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 (1)

0
votes
1answer
788 views

Finding a coterminal angle to $(13\pi)/7$ between 0 and $2\pi$

I need to find a coterminal angle to $(13\pi)/7$ between $0$ and $2\pi$. I'm not sure how to approach this problem as adding $2\pi$ would put me over the domain and subtracting $2\pi$ would put me ...
0
votes
1answer
191 views

Definite Integral with trigonometric identity answer

How does integral of $(\sin x)^2 (\cos x)^3 = {1\over5}(\sin x)^5 - {1\over3}(\sin x)^3$ manage to turn into ${1\over30}(\sin x)^3 (3\cos(2x)+7)$?
2
votes
2answers
3k views

Finding a side given 2 angles and a side (and rationalizing a denominator afterwards)

(In advanced, I apologize for not knowing how to make fractions) Here's the problem: A triangle has side $c = 8$ and angles $A = \pi/4$ and $B = \pi/6$. Find the length of the side opposite $A$. ...
0
votes
2answers
917 views

Finding all Trigonometric Solutions of an Equation within a Given Interval

My question is: How do I find all solutions in the interval $[0, 2\pi)$ of the equation $\cos 2x - \sin x = 1$? Any pointers into the direction I should be taking would be very helpful. Thank you in ...
2
votes
1answer
632 views

Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$

A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real. the question paper and ...
1
vote
1answer
18k views

Calculate width and height of a rectangle, given its diagonal and ratio

Well, I know, it's easy. We did it in class some time ago and I forgot it, I'm stupid because I can't figure it out: E.g. I have a 32" TV with 16:9 ratio and I want to know its width and height. I'd ...
2
votes
2answers
107 views

Compute $\cos'(0)$

I know that $\cos'(0)$ is $0$, but my work follows: $$\begin{align}\cos'(0) &= \lim_{\Delta x\to 0}\frac{f(0+\Delta x) - f(0)}{\Delta x}\\ &= \frac{\cos(0+\Delta x) - \cos(0)}{\Delta ...
5
votes
2answers
776 views

Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$

how to prove $\sup \{ \sin n \mid n\in \mathbb N \} =1$
3
votes
3answers
729 views

How to create 2x2 matrix to rotate vector to right side?

I have vector u=(x,y) and i need to create matrix M: M*u=(1,0). But that matrix has to rotate vector, instead of keep and ...
2
votes
1answer
87 views

Constructable Trigonometric Inverses

By doing some right triangle gymnastics, we can derive things like $\cos(\arctan x) = \frac{1}{\sqrt{1+x^2}}$, for $x>0$ $\cos(\arcsin x) = \sqrt{1-x^2}$ $\tan(\arcsin x) = ...
0
votes
1answer
237 views

Line Rotation Problems in Trigonometry

I have 2 lines that i draw like this: ...
1
vote
1answer
175 views

Calculate angle of a line at the point it intersects a circle

Given a line with end points $(x_1,y_1)$ $(x_2,y_2)$ and a circle centered at $(x_1,y_1)$ how do I calculate the angle of the line (in degrees) as it relates to the circle? If that doesn't make sense ...
7
votes
2answers
244 views

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities?

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities? In functions like $\tan^{-1}(a \tan(x))$, the angle wraps and the result is discontinuous. Is it legitimate to redefine the ...
4
votes
2answers
435 views

Proving $\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1$

I was stumped by another past-year question: In $\triangle ABC$, prove that $$\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1.$$ Here's what I have done so far: I tried to replace $C$, using ...
5
votes
1answer
99 views

Some trigonometric formula

How to prove that $1+2(\cos a)(\cos b)(\cos c)-\cos^2 a-\cos^2 b-\cos^2 c=4 (\sin p)(\sin q) (\sin r)(\sin s)$, where $p=\frac{1}{2}(-a+b+c)$, $q=\frac{1}{2}(a-b+c)$, $r=\frac{1}{2}(a+b-c)$, ...
3
votes
2answers
238 views

Weird derivative of $\tan^{-1} x$

I've seen this in Stewart calculus book: $$\frac{\mathrm d \tan^{-1} x}{\mathrm dx} = \frac1{1+x^2}$$ But how do I get it? If I do it myself, $$\frac{\mathrm d \tan^{-1} x}{\mathrm dx} = ...
0
votes
1answer
8k views

How to find the Period and Phase angle?

