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

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2
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1answer
18 views

Prove in triangle :$II_1 = a\cdot \sec \frac{A}{2}$

Prove that $II_1 = a\cdot \sec \dfrac{A}{2}$. $I$ is center of incircle, $I_1$ is center of excircle. What I did is : Drop $ID \perp AB$, & $I_1F \perp AF$ at $F$ So $ID\parallel I_1F$ $\dfrac{...
0
votes
2answers
41 views

value of $\alpha$ in rational expression.

For all real values of $x\;,$ Given that $$\frac{4x^2+1}{64x^2-96x\sin \alpha+5}<\frac{1}{32}\;,$$ Then $\alpha$ lie in the interval $\bf{My\; Try::}$ We can write it as $$\frac{32(4x^2+1)-(64x^2-...
6
votes
1answer
62 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$$ I believe that we must use the Dirichlet integral $$\int_0^{\infty} {\frac{\sin{x}}{x}}\ ...
-1
votes
0answers
37 views

Trigonometry inequality 1

Let $n, r, s$ be the non negative integers such that $(\frac{1}{2})^2+(n+\frac{1}{2})^2=(r+\frac{1}{2})^2+(s+\frac{1}{2})^2$. Then prove that $\cos (\theta/2)-\cos(r+1/2)\theta\geq \cos(s+1/2)\theta- \...
1
vote
0answers
23 views

Inequality involving inverse tangent (arctan)

Suppose $-\frac{\pi}{2}+\epsilon\leq\theta\leq\frac{\pi}{2}-\epsilon$ for some $\epsilon>0$ and $x,y\in\mathbb{R}$. I am wondering whether there exists $\delta>0$ such that: $$\left|\tan^{-1}\...
0
votes
1answer
26 views

Relationships between trigonometric functions and inverse trigonometric functions

Here is a link to the Wikipedia site which has a chart of all the relationships between theta and the different trig functions: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions My ...
7
votes
2answers
122 views

What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) = \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\Big(\frac{2\pi }{7}\Big),\;x_2 =2\cos\Big(\frac{4\pi }{7}\Big),\; x_3 = 2\...
0
votes
2answers
27 views

How to prove main argument formula for any $z\in\mathbb C^*$

I would prove that for any complex number $z \in \mathbb C^*$ such that $z = x + \mathbb i y$ with $(x,y)\in\mathbb R^2$ and $x+\vert z\vert \neq 0$: $$ \arg z = 2\arctan\left(\dfrac{y}{x+\vert z\vert}...
0
votes
1answer
30 views

Trigonometric substitution [illustration / right triangle derivation]

Real quick: If I have the function $$\int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$$ I can easily substitute by setting $x$ equal to $a\sin \theta$. But why actually is that? If I draw a right ...
1
vote
1answer
21 views

Graphs of the hours of daylight in certain latitudes

Problem Attempt We are given that Graph A is North. Since South is more "dark" when North more "light" I picked the graph that is mirrored horizontally: Graph D. For East and West I can only ...
-3
votes
1answer
47 views

Prove this trigonometric inequality [closed]

We know this : $0<\theta <(\pi/2)$ How to prove this inequality : $0<\sin^6 \theta + \cos^6 \theta<1$ I used basic inequalities but I can't get answer. Please help!
3
votes
2answers
743 views

What is $\int x\tan(x)dx$?

I have a problem when trying to solve this question Question. What is the answer of the indefinite integral $$\int x\tan x \; dx?$$ Maple gives a complicated answer based on the series. Is there any ...
1
vote
2answers
828 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
1answer
1k views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which $\...
0
votes
4answers
40 views

Solve for $\theta$: $2 \sin \theta = 2 - \cos \theta$

Actually , I'm new to trigonometry .. So i want help in this question $$2 \sin \theta = 2 - \cos \theta $$ My attempt -> $$\begin{align} 2 \sin \theta &= 2 - \cos \theta \\ 2 \sin \theta &...
2
votes
2answers
32 views

