Tagged Questions

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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1answer
24 views

Existence of roots of $A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$

It seems very intuitive that $$f(t)=A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$$ has roots, but how to prove it? $A_i>0$, $\omega_i>0$ and $\phi_i\geqslant0$ (even though these ...
0
votes
1answer
33 views

Finding theta of trigonometry

I'm not quite sure how to ask this question, so I will give an example instead. I know that $\cos(\theta)=0$ if $\theta$ is $\pm\frac\pi2$, or $\sin(\theta)=0$ if $\theta$ is $0$, $\pi$, etc. as those ...
1
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1answer
26 views

How can I measure a frustum inside a frustum?

If I know the measurements of a frustum A, how can I find the measurements of frustum B if I only know ...
2
votes
2answers
40 views

Expansion of function , defined on a open interval containing $0$ , in terms of $\sin$ function

Let $I$ be an open interval in $\mathbb R$ containing $0$ and $f:I \to \mathbb R$ be a twice differentiable function , then is it true that $$\lim_{x \to 0}\dfrac {f(x)-f(0)-f'(0)\sin x - ...
3
votes
1answer
82 views

Find one of two forces, given their resultant

The resultant of two forces acting on a rock is 107N and it makes an angle of 38 degrees, and 40 minutes with the first force of 51N. Find the magnitude of the second force and the angle it makes ...
2
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1answer
64 views

Are $\sin(\alpha\beta)$ and $\sin(\alpha^{\beta})$ expressible in terms of $\sin(\alpha)$ and $\sin(\beta)$?

There is a well known formula for expressing $\sin(\alpha+\beta)$ just using $\sin(\alpha)$ and $\sin(\beta)$. It is enough to replace $\cos$ in the formula ...
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2answers
50 views

Trigonometric problem: find the angle [closed]

In the adjoining figure, find the measure of angle CAB.
0
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1answer
48 views

This question is hard to understand

In London, the shortest day in 2014 was 21 June 2014, (the 172nd day of the year) when sunrise was at 06:37 (6.62 hours after midnight) and the longest day will be 23 December 2014 when sunrise will ...
1
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2answers
43 views

Trigonometric differential equation

Is it possible to solve the following ordinary differential equation: $\theta'(t)=x(t)\sin(\theta(t))-y(t)\cos(\theta(t)),\ \forall t\in I$, $I-$ interval from $\mathbb{R}$, where ...
1
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1answer
32 views

Trigonometric function solutions within an interval

I'm just wondering if anybody can check my solution to the given problem. The problem is: Find the exact values of $x$ in $[0,750]$ that satisfy the equation $sin(x) = 1.$ My approach: The ...
1
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4answers
68 views

Evaluate $\lim_{\theta\to 0}{\frac{1-2\cos\theta+\cos^2 \theta}{\theta \sin \theta}}$ [closed]

Evaluate the limit $$\lim_{\theta\to 0}{\frac{1-2\cos\theta+\cos^2 \theta}{\theta \sin \theta}}$$
3
votes
5answers
97 views

How to evaluate the integral $\int\frac{\sqrt{x^2-9}}{x^3}\;\mathrm d x$?

$$\int\frac{\sqrt{x^2-9}}{x^3}\;\mathrm d x$$ The question is ask me to evaluate the integral but I have no idea how to start? If there are any formulas required for this question, can you please ...
0
votes
1answer
24 views

How to solve for the chord being tangented by the 2 lines?

Radius = 2 Please help me. I came up with an equation. I used pythagorean theorem to solve for the (h) and (r - h). I got the correct answer. Im thinking, what if the green line is smaller. Does the ...
1
vote
0answers
19 views

What function has a 3D graph that will look like a spiral into a singularity?

I am trying to draw text spiraling into a black hole, from a more interesting slightly off-orthogonal viewpoint. I think a function that defines a black hole/singularity surface might look something ...
1
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2answers
123 views

Prove that: $\lim_{x\to 0}\frac{x}{\sin^2(x) + 1} = 0$

Prove $$\displaystyle \lim_{x\to 0} \frac{x}{\sin^2(x) + 1} = 0$$ The proof: Let $$|x| \le 1 \implies -1 \le x \le 1$$ $$\displaystyle \frac{|x|}{|\sin^2(x) + 1|} < \epsilon\text{ for ...
0
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1answer
18 views

Implicit differentiation of trigonometric function $x\tan(y)=6-x^2$

I am asked to find the equation of the tangent and the normal to the curve $$x\tan(y)=6-x^2$$ at the point $(2,\pi/4)$. Rearrange to $$x\tan(y)+x^2-6=0$$ By product rule $$\frac{d}{dy}( x\tan(y)) = ...
0
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1answer
14 views

trigonometric presentation inside a diagonal matrix.

