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

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4answers
68 views

$\text{Prove that}$ $\frac{\sin(\frac{n+1}2)*\cos(\frac n2)}{\sin\frac 12} \ge\frac n2$

Prove that$$\frac{\sin\left(\frac{n+1}2\right)\times\cos\left(\frac n2\right)}{\sin\left(\frac 12\right)} \ge\frac n2$$ So far I've switched up the problem and gotten it down to all sin functions. I ...
1
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4answers
88 views

How can I prove this question concerning trigonometry?

Prove that, for some constant $B$, $$4\cos(x) - 3\sin(x) = 5\cos(x+B).$$ Then, estimate the value of $B$.
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0answers
8 views

Simultaneously solving trigonometric equations

Let $N\in\mathbb N$. Given $\theta_1,\ldots, \theta_N\in [0,2\pi)$ I would like to prove that there exist $\rho\in\mathbb R_+$ and $\varphi\in[0,2\pi)$ such that $$ f_\ell(\rho,\varphi):=\theta_\ell ...
1
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2answers
32 views

How do you solve this trig/geometry question?

In a quadrilateral $ABCD$, if $\sin\left(\frac{A+B}2\right)\cos\left(\frac{A-B}2\right) + \sin\left(\frac{C+D}2\right)\cos\left(\frac{C-D}2\right) = 2$ then $\sin\left(\frac A 2\right) ...
0
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1answer
31 views

Calculating the points of a annular sector type shape.

The problem involves a circle inside a square sharing a common center point. The circle is always smaller than the square so that their edges never intersect. Then an annular sector (see cyan shape in ...
-1
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1answer
44 views

Prove this trigonometric identity? [on hold]

Prove that $(\tan^2 \theta -\sin^2 \theta) = (\tan^2 \theta) \cdot (\sin^2 \theta)$
0
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2answers
11 views

Partial derivative of trig function

I need some assistance on the following calculus problem: Let $$w = 2\cot(x)+y^2z^2$$ $$x = uv$$ $$y = \sin(uv)$$ $$z = e^u$$ Find $\frac{\partial w}{\partial u}$ for $u = \frac{1}{4}$ and $v = ...
2
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3answers
33 views

Proving that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$

Edit: got it, silly mistakes :) I need to prove that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$ $$=\frac{\tan x+\tan y-\tan x+\tan^2x\tan y}{1-\tan x\tan y+\tan^2x+\tan x\tan y}$$ ...
0
votes
2answers
28 views

How to write $a^{ix}$ in terms of $\sin(x)$ and $\cos(x)$?

We know that $e^{ix} = \cos(x) + i\sin(x)$ and the plot of $2^{ix}$ seems to have sinusoidal behavior. http://goo.gl/Xfg2wp Can we claim that we can write $a^{ix}$ in terms of $\sin(x)$ and ...
0
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1answer
437 views

Solving negative domain trigonometric equations with unit circle

How would you solve a trigonometric equation (using the unit circle), which includes a negative domain, such as: $$\sin(x) = 1/2, \text{ for } -4\pi < x < 4\pi$$ I understand, that the sine ...
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1answer
54 views

Fermat's Last Equation

Sorry this is an amateur question but I was wondering since Andrew Wiles solved Fermat's Last Theorem what effect does this have any impact on Geometry. Does this prove in a sense Higher Order right ...
1
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1answer
16 views

Calculating quadrant facing from a rotational matrix and two 3d vectors

I am working on a space-ship simulator, and having trouble with facing arcs between two space objects. Each object has a rotation matrix defined as follows: ...
1
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1answer
44 views

Trying to find an $\arctan(x/y)$ identity.

I have this equation : $$\theta = \arctan\left(\tfrac xd\right) + \arctan\left(\tfrac yd\right).$$ $\theta$ is an angle and I am trying to express $d$ as a function of $\theta$. So is there a way ...
1
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1answer
631 views

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side.

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side. The direction of rotation is counter clockwise. The bottom of the Ferris wheel is 2 m off the ground. It rotates ...
0
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1answer
20 views

equation for the radius of a circle that is tangent to two lines and passing through a specific point on one of the lines?

I'm interested in finding the equation for the radius (and optionally the center point) for a circle that is tangent to two lines and passing through a specific point on one of the lines. So far, I've ...
0
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1answer
17 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 ...
1
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2answers
41 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

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 $m, n, l >0$.
3
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6answers
95 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 ...
3
votes
5answers
120 views

Proof of trigonometric identity $\cot \theta \sec\theta= 1/ \sin\theta$

Is this trigonometric identity provable? $$\color{red}{}\;\color{navy}{\cot \theta \sec \theta = \dfrac 1 {\sin \theta}}$$ I can't seem to get passed: $\dfrac{1}{\tan\theta \cos\theta}$
1
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3answers
22 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. ...
2
votes
2answers
20 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 ...
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2answers
139 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 ...
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2answers
78 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
votes
2answers
60 views

What is the value of $\alpha$ in, $\tan\theta=\frac{Q\sin\alpha}{P+Q\cos\alpha}.$

In Vector chapter i found the formula, $$ \tan\theta=\cfrac{Q\sin\alpha}{P+Q\cos\alpha} $$ Suppose I have the values of $$ Q = |\vec{Q}|\\ P = |\vec{P}|\\ \theta=angle\ between\ \vec{P}\ and\ ...
1
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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 ...
4
votes
4answers
155 views

Trigo Problem : Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$

Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ My approach : I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $ $\Rightarrow ...
1
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1answer
32 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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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?
2
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1answer
34 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 + ...
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 ...
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0answers
36 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 ...
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1answer
38 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 ...
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, ...
1
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1answer
109 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
0answers
37 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 ...
12
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3answers
438 views
+300

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
6
votes
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: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
15
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7answers
1k views

Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$

How do I show that: $$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$ This is actually problem B $4371$ given at this link. Looks like ...
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 ...
0
votes
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 ...
2
votes
6answers
81 views

How to find $\lim_{x\to 0}\frac{\tan 3x}{\tan 5x}$?

I am asked to find the following limit: $$ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x}$$ My problem is in simplifying the function. I followed two different approaches to solve the problem. But both ...
2
votes
1answer
32 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
votes
1answer
15 views

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

Find all solutions in the interval $[0, 2 \pi)$ for $$ 2 \cos x \csc x - 4 \cos x - \csc x + 2 = 0 \,? $$
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votes
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 ...
0
votes
2answers
30 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 ...
1
vote
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 ...
1
vote
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.