I don't know if I'm just randomly blanking or if I never really knew and have just been going with the flow, but I'm not sure what x represents.  In early high school they were degrees, eg. $\sin(30)$ which equaled $0.5$. Later on we learned about radians and $\pi$, and how $\sin {\pi \over 3}$ was equal to $\sqrt 3 \over 2$.  Now I'm in Uni and I'm discovering that I maybe don't know trigonometry as well as I should.  For example, when doing the squeeze theorem, and I'm asked to find the limit of $\sin(n) \over n$, what is the $n$? Is it in radians? degrees? It's for graphing so what should I visualize? $n$ as just a $x$ value to try and find any $y$??
124233
I was just wondering if you use degrees or radians in trig functions. For example if I have a degree of 0.5 would I do: Sin(0.5) or would I have to convert that to radians?  Or does it not matter either way?
18581
I am a little weak in trigonometry. I have two questions:   Find the value of $\cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $ Find the value of $\sin 28^{\circ}+ \cos 17^{\circ} + \cos 28^{\circ} + \sin 17^{\circ} $   I am asking these questions because: 1. I am weak and unable to solve these. 2. I want to know the difference in these questions.
91199
I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation. Here is what I understand so far from what they told me:   1)Treat the top variable as as variable when finding the derivative 2)Treat the bottom variable as a constant when finding derivative 3)It it said "Find x with respect to y", but what exactly does that mean? What does it mean for something to be in respect to something else?  It seems like $\frac{dx}{dy}$ notation changes according to values in the problem. For instance, If $y = x^3 + 2x$ and $\frac{dx}{dt} = 5$, find $\frac{dy}{dt}$ when $x=2$. Why do the values of $\frac{dx}{dy}$ change in this problem and how do I solve this?
102490
The question  is to find $x\in\left(0,\frac{\pi}{2}\right)$:  $$\frac {\sqrt3 - 1}{\sin x}  + \frac {\sqrt3  + 1}{\cos x} = 4\sqrt2 $$  What I did was to take the $\cos x$  fraction    to the right and try to simplify ;  But it looked very messy and trying to write $\sin x$  in terms of $\cos x$ didn't help.   Is there a more simple (elegant) way to do this.
90685
I have to write a paper about the unit circle and I'm trying to uncover some of its origins.   Also, when the trig functions were expanded to angles greater than $90^{\circ}$ and what was the rationale behind it? Also, why mirror the right triangle along the axis instead of just moving the base of the right triangle to the next axis $(+x\to+y\to-x\to-y\to+x)$?  I had no luck trying to find that online, and all the books that I looked into would jump right into the practical part of it.
78855
Can someone please explain me what is a ?  Also, how does putting $$X_a(\psi) ~=~ x_a\star\psi$$ realise $$[X_a,X_b]=i\theta_{ab}{\bf 1}?$$   Ref:
79871
One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does  $$ x_0 = \sin 1, \space  x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ and after a while, $$ x = \pi $$  I know this to be true because I have evaluated this on my TI-84 and more deeply evaluated it with this program I made:   To $100000$ Iterations (which I believe is enough)   So what's the math behind it? Why does adding $\sin 1$ in this manner produce $\pi$? Why doesn't doing the same thing using $100$ produce $\pi$? For example,  $$\sin(1) = 0.841470...,\space \sin(1) + \sin(0.8414...), \sin(1) + \sin(0.841470...) + \sin(1.587095126...),\space ... $$ $$ = \pi$$  My question is different from the mentioned because my function is not taking the sin of a sin consecutively, but is this following function: $f(x) = x + \sin x$ and is not $f(x) = \sin x$ and therefore can exhibit different properties I need explained.
116568
Determine the smallest positive value of x(in degrees) for which: $\tan(x+100^{\circ}) = \tan(x+50^{\circ})\tan (x)\tan(x-50^{\circ})$  I tried to apply the formula of $\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A \tan B}$ but that led me nowhere resulting in a huge equation.  Please help.
