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

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-1
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0answers
11 views

For which value of $ r \in \mathbb N $ is satisfied that $r\sin (\frac{1}{r})\leq sin(\frac{1}{r+1})(r+1)$ [on hold]

For which value of $ r \in N $ is satisfied that, $\qquad r\sin (\frac{1}{r})\leq sin(\frac{1}{r+1})(r+1)$
0
votes
0answers
33 views

How prove this konw$\sin{A}:\sin{B}:\sin{C}$ then Find $\sin{(2A)}:\sin{(2B)}:\sin{(2C)}$

Question: let $x,y,z>0$ is give numbers, and the postive number $k$ such $$\dfrac{x^2}{x^2+k}+\dfrac{y^2}{y^2+k}+\dfrac{z^2}{z^2+k}=1$$ in $\Delta ABC$, ...
0
votes
0answers
4 views

Solving/simplifying a trig function

Given: $\tan\:a=\frac{5}{12}$ and $a\in QIII$ $\csc\:B=-\frac{5}{4},\:\cos\:B>0$ $\cos\:\theta =-\frac{8}{17},\:\frac{\pi }{2}<\theta <\pi $ Solve: ...
0
votes
1answer
7 views

Solving/simplifying a trig expression

My problem sheet says that $\tan a= 5/12$ and $a \in {\rm Q\,III}$ ($a$'s in quadrant III). Using this information, I am to solve/simplify the expression $\quad \quad \cos\left(\frac{1}{2}a\right)$ ...
3
votes
1answer
44 views

Method of proof of $\tfrac{56700}{19}\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\pi^7$

The following formula was stated by Ramanujan: $$\frac{56700}{19}\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\pi^7$$ Does anybody know the method of proof of this formula? I know that typically ...
4
votes
2answers
198 views

Could Trigonometry exist in one dimension

Even though trigonometry is based on circles, and angles, both of which commonly exist in two dimensions, could it also exist in one dimension? This question probably sounds really weird to you, but ...
0
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2answers
26 views

Number of values that satisfy $2\sin ^2(x) - 3 = 3 \cos (x), \: 90^{\circ} < x < 270^{\circ} $

Graphing this function is difficult as many overlaps exist and finding a viewing window is hard. What's a good algebraic method to solve this problem?
-3
votes
2answers
50 views

High school Math Team problems. [on hold]

In triangle $\triangle ABC$, $AB=5,BC=6$ and $AC=7$. The circle with diameter $AB$ intersects $BC$ at $D$, ($D \neq B$). Compute $BD$.
0
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0answers
14 views

Tangent to circle + bisecting an angle

Given a triangle ABC, and a circle centered at A such that B & C are outside the circle. How can I find a point Q on the circle such that QM is tangential to the circle, and bisects angle BQC? ...
0
votes
1answer
18 views

$−8\sin 3x+5\cos 3x=4.3$ for $0<x<360.$

can you tell me please how to solve $−8\sin3x + 5\cos3x = 4.3$ for $0<x<360$? I find 6 solutions! and I don't know if they are the correct, although they seem to fit in the equation! thanks!
0
votes
2answers
20 views

Finding the domain of this trigonometric function

How can I find the domain of this function? $$f(x)=\frac{x\sin(x)+\cos(x)}{1-\cos(x)} + \frac{|x|-2}{x^2-4}$$ I assume we don't want the denominator to be zero, but do we have to combine the ...
2
votes
3answers
95 views

Solving $\;2^{\large \cos x} = \sin x$

$$2^{\large \cos x} = |\sin x|$$ Solve the equation. I found just one solution $\cos x= 0$ and are there any other solutions. Right hand side is modulus $\sin x$.
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votes
3answers
33 views

Easy Question for Trigonometry. [on hold]

If $\tan 63 = m$ and $\sin 63 = n$ then what is the value of $\cos 63$ ?
6
votes
3answers
233 views

Why doesn't $\arccos x = -\tfrac12\sqrt{3}$ have any solutions?

I have this exercise with an unclear answer. The question is this: $$\arccos x = -\frac{\sqrt3}{2}\,.$$ The answer is this: $$\begin{gather*} \varphi(x)= \arccos x\\ V_\varphi = [0,\pi]\\ ...
1
vote
3answers
70 views

Trigonometry General Solution

I've been working out some general equations recently, whilst the simple ones are fairly easy to work with, the more advanced ones (for me ofcourse) seem to somewhat confuse me since I have no ...
1
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0answers
30 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
0
votes
0answers
25 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
0
votes
0answers
8 views

Least period of a nonzero sum of two sinusoidals.

Let $a,b,\omega_1,\omega_2,\phi_1,\phi_2\in\mathbb R$ where $a,b,\omega_1,\omega_2>0$. Let $f,g\colon\mathbb R\to\mathbb R$ by $f(t)=a\cos(\omega_1t-\phi_1)$ and $g(t)=b\cos(\omega_2t-\phi_2)$. ...
0
votes
1answer
4 views

Finding the domain of this trigonometric function

how can I find the domain of this function? f(x) = (xsin(x) + cos(x) / 1 - cos(x)) + (|X| - 2 / x^2 -4) I assume we don't want the dominator to be zero so f(x)1 ...
-4
votes
1answer
33 views

equation solver online

can you tell me please if is there an online or software tool that will solve equations like $-8sin3x + 5cos3x = 4.3$ for $0< x <360$? that I will just type equations like the above and it will ...
4
votes
2answers
49 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
0
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2answers
26 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
6
votes
1answer
80 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
-1
votes
1answer
22 views

Compound Angles with an unknown [on hold]

$4.5 \sin\theta + 1.5 \sin(\theta + \alpha) = V_m \sin(\theta + \phi)$ Can this equation be solved without the value of $\alpha$ ? I'm looking for the values of $V_m$, $\theta$ and $\phi$.
3
votes
1answer
45 views

Solution to trigonometric derivative

Version 2 For \begin{align} &x(t)\text{:=}\cos (t)+\cos (2 t)+1&\\ &y(t)\text{:=}\sin (t)+\sin (2 t)&\\ \end{align} how would I go about proving that the solutions to \begin{align} ...
0
votes
2answers
14 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
1
vote
4answers
64 views

Trigonometric Expression Simplification

Could someone explain how to simplify $(\cos(x)-\csc(x))/(\sin(x)-\sec(x))$? Any help would be appreciated.
1
vote
2answers
69 views

Why is $x^5 \sin x$ an odd function?

