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

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5
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4answers
632 views

How do I find the exact value of $\cos\frac{\pi}{12}\cos\frac{5\pi}{12}\cos\frac{7\pi}{12}\cos\frac{11\pi}{12}$?

I know that $\cos(6\phi)\equiv32c^6-48c^4+18c^2-1$ where $c=\cos\phi$. I also know that when $\cos(6\phi)=0$, then $\phi=\frac{k\pi}{12}$ ($k = 1,3,5,7,9,11$). How do I find the exact value of: ...
1
vote
1answer
268 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
1
vote
5answers
156 views

How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?

$e^{ix}=\cos(x)+i\sin(x)$ $Ae^{4ix}=A(\cos(4x)+i\sin(4x))$ $Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$ What am I doing wrong? I am trying to find the complimentary function of $\frac{d^2y}{dx^2} ...
0
votes
3answers
82 views

Trigonometry - Addition Theorem Finding Another Trig function

Using the expansion of $\sin(A + B)$, prove that $\tan 75^\circ = 2 + \sqrt 3$
1
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1answer
1k views

How do I input these interval equation in to wolfram to show me the solution

How do I enter these equation into wolfram to show me the interval type equation which gives the solution as exact values.. Question 1 : $2\tan x\cos x = \tan x$ where $0\leq x< 2\pi$ Question 2: ...
3
votes
2answers
102 views

What stops me from making this conclusion?

Suppose I want to find $\sin^6x+\cos^6x$. What stops me from saying that $\sin^2t=\sin^6x$, and $\cos^2t=\cos^6x$? Of course this is wrong because $\sin^2t+\cos^2t=1$ and $\sin^6x+\cos^6x$ does not ...
2
votes
2answers
272 views

Determine the general solution for $2\cos 2x−5\cos x+2=0$

Determine the general solution for $2\cos 2x−5\cos x +2=0$ my answer I got was : $1.05+n\pi, 4.19+n\pi$
4
votes
6answers
332 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
3answers
196 views

How to find $(2-\sec^2 1^\circ)(2-\sec^2 2^\circ)\cdots \overline{(2-\sec^2 45^\circ)}\cdots(2-\sec^2 89^\circ)$

Evaluate $$(2-\sec^2{1^{\circ}})(2-\sec^2{2^{\circ}})(2-\sec^2{3^{\circ}})\cdots(2-\sec^2{44^{\circ}})(2-\sec^2{46^{\circ}})\cdots(2-\sec^2{89^{\circ}})$$ This same problem come from Problem 21:But ...
3
votes
1answer
56 views

What am I doing wrong with solving $2\tanh^2x-\text{sech}~x=1$?

$2\tanh^2(x)-\text{sech}(x)=1$ $\tanh^2(x)=1-\text{sech}^2(x)$ $2(1-\text{sech}^2(x))-\text{sech}(x)=1$ $2\text{sech}^2(x)+\text{sech}(x)-1=0$ $\text{sech}(x)=\frac{1}{2} $ Not possible. And ...
1
vote
4answers
115 views

The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$

There is an exercise on $\sin x$. How could I see that for any $0<x< \frac \pi 2$, $\frac 2 \pi x \le \sin x\le x$? Thanks for your help.
16
votes
2answers
805 views

Adriaan van Roomen's 45th degree equation in 1593

Adriaan van Roomen proposed a 45th degree equation in 1593(see this book, picture reference as follows): $$ \begin{gathered} f(x) = x^{45} - 45x^{43} + 945x^{41} - 12300x^{39} + 111150x^{37} - ...
-2
votes
3answers
187 views

Calculate cosh 1 correct to 6 decimal places.

Can anyone help me with this problem? Calculate $\cosh 1$ correct to 6 decimal places.
5
votes
1answer
178 views

Determine the general solution for $2\cos^2x-5\cos x+2=0$

Determine the general solution for $ 2\cos^{2}x-5\cos x+2=0$ My attempt: $2u^2 - 5u + 2 = 0$ $(2u - 1)(u - 2) = 0$ $u = \frac{1}{2}$ or $u = 2$ $\cos(x) = \frac{1}{2}$ or $\cos(x) = 2$ The ...
2
votes
1answer
204 views

A cosine function has maximum value of 14 and a minmum value of 4, a period of 7, and a phase shift of 12.