77772
My friend and I were discussing this and we couldn't figure out how to prove it one way or another.  The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ is some product of a fraction of $\pi$.  Is $\sin(x) $ (or other trigonometric function) necessarily irrational if $ x $ is rational?  Edit: Excluding the trivial solution of 0.
53169
It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$  Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$  I was thinking of making polar transformations $X=r \cos \theta, Y=r \sin \theta$  Then I am getting stuck at ranges of $\theta$
121759
How can I find the following product using elementary trigonometry?     Suppose $0 \lt x \lt \frac{\pi}{2}$   is an angle measured in radians. Use the trigonometric circle and   show that $\cos(x) \le \frac{\sin(x)}{x} \le \frac{1}{\cos(x)}$.   I have been trying to solve this question. I can't figure out whether or not the solution requires a trigonometric circle or if it can be done using another method.
115343
$$\frac{dy}{dx} = \sin^{-1} (y)$$ The above equation is a form of $\frac{dy}{dx} = f(y)$, so degree should be $1$.  But if I write it as $$y = \sin\left(\frac{dy}{dx}\right)$$ then degree is not defined as it is not a polynomial in $\frac{dy}{dx}$. Please explain?
73606
$\int_{-2}^0\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} dydx$  y describes the circle of radius 2, and is positive and negative, so doesnt limit any quadrants in our integral, however x can only be negative, which means the left half of our circle is what we are interested in.  This means, we need the integral:  $\int_{\pi/2}^{3\pi/2}\int_{0}^{2} rdrd\theta$  Giving us a value of $2\pi$.  Is this a correct assessment? I wasnt sure if making  $0$ and $2$ the limits of $r$ was entirely correct, since it describes the entire circle of radius $2$, and not just one half. Then again, a circle of radius $2$ does have an area of $4\pi$ which makes me feel I am along the right lines.   Any clarification would be fantastic. Thank you!
11545
In $(0 \:\:\pi)$Find the number of solutions of $$\sin x+2 \sin 2x- \sin 3x=3$$  The equation can be written as  $$\sin x+4 \sin x \cos x=3+\sin 3x$$ i.e.  $$\sin x(1+4\cos x)=3+\sin 3x$$ i.e.,  $$\sin x(1+4\cos x)=3+\sin x(3-4\sin^2 x)=3+\sin x(4\cos ^2x-1)$$  so  $$\sin x(4\cos^2 x-4\cos x-2)=-3$$ Any hint from here?
19596
What is the reason that  for $x&lt;0.5$, $\sin(x)\approx x$?  Are there more known properties of these kind for other trigonometry functions?
69099
In  answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.  In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumference; because the perimeter of the $n$-gon is always less than $2 \pi$, the single side must always be less than $2 \pi / n$.  The inequality  $$1 - \cos (x) \leq \displaystyle \frac{x^2}{2}$$ (1)  is used and the proof is completed with   $$2(1 - \cos(x)) \leq (2 \pi / n)^2$$  $$\sqrt{2(1 - \cos(x))} \leq 2 \pi / n$$  But it is well known that the cosine is a function $f(x) \in [-1;1]$, so $1 - \cos (x) \in [0,2]$. By using this information, we would obtain  $$1 - \cos (x) \leq 2$$ (2)  The proof would provide  $$2(1 - \cos(x)) \leq 4$$  $$\sqrt{2(1 - \cos(x))} \leq 2$$  which is a completely different result.   Why in that case it is preferable to use (1) instead of (2)? How to choose when it is convenient to use (1) and when to use (2) in a proof?
2958
Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$? How can we find the RHS if we don't know what it is? (instead of proving the identity itself) I could find a geometric solution in , but is there any solution that doesnt require drawing something? Edit: I now saw some nice proofs using $e$ and Euler's identity. I would appreciate anything NOT using them too, as a change. Out of curiosity, Is there a proof that says you actually "can't" prove the identity using only other simple 1-variable identities like $\sin^2\alpha+\cos^2\alpha=1$?