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

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4
votes
7answers
246 views

Exact value for $\cos 36°$

Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$ The problem reads as follows An ...
4
votes
2answers
140 views

In triangle, $\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$

To prove $$\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$$ My approach : $$ \begin{align} \text{L.H.S.} & = ...
19
votes
2answers
449 views

integrate square of $\arctan x$. Tricky

$$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$ I ran across an integral I am having a time solving. The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
12
votes
2answers
12k views

How does a calculator calculate the sine, cosine ,tangent using just a number?

Sine Θ = oposite/hypotenuse Cosine Θ = adjacent/hypotenuse Tangent Θ = oposite/adjacent So in order to calculate the Sine or the cosine or the tangent I need to ...
9
votes
3answers
260 views

How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$

Is there an easy way to prove the identity? $$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3} \right ...
8
votes
2answers
222 views

Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$

My Question is: Examine the continuity of $$f(x) = \lim_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \qquad \text{for} \ x \in \mathbb{R}$$ How can I do this? Honestly, speaking I have $\text{no ...
7
votes
1answer
182 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-mx+N = ...
6
votes
2answers
327 views

Reference for a tangent squared sum identity

Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$ \sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+. $$ I ...
5
votes
4answers
218 views

Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$

I have a problem with this integral. It seems that solution has to be simple, but I couldn't find out. $$I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$$ I tried using integration by parts and ...
4
votes
4answers
234 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 ...
4
votes
2answers
2k views

Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area

In other words, if I had a rectangle of $10\times 10$ and an angle of $45$, and the allowed area was $100\times 100$, the rectangle would be about $70\times 70$. The allowed area is $100\times 100$ ...
3
votes
4answers
2k views

Memorizing the unit circle?

I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the ...
1
vote
2answers
223 views

Finding an unknown angle

Geometry: Auxiliary Lines As shown in the figure:
12
votes
2answers
504 views

Evaluating $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

I have to evaluate: $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx $$ I can't get the right answer! So please help me out!
11
votes
4answers
863 views

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$ I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
11
votes
6answers
734 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
11
votes
5answers
746 views

Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$

I am trying to study for a test and the teacher suggest we memorize $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, $\cos (A-B)$, and then be able to derive the rest out of those. I have no idea how to get ...
10
votes
3answers
9k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
9
votes
4answers
1k views

Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...
7
votes
1answer
227 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
6
votes
3answers
916 views

Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$

I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me: Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers ...
5
votes
1answer
2k views

Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$ [duplicate]

For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
4
votes
2answers
184 views

proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$

How can I prove the following identity using complex variables $$ \begin{align*} 1) & \csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8 ...
4
votes
7answers
4k views

Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$

I've tried following this way, but I haven't succeeded. Thank you!
4
votes
4answers
1k views

Resizing a rectangle to always fit into its unrotated space

(For those coming here looking for answers to rectangle problems it may help to see the related (and solved) question: Given a width, height and angle of a rectangle, and an allowed final size, ...
3
votes
3answers
134 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
3
votes
4answers
243 views

Finding $\tan t$ if $t=\sum\tan^{-1}(1/2t^2)$

I am solving this problem. If $$\sum\limits_{i=1}^{\infty} \tan^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t$$ then find the value of $\tan{t}$. My solution is like the following: I can rewrite: ...
2
votes
3answers
172 views

Finding the exact value of $\tan(\pi/5)$

Hi, I realise there has been a question already asked regarding this particular exact value, but this question requires for it to be done under different conditions, which is the part I require help ...
2
votes
4answers
491 views

Proof of $\arctan{2} = \pi/2 -\arctan{1/2}$

I am studying undergraduate complex analysis, and in my Textbook the author claimed that $$\arctan(2)=\frac{\pi}{2}-\arctan\left(\frac{1}{2}\right)$$ when he was doing an example regarding to ...
2
votes
1answer
119 views

Is there a way to solve for $x$ in $\cos^{-1}(ax) / \cos^{-1}(bx) = c$?