Prove: $\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$, for circumradius R, inradius $r$, and exradii $r_x$ [on hold]

In $\triangle ABC$, prove: $$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$ for circumradius $R$, inradius $r$, and exradii $r_a$, $r_b$, $r_c$ in the standard ...
1
vote
4answers
44 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
1
vote
2answers
42 views

Evaluate using complex numbers: $\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$, where $m=2n+1$

Evaluate using complex numbers: $$\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$$ where $m=2n+1$. $\bf{My\; Try::}$ Let $\displaystyle P = \prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right).$ Now let $\...
0
votes
2answers
8 views

Given point before and after rotation at given axis calculate the angle of rotation

I have a point at 2d space in only positive x and y axis, point P(x1, y1) is rotated along axis point C(x3, y3) to reach at point P2(x2, y2). Now I just need to calculate the angle of rotation. If ...
0
votes
2answers
24 views

Difference between $\cos(x)$= positive and $\cos(x)$= negative

I'm confusing myself here so need some clarification. If I was to work out the solutions to $\cos(x) = 1/\sqrt2$ , I know the solutions would be $\pi/4, 2\pi - \pi/4, 2\pi + \pi/4...$ etc. What is ...
-2
votes
0answers
30 views

Area of sector.

For a circle with radius $7$ and a chord $AB=10cm$ the area of the sector $AOB$ I keep obtaining is $38.955cm^2$ while the answer in my book is $14.5cm^2$ I don't know where I keep going wrong.
-2
votes
2answers
34 views

$3A$ and $4A$ are supplementary angles…

If $3A$ and $4A$ are supplementary angles,then find the exact value of : $4\cos A - \sec 2A$ I tried breaking the angles into $ A + 2A + 4A$ and then using the formulas for angles of a triangle. ...
-1
votes
2answers
40 views

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos(2\beta)$, then prove that $\tan(\alpha)$, $\tan(\beta)$ and $\tan(\gamma)$ are in G.P.

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos(2\beta)$, then prove that $\tan(\alpha)$, $\tan(\beta)$ and $\tan(\gamma)$ are in G.P.
4
votes
1answer
114 views

If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$,show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is multiple of $\pi$.

This question is from SL Loney. If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$, then show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is a multiple of $\pi$. My try: ...
2
votes
1answer
702 views

Real life math to explore/solve [on hold]

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
1
vote
4answers
216 views

why $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?

Why does $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$? Breaking up the limit: $\lim\limits_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3 $\lim\limits_{x\to-\...
3
votes
0answers
89 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
10
votes
4answers
4k views

Solving trigonometric equations of the form $a\sin x + b\cos x = c$

Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos ...
2
votes
1answer
32 views

Is there a proof that the sum of the trihedral angles of a tetrahedron is minimal when the latter is regular?

Since the sum of the 6 dihedral angles is always 1 sphere more than the sum of the 4 trihedral angles, both sums are maximized or minimized at the same time. I showed that for all four extremal ...
-3
votes
0answers
34 views

Clock angles question. [on hold]

At 4:00 oclock, what will be the angle between the two pointers ? Adding a picture :
0
votes
2answers
45 views

How can I solve this trigonometry question? [on hold]

$ABC$ is a triangle $m( A \widehat BC) = m( A \widehat CB) + 90^\circ$ $3 \lvert AC \rvert = 7 \lvert AB \rvert$ area of the $ABC$ triangle is $4{,}2$ cm$^2$ $\lvert BC \rvert =$ ?
4
votes
2answers
85 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
2
votes
1answer
517 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
17
votes
5answers
1k views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= \sqrt[3]...
0
votes
4answers
49 views

How do I remember trigonometric angles?

