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

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Tips for understanding the unit circle

I am having trouble grasping some of the concepts regarding the unit circle. I think I have the basics down but I do not have an intuitive sense of what is going on. Is memorizing the radian ...
4
votes
3answers
348 views

How to prove this trignometrical Identities?

The following two identities comes from my trigonometry module without any sort of proof, If $A + B + C = \pi $ then, $$\tan A + \tan B + \tan C = tan A \cdot tan B \cdot tan C$$ and, $$ \tan ...
4
votes
3answers
4k views

Find the coordinates in an isosceles triangle

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
3
votes
3answers
184 views

Computing $\cos\frac{\pi}{7}$

Assume the 7-gon below is regular. Each of the angles marked with red below is $\frac{\pi}{7}$. Troughout this question I will use $\gamma$ to mark $cos\frac{\pi}{7}$. By the cosine law we have $b = ...
3
votes
0answers
63 views

Reference request on some trigonometric identities

Reference request: Where is this trigonometric identity found? Here I will somewhat extend this earlier question (linked above) that I asked. \begin{align} \frac{a\tan\theta+b}{c\tan\theta+d} = ...
3
votes
3answers
117 views

Parametric equation - of a hyperbola

I know that the parametric equation for points on a hyperbola($\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$) is: $$x = a\sec \theta$$ $$y = b\tan \theta$$ However, what does the parameter $\theta$ actually ...
3
votes
1answer
48 views

Equation of the form $tan(\alpha)=cos(\alpha+C)$ where $C\in\mathbb{R}$

I have seen the following math problem posed online by a high school student (knowing their material, most likely it wasn't given as an exercise): Find the solutions for the equation ...
3
votes
2answers
272 views

What are some rigorous definitions for sine and cosine?

Here are some of my ideas: 1. Addition Formula: $\sin{x}$ and $\cos{x}$ are the unique functions satisfying: $\sin(x + y) = \sin x \cos y + \cos x \sin y $ $\cos(x + y) = \cos x \cos y - \sin x ...
3
votes
4answers
64 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
3
votes
3answers
263 views

Limit of $\dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ as $x \rightarrow 0$

Find $\lim_{x \to 0} \dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ I came across this limit a long time ago and could easily obtain a straightforward solution ...
3
votes
4answers
737 views

solve a trigonometric equation $\sqrt{3} \sin(x)-\cos(x)=\sqrt{2}$

$$\sqrt{3}\sin{x} - \cos{x} = \sqrt{2} $$ I think to do : $$\frac{(\sqrt{3}\sin{x} - \cos{x} = \sqrt{2})}{\sqrt{2}}$$ but i dont get anything. Or to divied by $\sqrt{3}$ : $$\frac{(\sqrt{3}\sin{x} - ...
3
votes
2answers
232 views

Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $

Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan ...
3
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3answers
129 views

To show inverse of tan x

It quite confuses me. Where do I start? Please help.
3
votes
5answers
47k views

How to find the equation of a line tangent a circle and a given point outside of the circle

I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outside ...
3
votes
6answers
399 views

How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$

Prove the following equation. \begin{eqnarray} \\\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1\\ \end{eqnarray} I can't prove it by many methods I use. Please give me some hints. Thank you ...
3
votes
2answers
537 views

If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$

Problem : If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$ Solution: Adding $ \cos x +2 \cos y+3 \cos z=0$ and $\sin x+2 ...
3
votes
4answers
1k views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
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2answers
2k views

Divergence of the sequence $\sin(n)$ [duplicate]

Possible Duplicate: Prove the divergence of the sequence $(\sin(n))_{n=1}^\infty$. How can I show that the sequence $$ a_n = \sin(n) $$ is divergent? I tried to show that $\sin(n+1) - ...
3
votes
5answers
2k views

Prove that $\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(k-1)\pi/n)=-1$

How do you prove that $$\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(n-1)\pi/n)=-1,$$ where $n \geq 2$?
3
votes
1answer
5k views

Find the area enclosed by the curve $r=2+3\cos \theta$.

the question is Find the area enclosed by the curve: $r=2+3\cos \theta$ Here's my steps: since when $r=0$, $\cos \theta=0$ or $\cos\theta =\arccos(-2/3)$. so the area of enclosed by the curve ...
3
votes
1answer
957 views

How do you find the angle of circle segment formed with points (x,y) and (radius,0)?

I've been learning about the unit circle, sine, cosine, and the like in my introduction to trigonometry course, but I'm drawing a blank here. If I have a circle centered at the origin, with radius r ...
3
votes
1answer
4k views

how to find mid point of an arc?

I have start point $(x_1,y_1)$ and an end point $(x_2,y_2)$ and radius of arc. How to calculate the co-ordinates of mid-poing of arc? The arc is the part of a circle. Known Values ...
3
votes
1answer
6k views

Calculations of angles between bonds in CH₄ (Methane) molecule

In my high school chemistry class, we talked about the angles between bonds in molecules. One that caught my attention was the CH₄ molecule. I asked my teacher how to calculate this result, he said ...
2
votes
3answers
51 views

Find all $(a,b)$, such that $y = \cos^2 x + \cos^2 (x+a) + 2\cos x\cos (x+a)\cos b$ is constant for all $x \in \mathbb{R}$.

