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

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1answer
18 views

Regarding Degrees and Calculation of time

How to find the Angle between the time 4.30 pm in clock, what is the Angle? please share the Answer answer like 40 degree, 60 degree etc like that
2
votes
5answers
71 views

Understanding a Cosine Derivative

Let $c$ be a constant. Why is it that $$ D_x \left(- \frac{\cos(cx)}{c} \right) = \sin(cx)? $$ I understand that $D_x \cos(x) = - \sin(x)$. So what trigonometric identity is allowing us to infer ...
0
votes
2answers
38 views

Trigo equation $3\cos^2(2x)=1+\sin x$?

How to solve $3\cos^2(2x)=1+\sin (x)$? $0\leq x \leq 360$ I've been figuring this question for a such time. My lecturer told me this was a wrong question, I didn't know why.
0
votes
0answers
14 views

Discrepancy between range of amplitude of complex number and range of $\arctan$ function

I have a conceptual doubt regarding the amplitude of complex numbers. My teacher said that in general for a complex number $x+iy$ the amplitude can be given by $\tan^{-1}({\frac{y}{x}})$ in case we ...
5
votes
1answer
108 views

Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
2
votes
1answer
25 views

Question regarding the earth's spherical geometry

Hello I have a question and it is as follows: What is a general formula to derive an angle from true north such that I know how to face a certain object assuming I know the longitude and latitude of ...
0
votes
1answer
26 views

How do I represent the graph of $\,\sin (x-2)\,$?

What does the graph of $\,\sin(x-2)\,$ look like? I have tried putting values in $x$ but its getting complicated to represent it as a graph.
1
vote
1answer
51 views

How to express $\cos(20^\circ)$ with radicals of rational numbers?

In showing that the trisection of an angle with ruler and compass is not possible in general one shows that $\cos(20^\circ)$ cannot be constructed (thus the angle $60^\circ$ cannot be trisected) by ...
1
vote
1answer
69 views

Inner Product Examples, what is the points?

Example: For $ -\pi<x<\pi$, $$x =-2 \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sin(nx)$$ and $$x^3 =-2 \sum_{n=1}^{\infty} \left( \frac{\pi^2}{n}-\frac{6}{n^3} \right)(-1)^n \sin(nx)$$ by ...
0
votes
0answers
27 views

Trigonometric identity (Backlund permutability theorem)

I have been studying Backlund transformations using Rogers and Schief, and I am now reading about the permutability theorem. I understood everything up to the very last part for the permutability. It ...
6
votes
1answer
32 views

Cosine Law Duality in Hyperbolic Trigonometry

From setting up a hyperbolic triangle with hyperbolic side length $a,b,c$ and corresponding angles $A,B,C$, it is not hard to prove the following law of cosine: $$\cos A= \frac{\cosh b \cosh c -\...
3
votes
2answers
55 views

Prove this inequality with trigonometry $9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$

let$x,y\in R$,show that $$9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$$ Maybe use Cauchy-Schwarz inequality can solve it?and I can't Adit it:I think the right hand can replace constant $9$ ...
0
votes
0answers
16 views

Fuzzy variant of cos²(x)

For a simulation, I need to find a variant fuzzy function of cos²(x) For the intented purpose, with n from 5 to 20 and L small , the function $fuzzyCos(a) = \prod_{i=1}^{n}{ cos^2( a + i \frac{L}{n} -...
1
vote
0answers
36 views

maximising sinusoidal functions

I have come across a maximisation problem that I do not know how to handle. I have posted the question here in the past. I have the following function to maximise for $x,y$ $$f(x,y)=a_1 \cos(x) +b_1 ...
-5
votes
1answer
42 views

Unable to solve the question.. Help [on hold]

(Sin1)(sin3)(sin5).... =2^(-n) Value of n, where angles are in degree
4
votes
3answers
121 views

Should we re-define Sine?

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these ...
1
vote
2answers
50 views

Evaluation of $\sum^{\infty} _{n=1} \tan^{-1} \frac{4n}{n^4-2n^2+2}$

Evaluate $\sum^{\infty} _{n=1} \tan^{-1} \frac{4n}{n^4-2n^2+2}$. I know we know to convert it in the of $\tan^{-1} \frac{a-b}{1+ab}$ but I am not able to do so here. Could someone give me some hint?
3
votes
3answers
46 views

Is this a valid proof that sine is continuous at the origin?

$$ \text{Let } \left|\sin x - 0\right| < \epsilon. \\ -\epsilon < \sin x < \epsilon \\ \arcsin (-\epsilon) < x < \arcsin (\epsilon) \\ -\arcsin \epsilon < x < \arcsin \epsilon \\ \...
4
votes
4answers
99 views

Sum to infinity of trignometry inverse: $\sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right)$

If we have to find the value of the following (1) $$ \sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right) $$ I know that $$ \arctan \left(\frac{4}{r^2+3} \right)=\arctan \left(\frac{r+1}2 \right)-\...
0
votes
2answers
33 views

How to rotate a line based dimensions of a piece of paper

I have a line where I know the start and end point on a piece of paper with the dimensions of 8 1/2 inches x 11 inches. the start point is 5.6 inches from the right of the paper and 4 inches down ...
1
vote
0answers
47 views

Find the length of the side of a right angle triangle inside a circle

Hello Stack Exchange. I have a question which has really been preventing me from making a certain program.In my program I need to find the length of AC using only AB and BD.The triangle is right-...
2
votes
3answers
50 views

How does one plug radicals with non-perfect squares and variables into the Pythagorean theorem formula?

