A+B)\cdot \sin(A-B)$?
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3answers
48 views

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ How do I simplify this limit? I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give ...
0
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0answers
16 views

Fitting a sinusoidal function to three known points

I have 3 points from a sine wave and I need to determine the sine function from this. There is a very similar question, but this question is with $-30°$, $0°$ and $+30°$: Fitting a sinusoidal ...
2
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5answers
102 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
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1answer
64 views

Write $20 \sin \theta + 17 \cos \theta$ as a single cosine

Write $20 \sin \theta + 17 \cos \theta$ as a single cosine with phase displacement. I don't know how to start this one. If somebody could give me the formula or a sample that would be amazing!
2
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1answer
32 views

Generalization of $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$

I'm looking for a generalization of the following statement $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$ In particular, I want to find $\sup \limits_{\theta} (a \sin ...
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2answers
24 views

solving trigonometry equation $90$ for $ x$

*Solve each equation for all values of $x$: $3\sin x+3=\cos^2 x$ I've tried changing trig values but I don't think its right.
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1answer
20 views

What is the optimal way to detect a collision between an AABB figure and a non-AABB figure?

Background I'm looking to do this programmatically in Java, but if desired you can post solutions in C/C++ or plain English instructions if you're not a programmer, but I would appreciate an ...
2
votes
1answer
45 views

Trigonometry Identity (Proving)

How to prove this identity? $$\frac{\cos 2\alpha+\cos 2\beta}{1+\cos 2(\alpha+\beta)}=\frac{\cos (\alpha-\beta)}{\cos (\alpha+\beta)}$$ I've tried solving from L.H.S and R.H.S. But failed. Anyone ...
3
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3answers
64 views

Finding the period of $f(x) = \sin 2x + \cos 3x$

I want to find the period of the function $f(x) = \sin 2x + \cos 3x$. I tried to rewrite it using the double angle formula and addition formula for cosine. However, I did not obtain an easy function. ...
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2answers
29 views

Prove trigonometric identity, hence or otherwise find the general solution

The following question requires one to prove the below trigonometric identity $$\cos 3x = 4\cos ^3 x - 3\cos x$$ Hence, or otherwise, find the general solution of the following equation $$(4\cos ^2 x ...
0
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4answers
53 views

Problem Verifying Two Challenging Trig Identities

My math teacher gave us an equality involving trigonometric functions and told us to "verify" them. I tried making the two sides equal something simple such as "1 = 1" but kept getting stuck. I would ...
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1answer
17 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
2
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2answers
37 views

Finding all values of $\theta$ which describes a straight line

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed ...
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0answers
21 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
0
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3answers
19 views

Trig algebra problems, taking out a factor of tan

$$ \sin\theta-\cos\theta=0 $$ ${\sin\theta\over\cos\theta}=\tan\theta $ $$ \cos\theta (\tan\theta-1)=0$$ $$\tan\theta=1$$ $$\cos\theta=0$$ $$\theta=45, 90$$ However the second solution is not true ...
2
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1answer
47 views

Trigonometry express $4\cos x+3\sin x$ in the form $R \cos (x+a)$.

I have been asked to express $4\cos x+3\sin x$ in the form $R \cos (x+a)$. I know that the formula to express it in that form is $a \cos x+b\sin x=R \cos (x-a)$. But as the question is asking me to ...
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1answer
61 views

Integration a trigonometric expression

How would you evaluate the following indefinite integral? $$ \int \frac {\ln{(x)} \cdot \cos{(x)}}{\sin^2 {(x)}} dx $$
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2answers
28 views

If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$ [on hold]

Could someone solve this step by step so I can wrap my head around the process?? If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$.
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2answers
180 views

Proving an Integral

The table of integrals says that \begin{equation*} \int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C \end{equation*} where $C$ is a constant. What's wrong with my proof? $$ \begin{align*} ...
5
votes
3answers
70 views

$\sin(x^2)$ in terms of $\sin(x)$ and $\cos(x)$

One of my students asked me "Can you write $\sin(x^2)$ in terms of $\sin(x)$"? I said I'd think about it. Having thought about it for a while, I now know that I definitely don't know the answer! Lets ...
2
votes
1answer
40 views

Prove that $1/(\sin x + 1) - 1/(\sin x - 1) = 2 \sec^2 (x)$

Can anyone solve this for me? Prove that $\frac1{\sin x + 1} - \frac1{\sin x - 1} = 2 \sec^2 (x)$. This is as far as I went: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...
-7
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0answers
23 views

Trigonometry related question [on hold]

Using the parent function, explain how we can graph $g(x) = -2\cos(\pi{x} + \frac{2\pi}3) $using transformation. Specify the amplitude, period and phase shift for $g(x) = -2\cos(\pi{x} + ...