I am really stuck here, I need help remembering the trigonometric angles, Can someone point me to a good place to learn them? I often confuse when trying to implement them in my problems
5
votes
3answers
134 views

Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$

How can we evaluate the following integral: $$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$ I know basically how to calculate by using the substitution $x=\tan{\theta}...
0
votes
1answer
13 views

Sin wave that has half wave length of $ a - b$

How can I make a sin wave that has double the wavelength of $a - b$, such that two consecutive zero points on the line are through a and b. And that the peak of the wave in between $a$ and $b$ is at $$...
-2
votes
0answers
47 views

how to solve $\sin ( 2 \theta)=\tan (2\theta + 2 \beta)$

if we have the equation : $$ \sin ( 2 \theta)=\tan (2\theta + 2 \beta )$$ then, what's the direct relation between $ \theta$ and $ \beta $? What do I do to solve for $ \theta$ as a function of $ \...
-1
votes
1answer
43 views

Calculating height of a viewpoint given 3 angles of depression [on hold]

Can not solve this problem I can help English Translation: From the top of a hill, a person finds the angles of depression of three consecutive kilometer markers on a straight road to be $\alpha,...
2
votes
1answer
424 views

how to solve $a\sin x+b\cos x$

Let's solve: $\sqrt{3}\sin x - \cos x=2$ The left hand side may be expressed as $R\sin(x+ \phi)$ We know that $R=\sqrt{3+1}=2$ We also know that $\tan \phi= \frac{-1}{\sqrt{3}}$ The solution to $\...
1
vote
3answers
7k views

“Show” that the direction cosines of a vector satisfies…

"Show" that the direction cosines of a vector satisfies $$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$ I am stumped on these things: "SHOW" that the direction cosines corresponds to a given ...
2
votes
2answers
57 views

Need help solving a trigonometric equation

I am preparing for finals and there is one exercise in my book that i don`t know how to solve. $$\frac{\sin a}{\sin \frac{a}{b}}=b$$ I just need to solve this for b. I tried wolfram alpha but it does ...
4
votes
4answers
42 views

Converting $\cos\phi$ into $\frac{1−t^2}{1+t^2}$, given that $t = \tan\frac{\phi}{2}$

I have to figure out the working to convert $\cos\phi$ into $\dfrac{1−t^2}{1+t^2}$, given that $t = \tan\dfrac{\phi}{2}$. It would be amazing if someone could help I've been trying to do it for ...
1
vote
3answers
1k views

Intuition around why domain of x of arcsine and arccosine is [-1;1] for “real result” & domain for arctangent is all real numbers

Context I'm working my way through basic trig (this question has a focus on inverse trig functions, specifically arcsine, arccosine and arctangent ), using Khan Academy, wikipedia and some of "trig ...
1
vote
1answer
36 views

Height of lighthouse based on angle difference

I have a question in my maths book: A lookout in a lighthouse tower can see two ships approaching the coast. Their angles of depression are 25° and 30°. If the ships are 100 m apart, show that the ...
-3
votes
0answers
22 views

Vectors applied on the arm [on hold]

I just received a maths question saying the body and its joints are subject to significant forces under load. Your challenge is to redesign a part of the body, and using vectors, explain how it could ...
2
votes
0answers
32 views

Number of solutions of some trigonometric equations

Let $N > 1$ and let $S$ be a subset of the integers in the (real) interval $[1, N]$. Can we prove that there are only finitely many solutions $x \in [1,N] \setminus \mathbb{Z}$ to the equation $$ \...
3
votes
1answer
52 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
2
votes
3answers
85 views

Find the values of $x$ such that $2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ is independent of $x$.

Find the values of $x$ such that $$2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ is independent of $x$. Checking for $x\in [-1,1]$ In the taken domain $\sin^{-1}\left(\frac{2x}{1+x^2}\...
2
votes
2answers
61 views

Simplify $\arccos\left(2\cos x\right)$.

Let $x\in[\pi/3,2\pi/3]$. We know that $\arccos (\cos x)=x$ but what we can say about $\arccos\left(2\cos x\right)$? Are there, for example, any "half-angle formula" also for inverse trigonometric ...