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

learn more… | top users | synonyms (2)

4
votes
5answers
49k 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 ...
4
votes
1answer
681 views

Why does there seem to be so much error in the laws of sines and cosines?

I've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convention ...
4
votes
1answer
146 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
4
votes
2answers
321 views

Maximum of a trigonometric function without derivatives

I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or ...
4
votes
6answers
514 views

What does $\lim\limits_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$ evaluate to?

What does $$\lim_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$$ evaluate to? This very likely needs substitution.
4
votes
2answers
494 views

Sum : $\sum \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right)$.

Calculate : $$ \sum_{k=1}^{n-1} \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right).$$
4
votes
1answer
102 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
4
votes
3answers
281 views

Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$

The first part of the question asks me to square both sides of the equation: $$3 \cos θ=2 − \sin θ$$ So that I can obtain and solve the quadratic: $$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ ...
4
votes
1answer
7k views

Intersection between a rectangle and a circle?

I have a poor working knowledge of math. I would like to calculate collision detection between a 2D circle and a 2D rectangle for a simple game of Pong. I thought of splitting the 2D rectangle into 4 ...
4
votes
3answers
651 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
4
votes
2answers
1k views

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor's Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How ...
4
votes
1answer
230 views

An unusual symmetric inequality of trigonometric functions

Given $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2 $. I have to prove that $ \left| \begin{matrix} \cos\alpha & \cos\beta & \sin\gamma\\\sin\alpha & \cos\beta & \cos\gamma\\\cos\alpha ...
4
votes
6answers
233 views

Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...
4
votes
1answer
263 views

How do you integrate $\cos(x^n)$, specifically for $n=-1$?

How does one integrate $\cos(x^{-1})$? I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put ...
4
votes
7answers
1k views

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
5k 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
1answer
76 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
3
votes
3answers
186 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
64 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
127 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
49 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
3answers
120 views

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I've been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles ...
3
votes
4answers
753 views

Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be ...
3
votes
2answers
280 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
753 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
236 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
votes
3answers
130 views

To show inverse of tan x

It quite confuses me. Where do I start? Please help.
3
votes
6answers
401 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
540 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...
3
votes
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
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
970 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
5k 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 ...
2
votes
2answers
96 views

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.
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
148 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
105 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
votes
3answers
97 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
108 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
votes
1answer
95 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
102 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
108 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
2
votes
2answers
195 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
68 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 ...