Is there a way to solve for $x$ in $\dfrac{\cos^{-1}(ax)}{\cos^{-1}(bx)} = c$? I guess it comes down to, are there any sine multiplication formulas I don't know about? The motivation for this is ...
1
vote
0answers
126 views

Trigonometric curiosity

How prove this $$-\tan\frac{10\pi}{41}+4\left(\sin\frac{2\pi}{41}+\sin\frac{4\pi}{41}+\sin\frac{12\pi}{41}+\sin\frac{20\pi}{41}-\sin\frac{26\pi}{41}-\sin \frac{30\pi}{41}\right)= ...
1
vote
4answers
403 views

Difficult trigonometric equation

Please, can you suggest something for solving this equation: I have to find the solutions included in interval $\left[3\pi/2, 2\pi\right]$: $$\sqrt{3\cos^2 x - \sin 2x} = - \sin x$$ This is what I ...
0
votes
2answers
229 views

Why does $\sin^{-1}(\sin(\pi))$ not equal $\pi$

And when does $\sin^{-1}(\sin(x)) = x$
13
votes
2answers
1k views

Is sin(x) necessarily irrational where x is rational?

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 $x$ ...
11
votes
3answers
212 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
9
votes
1answer
315 views

Calculating $\sin(10^\circ)$ with a geometric method

Excuse me if this is a simple question: What is a simple geometric method for calculating $\sin(10^\circ)$ using only the sines of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$? Generally, is ...
7
votes
2answers
227 views

Is $f(x)=10$ a periodic function?

I am not getting satisficatory explanation for this. Clearly $f(x+T) = f(x)$ for all values of $T$. If we assume it is periodic, does this mean period = $0$?
7
votes
1answer
274 views

Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$

I would like to show that: $$ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$$ We have: $$ \sum_{n=1}^N \arctan \left( \frac{2}{n^2} \right) =\sum_{n=1}^N \arctan ...
7
votes
4answers
467 views

Intuition for graphing Sine/Cosine

So I'm working my way through some basic trig (Khan Academy) - I'm trying to get a better intuition for what graphs of sine and cosine represent. I've seen the nice unit circle animations that do ...
7
votes
5answers
3k views

Solve $\sin x = 1 - x$

How would you be solve sin x = 1 - x, without drawing the graph and manually measuring the point of intersection?
6
votes
2answers
75 views

Another trigonometric proof…?

...sigh..another problem how shall I prove the following? $$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$ so what now? the following's what I've done: $$\cot A - \cot^2 A ...
6
votes
2answers
623 views

Method to find $\sin (2\pi/7)$

I just thought a way to find $\sin\frac{2π}{7}$. Considering the equation $x^7=1$ $⇒(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$ $⇒(x-1)[(x+\frac1 x)^3+(x+\frac1 x)^2-2(x+\frac1 x)-1]=0$ We can then get the ...
6
votes
8answers
3k views

Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$?

$$\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$$ Is there any formula that tells this or why is it like that?
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...
5
votes
2answers
563 views

Proof about $z\cot z=1-2\sum_{k\ge1}z^2/(k^2\pi^2-z^2)$

In Concrete Mathematics, it is said that $$z\cot z=1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}\tag1$$ and proved in EXERCISE 6.73 $$z\cot z=\frac z{2^n}\cot\frac z{2^n}-\frac z{2^n}\tan\frac ...
4
votes
2answers
189 views

Solve the following definite integral: $\int_{0}^{\infty}\frac{x^2dx}{({1-x^2})^2}$

Solve the following integral: $$\int_{0}^{∞}\frac{x^2dx}{({1-x^2})^2}$$ I know that substituting some trigonometric functions may help. But I was not able to solve. Can you give me some ...
4
votes
1answer
311 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
4
votes
3answers
1k views

How do we find specific values of sin and cos given the series definition

$\exp:x\mapsto \sum\limits_{n=0}^{+\infty}\cfrac{1}{n!}x^n$ $\cos:x\mapsto \Re\left(\exp \left(i x\right)\right)=\sum\limits_{n=0}^{+\infty}\cfrac{\left(-1\right)^n}{\left(2n\right)!}x^{2n}$ ...
4
votes
4answers
271 views

How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$?

How do I solve for $x$ in $$x\left(x^3+\sin(x)\cos(x)\right)-\big(\sin(x)\big)^2=0$$ I hate when I find something that looks simple, that I should know how to do, but it holds me up. I could ...
4
votes
4answers
903 views

Evaluate $\tan ^{2}20^{\circ}+\tan ^{2}40^{\circ}+\tan ^{2}80^{\circ}$

Evaluate $\tan ^{2}20^{\circ}+\tan ^{2}40^{\circ}+\tan ^{2}80^{\circ}$. Can anyone help me with this? Thank You!