How can I write an example of $2\times2$ matrix where $d_2$ belongs to the first quadrant and $d_1$ belongs to fourth quadrant, that has this form: $D = \pmatrix{d_1&0\\0&d_2},$ I wrote ...
3
votes
2answers
201 views

How do I determine if the following function is periodic?

A question in my textbook asks me to determine if the function $f(x)=\cos(3x)+\sin(x)$ is periodic. I do not believe this to be the case as the arguments of sine and cosine are different and as such ...
2
votes
2answers
49 views

Simple trigonometry

When solving a complex geometric problem, I came to a trigonometric equality that needs to be proven: $${{\sin{100^\circ} \over \sin {60^\circ}}+ {1 \over {1+2\sin{50^\circ}}}} = {2 \sin{50^\circ}}$$ ...
2
votes
4answers
52 views

Solve $\sqrt{3}\cos2\theta+\sin2\theta-1=0$

I tried using the identities $\cos2\theta=1-2\sin^2\theta$ and $\sin2\theta=2\sin\theta\cos\theta$. These give $\sqrt{3}(1-2\sin^2\theta)+2\sin\theta\cos\theta-1=0$ which doesn't seem to lead ...
1
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0answers
58 views

Did Wolfram Alpha make this term up?

So I'm currently taking a Trigonometry class and I'm using the iOS version of the Wolfram-Alpha app to assist me with some of the more mundane calculations (Pythagorean Theorem, etc) and I've come ...
0
votes
2answers
25 views

Defining all values for $X$ on trigonometric functions

So having a bit of a hard time understanding this part of the problem, I'm asked to graph $y = -3\tan(\pi x + 5\pi)$ So first I have to find if it's defined for all values of $x$... and I can't ...
0
votes
3answers
54 views

How to properly use this trigonometric identity?

I have to calculate the limit of: $$\lim_{x \to 0} \frac{\cos(3x-\frac{\pi}{2})}{x}$$ And it obviously indetermines with $\frac{0}{0}$. So I used trigonometric identities and converted it: ...
3
votes
1answer
80 views

What is cos and sin ACTUALLY doing?

I am having the hardest time figuring out what sin and cos are doing when you enter in calculator. What I do understand about them 1) They are both essentially finding the max and min values for ...
3
votes
1answer
30 views

Show that the function f solves the homogenous wave equation

This is a cleaned up and refined repost of my previous attempt, after I did some research on the subject. SOLVED The only problem here was that I was making tons of little mistakes, always watch ...
0
votes
5answers
33 views

Difference of Tangents

Express the $\tan(\arcsin(u) - \arccos(v))$ algebraically containing $u$ and $v$, but without using trig functions. I know that in order to express this I need to use the Difference of Tangents ...
18
votes
0answers
354 views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
-3
votes
2answers
53 views

How do I evaluate $\cos(x) + \cos (2x) +\cos (3x) + … + \cos (nx)$? [duplicate]

How to evaluate the above expression and express the answer in terms of $n$ and $x$?
0
votes
2answers
20 views

Help explaining steps in a solution (involving trig functions and algebra)

I was reading a textbook and was trouble deriving a particular equation. The last step involved this Can anyone explain why letting the constant be $-\frac{\pi}{2}$ leads to the final answer.
0
votes
0answers
46 views

How to find the desired angles ($\phi$ and $\theta$) in these two equations

I'm reading a paper in which the authors say from the following two equations, they can acquire the desired angles ($\phi$ and $\theta$) $$ x = \cos \phi \sin\theta \cos\psi + \sin\phi \sin\psi \\ ...
0
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0answers
24 views

stationary points for a function involving trigonometric function

I calculated the stationary points for a two-variable function in which $x$ only came part of $cos(x)$. So the function, one which I don't recall precisely, was something of this sort: $$y^2 * (5y + ...
0
votes
1answer
21 views

Solving for $x$. Trigonometry.

$\cos(56-3x)=-0.441$, where $0\leq x\leq 360$. I can't seem to do this question, as I knwo its CAST diagrams but I am not sure how to change the values and Find the correct answer. Please Help
2
votes
1answer
46 views

how to prove $\prod_{k=1}^{p-1} \sin(\frac{\pi k}{p}) = \frac{p}{2^{p-1}}$?

i found this relation whilst trying to evaluate the norm (over $\mathbb{Q}$) of $1-\zeta$ for $\zeta$ a primitive $p$-th root of unity ($p$ supposed prime) $$ \prod_{k=1}^{p-1} \sin(\frac{\pi k}{p}) = ...
3
votes
3answers
66 views

