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

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3answers
17 views

Implicit differentiation with trig function

I have the following expression which I need to implicitly differentiate: $$ xy^2 + x^2 + y + \sin(x^2y) = 0 $$ I'm a little confused as I'm not entirely sure what to do with the trig function. ...
1
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0answers
9 views

Calculating the points of a annular sector type shape.

I would like find out if there is a mathematical solution to a problem I am having, but so far everyone I ask say it is simply not possible to calculate or speak of all kinds of ways to do it without ...
0
votes
0answers
10 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
0
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1answer
37 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
2
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2answers
19 views

Inequality with trigonometric functions $\sin$ and $\cos$

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
2
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2answers
17 views

Definite integral of trig function

I'm looking for some assistance on the following problem: Let $$ T(x) = \int_{4r^3}^{4} tsin(t^3)dt $$ Find $$T'(r)$$ I'm struggling to find the antiderivative of the sine function, particularly as ...
3
votes
7answers
88 views

Find the first derivative $y=\sqrt\frac{1+\cosθ}{1-\cosθ}$

$$y=\sqrt\frac{1+\cosθ}{1-\cosθ}$$ my professor said that the answer is $$y'=\frac{1}{\cosθ-1}$$ she said use half angle formula but I just end up with ...
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4answers
48 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
12
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2answers
113 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 ...
0
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2answers
20 views

Slight help with inverse trigonometry question

I apologize for the lack of LaTeX, i will try to learn LaTeX and update this question as soon as possible. I am having some trouble with an inverse trigonometry question and was hoping that someone ...
0
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0answers
20 views

simplify and find domain for triginomic function

I am just doing some review and a question requires me to simplify and find the domain of this function, $\sin(2)\sin(2x)$ how do I find the domain?
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1answer
32 views

Equilateral Triangle Problem With Trig

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle P. The distance $|AP|=3 cm, |BP|=4 cm, |CP|=5 cm.$ What is the area of the ...
0
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0answers
15 views

Bounds on sum of sines/angles.

I have the following equalities: $$ \sin(\theta) + b_x = sin(\theta_a) \\ \cos(\phi) + b_y = cos(\phi_a) $$ My goal is to find upper bounds for the following: $|\theta - \theta_a|$, $|\theta + ...
-1
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0answers
33 views

Trigonometry problem - No right angles triangle [on hold]

I got a trigo problem I need to solve asap :p I've got a triangle, with no right angle. 1 of the side length is know, and the opposite angle is known too. I am spliting the triangle with a line ...
0
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0answers
33 views

How to determine $\cos(a) = \frac{2}{3}$? [on hold]

So I know that $\cos(a) = \frac{2}{3}$ Now I need to determine $\cos\left(\frac{13\pi}{2}\right)+a$ and provide it's exact value, but I don't know how to determine $a$ since it's not in the unit ...
1
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1answer
30 views

Constructing triangle using side length-median relationship

$$\begin{align} m^2_a&=\frac{2b^2+2c^2−a^2}4\\[4pt] m^2_b&=\frac{2c^2+2a^2−b^2}4\\[4pt] m^2_c&=\frac{2a^2+2b^2−c^2}4 \end{align}$$ Solving for $a$, $b$, $c$ in terms of $m^2_a$, $m^2_b$, ...
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1answer
36 views

Desmos.com simulating spinning orbital object

https://www.youtube.com/watch?v=U_VsPV1WJbg As shown in the video, the face with the eyes and mouth are orbiting an unplotted circle with radius = 4, but also spinning (rotating) in a circular motion ...
0
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0answers
34 views

How to find the components of a vector, given magnitude and angle?

Problem The velocity of an aeroplane is $100$ km/h at an angle $30$ degree from north toward west. Draw a vector diagram to obtain its north and east components. Progress The work I already tried ...
3
votes
1answer
66 views

Question of trigonometry

If $\cos^2 A=\dfrac{a^2-1}{3}$ and $\tan^2\left(\dfrac{A}{2}\right)=\tan^{2/3} B$. Then find $\cos^{2/3}B+\sin^{2/3}B $. I tried componendo and dividendo to write the second statement as cos A but ...
2
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1answer
29 views

Proving $\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi }{2-x}\right)$

How to solve the following question? If $n$ is an integer, show that \begin{eqnarray} \left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin ...
0
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1answer
57 views

Prove $\sin(x)< x$ when $x>0$ using LMVT

According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that ...
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2answers
76 views

Trig Equation - 2 years out of math & lost [on hold]

$$\cos^2(2x) + \sin^4(x) = 2$$ So lost on how to solve these things and it's already midnight. 3 days I've spent reviewing and doing practice, but I can't find any proper information on how to go ...
0
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1answer
14 views

Math question about solutions on intervals in trig. [on hold]

Find all solutions in the interval $[0, 2 \pi)$ for $$ 2 \cos x \csc x - 4 \cos x - \csc x + 2 = 0 \,? $$
1
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1answer
107 views

What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$? [duplicate]

How to compute $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$ I tried to rewrite it in terms of $\sin$ $$ ...
0
votes
2answers
29 views