I've tried using trig identities and $\frac{\partial y}{\partial x} = 0$ to no avail... I've proved that $\{(a,b)|a = b+\pi\}$ works, but I would like to know whether those are the only real ...
2
votes
3answers
140 views

How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$?

How to investigate the $\limsup$, the $\liminf$, the $\sup$, and especially the $\inf$ of the sequence $(\sqrt[n]{|\sin{n}|})_{n=1}^{\infty}$? Edit: The limit of this sequence is already investigated ...
2
votes
2answers
103 views

Longest pipe that fits around a corner. [duplicate]

While studying, I came upon the problem "Two corridors of widths $a$ and $b$ intersect at right angle. What is the length of the longest pipe that can be carried across the two corridors, touching the ...
2
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3answers
95 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
2
votes
4answers
124 views

Why the anti derivative of $\sec(x) \cdot \tan(x)$ is $\sec(x)$?

I have discovered that $$\sec(x) = \frac{1}{\cos(x)}$$ but I do not understand why the indefinite integral of $\sec(x) \cdot \tan(x)$ is $\sec(x)$. I am watching the following videos: ...
2
votes
1answer
105 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
2
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1answer
94 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
2
votes
4answers
99 views

Solve $\tan\left(\arccos\left(\frac{-\sqrt{2}}{2}\right)\right)$ without calculator

This is a question from the practice exercises of Barron's AP Calculus. The directions state that I cannot use a calculator. (No trig tables I think, since those are not allowed in the exam) So, ...
2
votes
2answers
192 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
2
votes
2answers
67 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
2
votes
1answer
64 views

Equilateral Triangle Problem With Trig

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle P. The distance $|AP|=3 cm, |BP|=4 cm, |CP|=5 cm.$ What is the area of the ...
2
votes
5answers
299 views

Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$

Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$ No idea how to even approach the ...
2
votes
1answer
975 views

$\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $?

How to prove, if $\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $ ? I am not familiar with trignometric identities. So any help will be appreciated. Thanks in Advance.
2
votes
2answers
93 views

On proving an identity given a system of trig equations

We are given the following: $$a^2 + b^2 + 2ab\cos\theta = 1 \tag1$$ $$d^2 + c^2 + 2cd\cos\theta = 1 \tag2$$ $$ac + bd + (ad + bc)\cos\theta = 0\tag3$$ It is required to prove that: $$a^2 + c^2 = ...
2
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2answers
928 views

Calculate coordinate of any point on triangle in 3D plane

I am really stuck and can't find right way to write a formula(s) that will calculate Z coordinate of point on triangle plane in 3D plane. I know all coordinates of triangle points ( Ax, Ay, Az, Bx, ...
2
votes
2answers
134 views

How to find the value of $4\cos(\frac{\pi}{26})+\tan(\frac{2\pi}{13})$

I have found in wolfram alpha that $\displaystyle 4\cos\left(\frac{\pi}{26}\right)+\tan\left(\frac{2\pi}{13}\right)=\sqrt{13+2\sqrt{13}}$. How to prove this identity ? Thank you.
2
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2answers
1k views

How to solve the following summation problem?

$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$ Find value of $n$ for which equation is satisfied.
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votes
0answers
95 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
2
votes
2answers
118 views

Convex Quadrilateral: $ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D $

Problem Let $ABCD$ be a convex quadrilateral with no right angles. Show that $$ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D. $$ ...
2
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0answers
105 views

conjecture regarding the cosine fixed point

context/motivation if the angle on a calculator is set to radians, then it is very easy to demonstrate that iteration of $cos x$ (for arbitrary initial x) converges - simply keep pressing the ...
2
votes
2answers
115 views

If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a\tan x +b\tan y$ equals

If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a \tan x +b \tan y$ equals ( options below ) (a) $(a+b) \cot\frac{x+y}{2}$ (b) $(a+b)\tan\frac{x+y}{2}$ (c) $(a+b)(\tan\frac{x}{2} ...
2
votes
1answer
564 views

Is sine of angles greater than 90 degrees a convention?

The sine function is defined as the opposite side of the angle in question over the hypotenuse of the $90^\circ$ triangle. $$\sin(â) = \frac{\text{opposite side}}{\text{hypotenuse}} ...
2
votes
1answer
130 views

$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$

Prove that (not use induction) $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) ...
2
votes
6answers
2k views

why is the square of this matrix with sin and cos equal to the identity matrix?

I have a question about why the square of the matrix Q, below, is equal to the identity matrix. Q = cos X -sin X sin X cos X My knowledge of ...
2
votes
1answer
56 views

Find all different integer exponents

Find all different integers that satisfy the following equality: $m(\sin^{n}x + \cos^{n} x- 1) = n(\sin^{m}x + \cos^{m}x - 1), (\forall) x\in\mathbb{R}.$ Case1: $m$ is odd, $n$ is even, then put ...
2
votes
3answers
226 views

Confusing Trigonometry Problem

Lets say at an intersection the words "STOP HERE" are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle ...
2
votes
0answers
115 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...