I'm currently brushing up my trig and found these two problems. I'm totally clueless on how to start. Please help. Find the period , amplitude , and phase angle, and use these to sketch a) ...
1
vote
1answer
156 views

2 solutions for, solve $\cos x = -1/2$? Answer sheet displays only one, does this mean there is only one?

$\cos x = -1/2$ can occur in quadrants 2 or 3, that gives it 2 answers, however the answer sheet only shows one. Does this mean im doing something completely wrong, or are they just not showing the ...
6
votes
1answer
347 views

Proving that if $\cos{x} = \cos{y}$ and $\sin{x} = \sin{y}$ then $x-y = 2\pi n$ for some $n\in \mathbb{Z}$

I was solving some exercises in complex analysis in preparation for a qualifying exam, and I came across a problem which asked me to prove that if $x, y \in \mathbb{R}$ then $$ e^{ix} = ...
0
votes
2answers
204 views

Solving $\sin{(\cdot)}, \cos{(\cdot)}, \tan{(\cdot)}, \cot{(\cdot)} \dots$ without a calculator.

For example solve $\cos{\left(\frac{5\pi}{4}\right)}$ without a calculator or solve $\cos{(x)} = -\frac{1}{2}$. I remember vaguely that the method involves referring to a triangle, but im not sure. ...
15
votes
3answers
434 views

Are there any natural occurrences of taking a trig function of a trig function?

In devising challenging exercises for my students, I am tempted to give them something like $\cos(3\sin(4))$, but then I get to wondering whether such a calculation would ever be encountered in ...
2
votes
2answers
134 views

Finding: $\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$

How to find: $$\lim_{x \to \frac{\pi}{2}} \frac{\tan{2x}}{x - \frac{\pi}{2}}$$ I know that $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^{2}\theta}$ but don't know how to apply it here.
0
votes
2answers
5k views

Deriving 2D Coordinate Rotation Formula

I'm trying to write out the steps in code for deriving the 2D coordinate rotation formula so I can understand it. ...
0
votes
0answers
77 views

Counteract preceding rotations

In a situation where I have two axis adjacent back to back (let's say a robotic arm) I can sometimes perform two rotations ($R_1, R_2$) such that the resulting position and direction is unchanged. ...
2
votes
4answers
1k views

Best way to find the Coordinates of a Point on a Line-Segment a specified Distance Away from another Point

I have 4 points: $Q, R, S, T$. I know the following Coordinates for $R$, $T$, and $S$; Length of $\overline{RQ}$ That segment $\overline{RT} < \overline{RQ} < \overline{RS}$; I need to ...
2
votes
2answers
268 views

Simplify using the Tangent Difference identity

I solved a problem to the point that I know the answer is $$\frac{2nr}{\tan\left( \frac{(n-2)\pi}{2n} \right)}$$ The question tells me that the answer is going to be ...
3
votes
0answers
114 views

Solving a linear trigonometric equation

Let $n$ be a natural number. For $a_i,\omega_i,\varphi_i \in \mathbb{R}$ how can one find solutions $x \in \mathbb{R}$ for the equation: $$\sum_{i=1}^n a_i \cos( \omega_i \cdot (x-\varphi_i)) = 0$$
3
votes
1answer
708 views

Integral of exponential function with trigonometric identities

I need help in solving the following definite integral. I could not find any example like this $$\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; ...
3
votes
3answers
1k views

Canonical to Parametric, Ellipse Equation

I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things. $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + ...
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, ...
6
votes
1answer
223 views

Trigonometric system

I would like to solve: $ x +y+z=\frac{11\pi}{6} $ $ \sin(x)+\sin(y)+\sin(z)= \frac{\sqrt{3}}{2} $ $ \cos(x)+\cos(y)+\cos(z)=\frac{1}{2} $ After eliminating $ z $ I get: $ ...
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 $100\times 100$ ...
2
votes
3answers
635 views

Limits with trig functions

I've been slacking off in calculus and I honestly don't know what to do with this problem. I got that it's in the indeterminate form, but I have no clue where to go from there. $$ \lim_{x \to 0} ...
33
votes
3answers
1k views

Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}\ ...
0
votes
1answer
362 views

I have equations for getting x,y,z given latitude, longitude, and altitude. How do I reverse them?