I am working on the following integral $$\int\left( 7x^2 - 3 \right)^{\frac 5 2} \, dx$$ I want to use the $\sqrt{u^2 - a^2}$ $u = a\sec\theta$ I know in order to get it into the form that will ...
-2
votes
2answers
30 views

Find $a$ and $b$ in the given equation below [on hold]

$$a\leq \frac{\sec^{2} \theta–\tan \theta}{\sec^{2} \theta+\tan \theta} \leq b$$ I'm confused. How do I do this? Please help.
-6
votes
1answer
51 views

Evaluate the following: [on hold]

$$ 1) \cos 2 \theta + \cos 2 \phi $$ and $$ 2) \sin(\theta + \phi) $$ If $$ \sin\theta + \sin\phi = a $$ and $$ \cos \theta + \cos\phi = b $$ please provide a detailed solution. I could not ...
-2
votes
1answer
49 views

Prove the following relation [on hold]

$$ (ax)^\frac{2}{3} + (by)^ \frac{2}{3} =(a^2 - b^2)^ \frac{2}{3} $$ if, $$\frac{ax}{\cos\theta} + \frac{by}{\sin\theta} = a^2 - b^2$$ and $$\frac{ax \sin\theta}{\cos^2 \theta} - \frac{by \cos\theta}{\...
-2
votes
2answers
26 views

Find values of $m$ and $n$ such that $m \leqslant 6 \sin x+ \cos (2x) -1\leqslant n$ [on hold]

$$m \leqslant 6 \sin x+ \cos (2x) -1 \leqslant n$$ I have no clue how to do it. please help.
1
vote
1answer
39 views

Prove $\tan(A+B)$ using $\cos(A-B)$ and $\sin(A-B)$

Use $\cos(A-B)$ and $\sin(A-B)$ to prove $$\tan(A+B)=\frac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}$$ It seems like we cannot simply change $A+B$ to $A+(-B)$ to prove it? Any ideas?
2
votes
12answers
141 views

Solve $\sec (x) + \tan (x) = 4$

$$\sec{x}+\tan{x}=4$$ Find $x$ for $0<x<2\pi$. Eventually I get $$\cos x=\frac{8}{17}$$ $$x=61.9^{\circ}$$ The answer I obtained is the only answer, another respective value of $x$ in $4$-th ...
-2
votes
1answer
41 views

$\cos^2 70^\circ + \cos 25^\circ.\sin 25^\circ$ [on hold]

Find the value of :- $\cos^2 70^\circ + \cos 25^\circ.\sin 25^\circ$ without using trigonometric table . Formulas may be used :- $\cos (180-A ) = -\cos A$ $\sin ( 180-A ) = \sin A $ $\sin 2A ...
9
votes
5answers
843 views

APICS Mathematics Contest 1999: Prove $\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$ is a constant function of $x$

This is question 3 from the APICS Mathematics Competition paper of 1999: Prove that $$\sin^2(x+\alpha)+\sin^2(x+\beta)-2\cos(\alpha-\beta)\sin(x+\alpha)\sin(x+\beta)$$ is a constant function of $x$...
0
votes
1answer
28 views

Trigonometric Equation $\tan x+\cot x=8\cos2x$

$$\tan{x}+\cot{x}=8\cos{2x}$$ How to solve it with the simplest way? I managed to solve by changing both the $\tan{x}$ and $\cot{x}$ into $\cfrac{\sin{2x}}{1-\cos{2x}}$ and $\cfrac {1-\cos{2x}}{\sin{...
2
votes
2answers
35 views

proving that triangles $ABC$, $A'B'C'$ are congruent

Given $AD$ is a median to $BC$ in triangle $ABC$, and $A'D'$ is a median to $B'C'$ in triangle $A'B'C'$, and $AD=A'D', AC=A'C', AB=A'B'$. How can i prove that triangles $ABC$, $A'B'C'$ are congruent? ...
1
vote
1answer
22 views

Equation of a circle in polar coordinates under a linear transformation

Let's say we translate a circle with origin $(0,0)$ on the x axis by some constant $c$. What would the new equation of the circle be in polar coordinates? I have tried subbing in the equation of the ...
0
votes
0answers
12 views

Nonlinear odd real sinusoidal functions

I need a class of odd nonlinear sinusoidal functions whose graphs are given here: I got some example functions: 1) $x = \cfrac{x_{\max}}{2}\times\sin(\cfrac{\pi y}{y_{\max}})$ where $x_{max}$ and $...
0
votes
0answers
33 views

