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

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13 views

Mandelbrot set of $c \cos(z)$

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 $$f_c(z) = c \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : n \in ...
0
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2answers
28 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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0answers
11 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 ...
3
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3answers
43 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 ...
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2answers
55 views

Solve: $\cos^2 θ - \sin^2 θ = \sqrt{3}/2$ [on hold]

Solve: $\cos^2 \theta - \sin^2 \theta = \frac{\sqrt{3}}{2}$ for $\theta$. Domain: $(0°, 360°)$ Calculator says: $π/12±2πn,11π/12±2πn,−π/12±2πn,13π/12±2πn$
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3answers
72 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 ...
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5answers
88 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
53 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!
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2answers
23 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
19 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
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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 ...
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2answers
58 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 ...
1
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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 ...
3
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1answer
81 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 ...
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4answers
52 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
16 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
36 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, ...
3
votes
3answers
62 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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0answers
14 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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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
46 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
59 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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7answers
667 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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3answers
68 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 ...
0
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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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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} + ...
2
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1answer
38 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: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...
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0answers
23 views

trigonometry related problem [on hold]

Explain a scenario where we can apply cosine rule to solve triangle instead of sine rule. Give examples if there are situations where we can apply only sine rule but not cosine rule. When does the ...
0
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0answers
27 views

Solutions of trigonometric equation $a\sin(x) + b\cos(x) = n$

Is there a solution of the equation $a\sin(x) + b\cos(x) = n$ in rational numbers (i.e. $a,b,n,x$ are rational) where $x$ is not of the form $90n^\circ$? (This question was also there on Integer ...
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3answers
45 views

If a = 3i + 2j and b = -7i + 4j, find a + b as…

"Trig functions enable you to make mathematical models of vector quantities:" If $\vec{a}$ = 3$\vec{i}$ + 2$\vec{j}$ and $\vec{b}$ = -7$\vec{i}$ + 4$\vec{j}$, find a + b as: A) a sum of two ...
1
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1answer
39 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
1
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1answer
29 views

A variant on Laplace's method $\int_{-\pi}^{\pi} \cos^{2n} (x/2) \cos( m x) dx \sim \frac{1}{\sqrt{n} }$

I am trying to estimate some probability using the inversion formula of characteristics of some discrete random variable and it finally boils down to the following integration which is very similar to ...
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2answers
40 views

Trigonometric double angle formulas problem

I want to simplify the answer to an equation I had to compute, namely, simplifying $\sin^2 (2y) + \cos^2 (2y)$. I know that $\sin^2 (y) + \cos^2 (y) = 1$ but is there anything like that I can use at ...
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1answer
36 views

If $\sin{x}+\sin{y}+\sin{z}= \cos{x}+\cos{y}+\cos{z}=0$, find the value of $\cos{2x}+\cos{2y}+\cos{2z}$. [on hold]

Is there any way to solve this question using complex numbers? I am trying the general way too but I am unable to solve the question.
-2
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0answers
39 views

Integer solutions of equation sin x + cos x = n [on hold]

If I have the equation $a.sin(x) + b.cos(x) = n$ , is there a solution where all of a,b,x,n are rational and where x is not of the form $90n$ (degrees)
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1answer
23 views

Find a possible polynomial under certain conditions

The polynomial of degree $5$, $P(x)$ has leading coefficient $1$, has roots of multiplicity $2$ at $x=5$ and $x=0$, and a root of multiplicity $1$ at $x=-5$. Find a possible formula for $P(x)$.
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2answers
48 views

Converting a word problem into an equation? Trignometry and calculus [duplicate]

The problem is question 3 of what I am about to download into this question. I drew a diagram of what the problem actually is, my professor has verified it's correct. I don't want an exact answer to ...
0
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0answers
37 views

Law of Cosines for SSA triangles

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles $ABC$ and $DEF$ such that $AB = DE$, $BC = EF$, and $\angle A = \angle D$, then we ...
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1answer
36 views

Ambiguous Triangles [on hold]

How can I determine whether an ambiguous triangle has has one answer, two answers, or none? I understand that an ambiguous triangle is that which presents two sides and one angle (SSA), I just don't ...
3
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1answer
53 views

Solving an integral with trig substitution

I'm looking to solve the following integral using substitution: $$\int \frac{dx}{2-\cos x}$$ Let $z=\tan\frac{x}{2}$ Then $dz=\frac 1 2 \sec^2 \frac x 2\,dx$ $$\sin x=\frac{2z}{z^2+1}$$ $$\cos x ...
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2answers
41 views

Limited partial sum of $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ are limited?

I'm wondering if it's true that $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ has limited partial sum. I know it has representation $\displaystyle ...
0
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1answer
23 views

How can i find the angle in the equation? [on hold]

Find the angle $θ$ in the equation bellow: $Y=Z\cdot(\sin θ+\cos θ)$ I know i can replace $\sin θ=(1-\cosθ)^{1/2}$. Thanks a lot
3
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0answers
32 views

$\cot^{-1}\frac{y}{\sqrt{1-x^2-y^2}} = 2\tan^{-1}\sqrt{\frac{3-4x^2}{4x^2}} - \tan^{-1}\sqrt{\frac{3-4x^2}{x^2}} $

Express $$\cot^{-1}\frac{y}{\sqrt{1-x^2-y^2}} = 2\tan^{-1}\sqrt{\frac{3-4x^2}{4x^2}} - \tan^{-1}\sqrt{\frac{3-4x^2}{x^2}} $$ as a rational integral equation between x and y. This is what I've ...
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1answer
22 views

What is this octagon constant and how do I calculate it for other 8*N-gons?

I'm drawing a circle with triangles in OpenGL and I am no good at maths. I've tried a couple of ways, one including the simple ...
1
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1answer
14 views

Find vectors at angle intervals from a reference vector

Referring to the figure, how can I find the vectors $V_1, V_2, V_3$ and so on that are subtended at an angle of $\theta$, $2\theta$, $3\theta$ and so on from $V_0$ respectively. The knowns are ...
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2answers
62 views

Getting different answers using different methods in a geometrical problem

Problem statement: Given a triangle with side lengths 4 and 6, their corresponding opposite angles have a 1:2 ratio. Find the length of the third side. I solved the problem in 2 ways and got as an ...
1
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0answers
44 views

L'Hôpital's rule exercise concerning trig function

I'd like to verify that my work on the following L'Hôpital's rule question is correct: $$\lim_{x \to 0}\,\,{\cot{x}\,(x^2+3x)} $$ As the limit evaluates to $\frac{0}{0}$, we take the derivative of ...
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1answer
42 views

trigonometry: prove that (tanA)(tanB)+(tanB)(tanC)+(tanA)(tanC)=1 [on hold]

Let $\angle A+\angle B+\angle C=90$. Prove that: \begin{equation*} (\tan A)(\tan B)+(\tan B)(\tan C)+(\tan A)(\tan C)=1 \end{equation*} please help. Thank you.
1
vote
3answers
94 views

Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$

Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$ $$$$ The solution was suggested like this:$$$$ SOLUTION: First of all its, quite obvious to have ...