A cosine function has a maximum value of 14 and a minimum value of 4, a period of 7, and a phase shift of 12. Write an equation representing this cosine function... Could someone tell me if I'am ...
1
vote
2answers
319 views

Numerical method to solve a trigonometric (cotangent) function - transient heat transfer problem

I was trying to develop a mathematical model for transient one-dimensional heat conduction of spheres using approximate analytical solution as mentioned in Cengel{refer page number 229 in that ...
12
votes
3answers
182 views

Can all points in the plane be represented like this?

Solving a task regaring affine geometry, I've come across a problem: Is it true that, for every point $(x,y)\in \mathbb{R}^2$, there exist $t\in \mathbb{R}, \alpha\in[0,2\pi]$, such that $$x = ...
3
votes
3answers
884 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...
0
votes
3answers
413 views

Solve system of equations involving cos and sin

I have come up with the following system, I want to solve it for $a$ and $c$: $ a \sin (x_0) - c \sin(x_0 - L) = 0\\ c \cos(x_0 - L) - a \cos(x_0) = 1 $ In this system $x_0$ and $L$ are arbitrary. ...
0
votes
1answer
1k views

Determine possible coordinates for point $P$ on the terminal arm of angle

a) If angle $\theta\\$ lies in Quadrant II and $\sin \theta ={3 \over {\sqrt {45} }}$. Determine possible coordinates for point $P$ on the terminal arm of angle $\theta$. b) Determine the Quadrant ...
3
votes
3answers
64 views

Simple trigonometry question

I am just wondering how can you get from $\cos(\pi t)=1/2 $ or $\cos(\pi t)=-1$ for $0<t<4$ to t = 1/3, 1 t = 5/3, 7/3, 11/3 ,3 I got $(\pi t) = \pi/3 +2k\pi $ and $(\pi t) = ...
4
votes
2answers
472 views

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

Calculate : $$ \sum_{k=1}^{n-1} \sin \left( \frac{(2\lfloor \sqrt{kn} \rfloor +1)\pi}{2n} \right).$$
0
votes
2answers
322 views

Get $x,y$ coord of a rectangle when rotation is $0$

I have this square div in the page. I need to get the $x,y$ coords (left/top) of the box based on rotation 0 even when the box is rotated at a different angle. Is there a formula to calculate the the ...
1
vote
0answers
122 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
1
vote
1answer
308 views

New x coordinate of a rotated line

I need help finding the equation to find $x$ I work in GIS and I'm working on a script that uses the new x coordinate of a rotated line. I havent work with trigonometry in a long long time so I ...
18
votes
2answers
397 views

Angular distribution of lines passing through two squares.

Let's say I've got two squares with side length $d$ that are held parallel at a distance $m$ apart. Suppose that particles are randomly falling from above in such a way that the polar angle ...
4
votes
4answers
134 views

Explanation of where this trig identity comes from

I'm working on a problem but it's been a while since I last saw trig identities so I'd love some help or being pointed in the right direction. Basically, I'd like to understand where this identity ...
0
votes
1answer
119 views

Location of roots of certain transcendental equations.

If $a$ and $b$ are two solutions of $\,e^x \cos x -1=0$, then how many solutions of the equation $ e^x \sin x-1=0$ lie between $a$ and $b$ ?
1
vote
1answer
1k views

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side.

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side. The direction of rotation is counter clockwise. The bottom of the Ferris wheel is 2 m off the ground. It rotates ...
4
votes
2answers
221 views

Proving $f(x) := \sin(ax) - \sin(bx)$ has absolute value greater or equal to 1 for some $x>0$

I am trying to prove that for any choice of reals $a$ and $b$ such that $a \neq b $ the function $f(x) := \sin(ax) - \sin(bx)$ has $|f(x)| \geq 1$ for some $x>0$ (This is not an exercise question ...
3
votes
7answers
874 views

period of a function $\lvert\sin x\rvert+\lvert\cos x\rvert$

I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$. But Wolfram says it is periodic with period $\pi$. Please tell what is correct.
2
votes
2answers
167 views

trigonometric representation of a complex number.

Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$. (The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where ...
15
votes
1answer
669 views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin ...
0
votes
5answers
4k views

Ratio of corresponding sides of similar triangles, given the areas.

The area of two similar triangles are 72 and 162. what is the ratio of their corresponding sides?
10
votes
3answers
980 views

Intuition of Addition Formula for Sine and Cosine

The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
2
votes
2answers
105 views

How can this property of definite integrals be true?

In this question, people are saying that the definite integral of $f(x)$ from $0$ to $a$ is equal to the integral of $f(a-x)$ from $0$ to $a$. How can that be true? Simple examples don't work.
0
votes
3answers
166 views

Finding the number of integer solutions, why is this wrong?

The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
1
vote
1answer
60 views

What do we call the angular arcs between two edges of triangles?

I've been trying to find a geometry library for java which is as high level as describing angles between adjacent sides of triangles given 3 sides. So, what do we call such kind of arcs. In many ...
0
votes
2answers
56 views

Geometric-Variational idea of Sine

I like to see sin(theta) as a property of a line making clockwise angle of theta with a horizontal axis.That property would dictate how much % the vertical component of each point in the line ...
3
votes
2answers
370 views

What is the actual geometric meaning of trigonometric operations such as adding cos,sine,tan

$$\sin(\pi/4)+\cos(\pi/4)=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}= \frac{2\sqrt{2}}{2}=\sqrt{2}$$ Thinking of trig components (cosine, sine) that I used to produce the result using the mechanics of ...
3
votes
5answers
3k views

What type of input does trigonometric functions take in

I see in my Book that 45 deg is equivalent of π/4 . Ι also do the conversion if I simply convert degrees into radians like this 45* π/180 = π/4 radians and ...
2
votes
1answer
151 views

How to compute $\int_0^\infty \frac{\sin t}{t^{s+1}} dt $?

How to compute $\displaystyle\int_0^\infty \frac{\sin t}{t^{s+1}}\;\text dt$ ? Here, the real part of the complex number $s$ is negative and greater than $-1$.
1
vote
2answers
167 views

confused about the limit of a trigonometric function

I am trying to calculate the limit of the following function for general $a$: $$\lim_{x\to a}[\cos(2 \pi x)-\sin(2 \pi x) \cot(\frac{\pi x}{a})]$$ I was believing this is infinite. But Mathematica ...
1
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1answer
70 views

Solution for Summation of $\cos^2x$

Can you give me the solution for the summation $$ \sum_{n=0}^{\infty} \cos^2(\pi n) $$ Edit: Please give me the explanation of how it is calculated and also final answer in integers.
7
votes
1answer
356 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
1
vote
0answers
101 views

Is there a continuous version of $tan^{-1}(\frac{y}{x})$ for the entire unit circle?

The fact that $tan^{-1}(\frac{y}{x})$ only "works" for the upper-right quadrant makes some calculations (for a physics simulator) impossible. I of course use $atan2(y,x)$ in the code, that's not what ...
2
votes
1answer
145 views

How find this maximum $x+y$

let $x,y\in [0,2\pi]$, and $2\sin{x}\cos{y}+\sin{x}+\sin{y}+\dfrac{1}{2}=0$,find the maximum $x+y$ my idea $$\sin{x}+\sin{y}=2\sin{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$$
1
vote
1answer
482 views

Find the terminal point when the distance is not in terms of $\pi$

From Stewart Precalculus 5th edi, P407 I am not sure what to do here, in the textbook, Steward didn't provide any example as to finding the terminal point when the distance $t$ is an integer. I ...
3
votes
3answers
239 views

Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc.

How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or ...
4
votes
5answers
706 views

How to express $2 \cos X = \sin X$ in terms of $\sin X$?

The Question was: Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only. I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed. ...