Non-trigonometric Continuous Periodic Functions

I've seen lots of examples of periodic functions, but they all have one thing in common: They all involve at least one trigonometric term (e.g. $\sin\theta$, $\cos\theta$, etc.). My question is ...
1
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1answer
26 views

If α, β are two values of θ satisfying equation cosθ/a + sinθ/b = 1/c then prove that cot ((α+β)/2) = b/a

What I did was $$b\ \cos (\theta) + a \sin (\theta) = \dfrac{ab}{c} \\ b\ \cos (\theta) = \frac{ab}{c} - a\ \sin (\theta) $$ Square both sides and using sum of roots and product of roots as ...
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2answers
41 views

If xsinθ = ysin(θ + 2π/3) = zsin(θ + 4π/3) then prove that Σxy = 0?

Please help! I don't know how to solve this question. I tried putting the whole thing equal to "k" and then calculating values of x,y and z in terms of k and putting there. But it messes up the ...
1
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1answer
57 views

Why do both trig functions have the same Macluarin series?

Both the degree version and the radian version of the trig functions have the same Maclaurin series, yet they are different. How is this possible? How can two different functions have the same ...
6
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5answers
122 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
-1
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0answers
29 views

trigonometric equation (proof answer) [on hold]

hi,all as you can see in the picture there are two parts that need to be proof. first is based on (b) and second based on (a) for the first equation, i already got the answer which is d3=2dm2. ...
1
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1answer
51 views

Roots of $f(x)=a_0+a_1\cos x+a_2\cos 2x+\dots+a_n\cos nx$

If $a_i$'s are nonzero real numbers such that $a_n > {\sum^{n-1}_{i=0}}|a_i|$ prove that the number of roots of $f(x)=a_0+a_1\cos x + a_2\cos 2x+\dots+a_n\cos nx$ is at least 2n.
0
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2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
2
votes
1answer
73 views

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$, compute $b^2-c^2$

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$. Then $(b^2-c^2) = $ $\bf{Options::}$ $(a)\;\; 4ac\;\;\;\;\;\;(b)\;\; a^2\;\;\;\;\;\;(c)\;\; 4bc\;\;\;\;\;\;(d)\;\; ...
0
votes
0answers
34 views

Proving the trigonometric identity with angles in GP [duplicate]

Prove $$\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}=\frac{\sqrt{7}}{2}$$ Attempt- I tried to use trignometric identities but couldn't get the result
1
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2answers
92 views

What is $\frac{2x}{1-x^2}$ when $x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$?

If $$x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$$ Find $$\frac{2x}{1-x^2}$$ I got till here by simplification by taking the previous value of x, ie, ...
0
votes
1answer
43 views

Trigonometric Inequalities

If $$a\sin A+b\sin B+c\sin C=k$$ then the minimum value of $$\sin^2A + \sin^2B + \sin^2C =?$$ ATTEMPT- I tried to use A.M-R.M.S inequality but it didn't help
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0answers
49 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
0
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2answers
32 views

Trigonometric idnetity

IF $\sin \alpha = 3 \sin (\alpha+2\beta)$, then the value of $\tan (\alpha+\beta)+2 \tan \beta=$? ATTEMPT: $\sin \alpha = 3 (\sin (\alpha+\beta) \cos \beta + \cos (\alpha+\beta) \sin \beta)$ ...
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1answer
46 views

how to prove that $\cos n\pi=(-1)^n$?

I'm asked to prove that $$\cos n\pi=(-1)^n\qquad n\in\mathbb {Z}$$ I'm not sure how to approach the problem, I want to know if there is a different way to use induction
0
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1answer
33 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
0
votes
1answer
35 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
5
votes
3answers
1k views

Ambiguity of notation: sin(x)^2

Several people have told me that $\sin(x)^2 = \sin(x^2)$. However, on several computing platforms, such as the TI-84 and Wolfram|Alpha, $\sin(x)^2 = \sin^2(x)$. Can I safely conclude that the notation ...
0
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1answer
22 views

Geometry Find the Radius of a circumcircle given the area of the triangle

Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle. I also know that to find the radius I ...
0
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1answer
27 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
1
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3answers
15 views

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg.

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg. I have tried adding 2x + x for the two legs, however ...
0
votes
1answer
9 views

Angle of Elevation and slope

Standing on top of a gentle 5degree slope I see the top of a tall building at an angle of elevation of 35degree 15’. I am 160 cm tall and it is 12 m from where I am standing to the foot of the ...
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1answer
37 views

Trigonometry question from “Quick Calculus” book [on hold]

I'm working through a book "Quick Calculus" by Daniel Kleppner and Norman Ramsey. I don't understand one of the questions (frame 55, pg.29) being asked in the book see below... In the figure both ...
1
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0answers
27 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
-4
votes
1answer
22 views

Trigonometry ramp problem [closed]

To load a war tank onto a transport trailer it has to go from ground level to $5$m high. The tank is limited to climbing ramps no more than $25^\circ$. Calculate the shortest ramp lenght that could ...
2
votes
1answer
34 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...