I am using equations that look like the following to get x, y, and z given latitude, longitude, and altitude. ...
2
votes
1answer
2k views

Find angle between two lines

I have two lines. I have each X,Y. I want to find the angle between them (please mark if your method returns radians or degrees) Aka a function F[x,y,x2,y2,Cx,Cy] that'll return their angle (at ...
7
votes
1answer
664 views

Trying to get the infinite product for $\sin x$

I start with the fact that the zeros of $\sin x$ are $ n\pi$, $n\in\mathbb{Z}$. Therefore, it should be possible to express it as an infinite product: $$\sin x = x ...
1
vote
2answers
155 views

Integration - Primitives - Antiderivatives

Please help to calculate: $$\int\sqrt {{r}^{2}-{x}^{2}}{dx},\quad x\in[0,r]$$ Do any method of trigonometric substitution? Thanks.
1
vote
1answer
305 views

Trigonometry, some true or false tasks about cosine-rule and sine-rule

I don't understand why the answer is as it says in the book. Let me write out the task first. A If the cosine-rule can be used to find an angle in a triangle, it is only one angle that fits this ...
2
votes
2answers
122 views

Prove that $\vert\sin(x)\vert + \vert\sin(x-1)\vert \ge \sin(1)$

While looking into the convergence of the series $\sum_{n=1}^{\infty}\frac{\sin(n)}{n}$ I stumbled into the inequality $\vert\sin(n)\vert + \vert\sin(n-1)\vert \ge \sin(1)$ for all $n\in\mathbb{R}$. ...
7
votes
3answers
688 views

What numerical methods can solve $\sin(x) + \sin(y) = \sin(xy)$

Here is a nice graph representing the solution: wolframalpha. I wish to draw such a graph myself but don't have any idea which methods exist and which of them are more appropriate for equations of ...
1
vote
1answer
866 views

Find the maximum area possible of equilateral triangle that inside the given square

How can I find the maximum area possible of equilateral triangle that inside a square whose sides have length a. And how does that triangle look like? Can we construct it (with compass and ...
10
votes
3answers
1k views

Show that $A_n=\sum\limits_{k=1}^n \sin k $ is bounded?

Let $A_n=\sum\limits_{k=1}^n \sin k $ , show that there exists $M>0$ , $|A_n|<M $ for every $n$ .
11
votes
1answer
292 views

Trigonometric identity

It is well known (?) that if $\alpha+\beta+\gamma=\pi$ then $4\sin\alpha\sin\beta\sin\gamma = \sin(2\alpha)+\sin(2\beta)+\sin(2\gamma)$ (I think I've seen it in some late-19th-century books, and I ...
3
votes
2answers
212 views

Complex number + trigo : $-1 + \tan(3)i$ , find modulus and argument

I have $-1 + \tan(3)i$ and must find its modulus and its argument. I tried to solve it by myself for hours, and then I looked at the answer, but I am still confused with a part of the solution. Here ...
2
votes
2answers
57 views

Movement depending on angle value

I have an object and an angle value (clockwise) in my game. Depending on the angle, the object will move toward a certain direction. If the angle is 0, the object will move (0,1) per frame (x,y) ...
3
votes
1answer
4k 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 ...
3
votes
4answers
4k views

What is the maximum value of this trigonometric expression

What is the maximum value of the expression $1/(\sin^2 \theta + 3\sin\theta \cos\theta+ 5\cos^2 \theta$). I tried reducing the expression to $1/(1 + 3\sin\theta$ $\cos\theta + 4\cos^2 \theta)$. How ...
1
vote
1answer
372 views

Trigonometry, height of mountain based on angle difference?

The text translated: From a point A on a water we observe a antenna on a mountain. The antenna is 23m high. Find the height of the mountain. I believe the solution is in the differences between ...
3
votes
1answer
184 views

trigonometric system

In order to show that $ e^{ix}+e^{iy}+e^{iz}=0 \Longrightarrow e^{2ix}+e^{2iy}+e^{2iz}=0 $, I want to prove that $ \cos x+\cos y+\cos z=0 $ and $ \sin x+\sin y+\sin z=0 \Longrightarrow \cos 2x+\cos ...