Why is $x^5 \sin x$ an odd function? Is the result just wrong? Because $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
1
vote
1answer
49 views

Given the matrix $A^k$, how to get $A^{k+1}$?

Given: $$A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$$ $$A^{k+1} \overbrace{=}^? \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & ...
1
vote
3answers
33 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
1
vote
2answers
30 views

Simplfy trigonometric functions by only considering integer inputs?

I have the below function which only takes integer input, $$ 2 \sqrt{3} \sin \left(\frac{\pi t}{3}\right)+\sqrt{3} \sin \left(\frac{2 \pi t}{3}\right)-\sqrt{3} \sin \left(\frac{4 \pi ...
0
votes
3answers
23 views

finding and angle and coordinate point

"For a given angle $θ$ and a circle of radius $r$ and center $(h,k)$, recall that we can determine the Cartesian coordinates $(x,y)$ of the point on the circle determined by $θ$ and $r$, where ...
1
vote
1answer
20 views

Trigonometry graphs sinusoidal waves

i need help on this questions. I couldn't figure how to determine for both question A and B. But i have the answers for them, i just don't understand how the amplitude is 3 and so on.
4
votes
1answer
31 views

Exact value of polynomial at trigonometric argument

Given that $$\cos 8\theta= 128\cos^8 \theta −256\cos^6 \theta +160 \cos^4 \theta −32\cos^2 \theta +1$$ Find the exact value of: $$4x^4 −8x^3 +5x^2 −x$$ where $x=\cos^2 ...
1
vote
1answer
20 views

Finding the square roots of a complex number.

Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram. So far I've worked out the mod/arg form of the ...
0
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0answers
17 views

Lagging or Leading trigonometric functions.

Consider the function $f(x) = 2 * \sin(0.5 * x)$. Now suppose I want to create a function which is similar to the mentioned but to "lag" the mentioned function by $45$ degree angle, then which of the ...
1
vote
1answer
51 views

Is there a quicker way to write $\cos (n\theta)$ in terms of $\cos \theta$?

Im writing $\cos 8\theta$ in terms of $\cos \theta$ using De Moivre's Theorem $$\cos 8\theta= \Re {(\cos\theta+ i \sin \theta)^8}$$ Let $s=\sin \theta$ and $c=\cos \theta$ $$=c^8 ...
0
votes
2answers
35 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
0
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0answers
12 views

Inverse cosine of a complex number, take $\cos z=\sqrt{2}$ for $z$

If I am given $\cos z=\sqrt{2}$ for $z$ and asked to solve it using the following: $$ \cos^{-1} z =-i \log\sqrt{z+i(1-z^2)} $$ I've only gotten as far as taking $\cos z=\sqrt{2}$ and changing it to ...
0
votes
2answers
15 views

Product to sum formulas

Write the product as a sum. cos 4x cos 2x this is what i tried 2{cos2xcosx} = 2[1/2 cos(2x+1x)+ cos(2-1)] = 1[cos(3x)+cos(1x)] = cos 3x + cos x
0
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0answers
18 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
0
votes
0answers
23 views

for what value of x is arcsin(sin(x)) = x true

for what value of x is arcsin(sin(x)) = x or sin(arcsin(x))= arcsin(sin(x)) true I know that the value is between -1 and 1 Could someone explain me why?
1
vote
2answers
73 views

Why “$\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does not exist” is not an acceptable answer?

Find the limits: $\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ Since the numerator and denominator tends to infinity as $x$ tends to infinity, then applying Lhopital's rule: ...
2
votes
1answer
23 views

Is a trigonometric function applied to a rational multiple of $\pi$ always algebraic?

Specifically, just to talk about cosine, is it true that $\cos(\frac{a\pi}{b})$ is algebraic for integers $a$ and $b$? Looking at this post and the link to trigonometric constants in the comments, it ...
3
votes
3answers
75 views

How to prove that $\tan 55^\circ<\pi/2$

How to prove that $\tan 55^\circ<\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
1
vote
1answer
30 views

Which identity is being used to get $\sin(wa)\cos(wt)=\sin(w(a+t))+\sin(w(a-t))$?

Which identity is being used to get $\sin(wa)\cos(wt)=\frac{\sin(w(a+t))+\sin(w(a-t))}{2}$? Couldn't find it among the trigonometric identities.
1
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0answers
27 views

Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...
4
votes
3answers
37 views

Calculating a limit with infinitely many terms

I've encountered this limit : $$\lim_{n\to\infty} \frac1n \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right)+\cdots+\sin{\frac{(n-1)\pi}{n}}\right)$$ Wolfram gives the value: ...
0
votes
1answer
41 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
0
votes
1answer
8 views

What is and what represents a convergents function in polynomial form?

$$\mathbf{convergents}(cos(1), 20)$$ What exactly is a convergents function and what, that series of fractions is representing ? There is an use for this in numerical linear algebra ? Feel free to ...