What is the value of $\cos\left(\frac{2\pi}{7}\right)$? [duplicate]

What is the value of $\cos\left(\frac{2\pi}{7}\right)$ ? I don't know how to calculate it.
9
votes
2answers
89 views

A question on cosine integral

So I've read a book and found myself stumped in this integral: $$\int_{0}^{\pi} \frac{\cos(n\theta)}{b^2-a^2\cos(2\theta)}\, d\theta=\begin{cases} \,\,0 &,\quad\mbox{if}\,\, n\,\,\mbox{is ...
1
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0answers
40 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
1
vote
2answers
59 views

Simplify $\def\Arctan{\operatorname{Arctan}}f(x) = \Arctan(2x) + \Arctan(3x)$

$\def\Arctan{\operatorname{Arctan}}$ Simplify $f(x) = \Arctan(2x) + \Arctan(3x)$ I had a go at it and this is what I got to : We have: $-π<\Arctan(2x)+\Arctan(3x)<π$ Let $a=\Arctan(2x)$ and ...
10
votes
5answers
999 views

How do calculators evaluate inverse trig functions?

I know for simple inputs, you can just memorize the answer, but what if I wanted to find $\arcsin{0.554}$. My calculator instantly tells me that the answer is $0.5752 \ \text{radians}$. How can I do ...
2
votes
2answers
110 views

circular reasoning in proving $\frac{\sin x}x\to1,x\to0$

The classic proof for $\frac{\sin x}x\to1,x\to0$ is using a squeezing theorem based on arguments about areas of circles. But as far as I know, all deduction of formula of circles' area is based on ...
0
votes
2answers
16 views

Find the values of $a$ and $b$ ~ Trigonometry

The function $f$, where $f(x) = a \sin x+b$, is defined for the domain $0 \leq x \leq 2\pi$. Given that $f(\frac{1}{2}\pi)=2$ and that $f(\frac{3}{2}\pi)=-8$, find the values of $a$ and $b$. I know ...
2
votes
2answers
50 views

Rewrite the expression in the form $A \sin(x+C)$

Rewrite the following expression in the form $A \sin(x+C)$ $$4 \sin x + 4\sqrt{3} \cos x$$ This is what I have so far, and I'm not even sure it's the right approach. I just dont understand this ...
0
votes
1answer
32 views

How does a complex exponential turn into the sinc function?

Suppose I have a complex variable $j$ such that we have $f(u) = \frac{1}{ju}[e^{\frac{ju}{2}} - e^{\frac{-ju}{2}}]$. Could somebody please explain how this turns into a sinc function ? I know I ...
1
vote
1answer
43 views

How to calculate this $\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}$?

I'm stuck with the expression $$\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}.$$ I have no idea how to begin, please give me a hint! (The answer should be $\sqrt3/8$.)
4
votes
3answers
43 views

Limit with Arctan

Here's a hard limit I've been trying to answer for a while : $$\lim_{x\rightarrow 1} \dfrac{-2x\arctan{x} + \dfrac{\pi}{2}}{x-1}$$ I've tried all the tricks that the teacher has taught us and still ...
0
votes
1answer
31 views

Trigonometric identity involving half angles

Ok.here is the problem in the picture below. How do I get these results? Given that d equals
1
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1answer
63 views

How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi

Only for an acute angle triangle. $A$,$B$,$C$ are angles of a triangle. This isnt sine rule form. Ive tried Cauchy Schwarz theorem , A.M, G.M form but am unable to get the above result. Could someone ...
1
vote
1answer
23 views

Simplify a LHS of the trigonometric equation to obtain RHS

Is this equality correct? If so, how to simplify the following expression on the LHS to get RHS: $$\frac{\sin(x+\frac{nh}{2}) \sin(\frac{(n+1)h}{2})}{\sin\frac{h}{2}} ...
1
vote
1answer
29 views

Solving trancendental with variable argument. $20 = ax\sin(ax)$

Approaching transcendental equations is in general new to me. My experience with numerical methods is limited, and this equation seems to require such a method. But there's a catch - it contains an ...
1
vote
1answer
29 views

Trigonometric Functions on a unit circle

I have to find all solutions for $\theta$ in the given range: \begin{equation} tan (\theta) = \frac {-1}{\sqrt3}, -\pi \le \theta \lt 2\pi \end{equation} I said that if $(x,y)$ is on the unit circle ...
1
vote
1answer
29 views

Confused about the answer to the inverse of a cosine function

$$\arccos { (\cos { (\frac { 17\pi }{ 6 } ) } } )$$ No matter how I try and look at this problem, I end up with $\frac { 5\pi }{ 6 } $ I counted $\frac { \pi }{ 6 } $ 17 times counter clockwise ...