Alternative derivation of Euler's product formula for sine

Euler's product formula states that: $$\sin(x)=x\prod_{n=1}^{\infty}\left[1-\frac{x^2}{\pi^2n^2} \right].$$ There is also a very simple formula for another product representation for the sine ...
0
votes
1answer
34 views

Proving $\cos(a+b)=\cos a\cdot\cos b - \sin a\cdot\sin b$ [on hold]

I would like to prove the following $$\cos(a+b)=\cos a\cdot\cos b - \sin a\cdot\sin b.$$
0
votes
2answers
39 views

Deduce the relation from the given trigonometric relation

If $$\frac{\tan3A}{\tan A}=k$$ Then prove that $$\frac{\sin3A}{\sin A} = \frac{2k}{k-1}$$ I tried this, $$ \tan3A = \frac{3\tan A-\tan^3 A}{1-3\tan^2 A}$$ then divided by $\tan A$ on both sides ...
0
votes
2answers
46 views

prove the following relation,

If $$ xy + yz + zx = 1 $$ then show that, $$\frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} = \frac{4xyz}{(1-x^2)(1-y^2)(1-z^2)}$$ I have tried multiplying all three terms on the left side, and ...
0
votes
2answers
38 views

How to calculate A in sin Ax if sin x = sin Ax? [on hold]

if sin x = sin ax, is a 180/pi or pi/270, or 270/pi, or pi/180. How do I calculate the value of a? Arun
2
votes
0answers
35 views

Names for related pairs of angles

I seek the names (if they exist) of two relationships between angles. Two angles are complements of each other if they add up to a quarter circle. $\sin\alpha=\cos\beta$ and vice versa. Two angles ...
-1
votes
1answer
33 views

Value of a product of cosines and the floor of its reciprocal

$$ \mbox{The question states}\quad {a \over b} =\prod_{n = 1 \atop{\vphantom{\LARGE A}n \not= 9}}^{17}\cos\left(n\pi \over 18\right) $$ $$\mbox{And it is also provided that}\quad \left\lfloor{b \over ...
0
votes
1answer
26 views

Overlapping area of two circle's crossing it's center i.e., length of overlapping is greater than r of the circle. Circle's has equal area.

Let there be two circular coasters of equal area (and negligible height). The purpose of is to find how far the two coasters need to be moved on top of each other such that the area of the overlapping ...
-2
votes
4answers
55 views

Limit to infinity of trigonometry

\begin{align*}\lim_{n\rightarrow \infty}\frac{n\left(\left(1-\cos^2\frac{16}{n}\right)\sin\frac{16}{n}\right)^{1/3}}{4}=\lim_{n\rightarrow\infty}\frac{n\left(\sin^2\frac{16}{n}\sin\frac{16}{n}\right)^{...
-4
votes
2answers
182 views

Find the integer solutions of $\sin\frac \pi {2n} + \cos\frac \pi {2n} = \frac{\sqrt n} 2$ [on hold]

Let $n$ be a fixed positive integer such that $$\sin\dfrac \pi {2n} + \cos\dfrac \pi {2n} = \dfrac{\sqrt n} 2$$ then find the value of $n$. I have no clue how to do this sum. I couldn't even try it.
-4
votes
3answers
89 views

How to maximize $\cos\theta$?

I have a question about maximizing $\cos\theta$. I have the equation $y=H\cos\theta$, where $H$ is the fixed height of a triangle. The problem asks me to maximize $\cos\theta$, but I have no idea ...
7
votes
5answers
472 views

Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$

Find the value of $$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$ My attempt I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\...
-1
votes
1answer
28 views

How to prove such a hyperbolic sine cosine related equality? [on hold]

$$\ln \left(\frac{\left(1+\sqrt{5}\right)^2 \left(2+\sqrt{5}\right)}{4}\right)=\text{arcsinh }(2)+2 \text{ arccsch }(2)$$
1
vote
1answer
39 views

Can Someone help me with my trigonometry rotation, formula? [on hold]

I've been working on some code for a game to make a hit box, this question is just about the math though. Basically I'm trying to rotate an X, Y point(i guess according to the game it's Z,X Not sure ...
0
votes
1answer
60 views

Help simplifying $\sum_{n=0}^\infty \cos(n\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$

In a proof of $\sum_{n=0}^\infty \cos(n\theta)=\frac{1}{2}+\frac{\sin[(n+\frac{1}{2})\theta]}{2\sin(\theta/2)}$ I need help figuring out the identity used to simplify from red $ \color{red}{1}$ to $\...
0
votes
1answer
25 views

Inverse Trigonometric piece-wise functions

I was solving the equation $$2\tan^{-1}(2x-1)=\cos^{-1}x$$ Now while solving the question, the author of the book has written only the first case in the solutions manual. CASE I $2x-1 \ge 0$ $\...