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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2answers
231 views

Simplest way to integrate this trigonometric integral:

$$\int \frac{1}{1+\tan x}dx,$$ A substitution like $t = \tan x, \;dt = (1+t^2)dx$ etc. immediately comes to mind, but I find this method a bit lengthy with the partial fractions. Is there a more ...
0
votes
1answer
27 views

Step function from inverse tangent

from this link http://blog.wolfram.com/2008/01/19/mathematica-and-the-fundamental-theorem-of-calculus/ it shows that $$x+2\tan^{-1}{\left(\frac{\cos{x}}{2+\sin{x}}\right)}$$ and $$2\tan^{-1}{\left(\...
5
votes
3answers
75 views

Evaluating $ \int {e^x \sin (k \pi x) } dx $

I'm trying to integrate $$ I = \int {e^x \sin (k \pi x)} dx. $$ I've used Matlab and Wolfram Alpha, which have both given me the result $$ I = \frac{e^x(\sin (k \pi x) - \cos (k \pi x))}{k^2 \pi^2 +1 }...
1
vote
2answers
90 views

Question about right triangle and sin(2theta)

This is a pretty basic question but I just wanted clarification. I know that sin(theta) is opposite/hypotenuse regarding right triangles. But what would sin(2theta) be? would it be (opposite/...
0
votes
2answers
69 views

Why is the phase shift -c/b instead of -c

In a function like $\sin(2x + 3)$ why is the phase shift $\frac{3}{2}$ units to the left instead of 3 units to the left
1
vote
1answer
33 views

Transform $\tan$ to be continuous between $0$ and $1$

I'm trying to create a $\tan$ function which has asymptotes between $0$ and $1.$ This is the closest I have gotten, but I can see that the asymptote is not actually at $1$ and when $x=0.5,\; y=0.02$. ...
1
vote
2answers
79 views

$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

In an acute triangle with angles $ A, B $ and $ C $, show that $ \cos {A} \cdot \cos {B} \cdot \cos {C} \leq \dfrac{1}{8} $ I could start a semi-proof by using limits: as $ A \to 0 , \; \cos {A} \...
3
votes
1answer
125 views

Prove that $\pi > 24\small{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}}$

Prove that $\pi > 24\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. I tried using trig but I couldn't solve it. A hint I was given is to use half angle identities. This should be easy for someone who is ...
14
votes
3answers
629 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since $...
1
vote
1answer
81 views

Trigonometric Integration.

Q. $$\int _0^{\frac{\pi }{4}}\:\left(\frac{1}{\left(\cos^4x-\cos^2x\sin^2x+\sin^4x\right)}\right)\:dx$$ My method: =>$$\int _0^{\frac{\pi }{4}}\:\left(\frac{1}{\left(\left(\cos^2x+\sin^2x\right)^2-3\...
0
votes
1answer
406 views

Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit $5$...
0
votes
1answer
321 views

Trigonometry: Find the side of a triangle within a triangle

Please help. I found a solution to this problem on yahoo answers but I do not understand the answer. I would use the laws of cosine but I have to be able to answer this without a calculator If AB = ...
1
vote
1answer
57 views

Solving a particular system in three variables

I am trying to analytically solve these equations for the three variables of $\theta$, $L_p$, and $R_c$. Matlab can not solve them! I am wondering if there is any solution for this at all? And how I ...
2
votes
1answer
38 views

Solve for $x$ in the following trigonometric equation: $3\cot^2(x) = 1$ in the domain $x$ is an element of $[0, 2\pi]$

My question lies in the third step of my solution, is taking the square root of the term $\cot^2(x)$ valid? $$\cot^2 x = 1/3$$ $$\cot^2 x = (\cot x)^2$$ Which implies $$ \sqrt{\cot x}^2 = \sqrt{1/...
17
votes
2answers
459 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
3
votes
2answers
154 views

Trigonometric equation $\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X)$

A trigonometric equation is to be solved, the solution ($X=10^\circ$) is very clear but I need a proper method $$\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X).$$
0
votes
5answers
82 views

The limit of $(x^2-\tan 2x)/\tan x$ as $x\to0$

I'm stuck in finding the following limit: $$\lim _{x\to 0}\left(\frac{\left(x^2-\tan\left(2x\right)\right)}{\tan\left(x\right)}\right)$$ I am not sure how to do this one help will be appreciated.
1
vote
2answers
35 views

Proof regarding the function $\cos(1/x)$

Prove that for every number $a>0$ there exists 2 numbers $x,y$ with $0<x,y<a$ for which $f(x)>0$ and $f(y)<0$ with $f = \cos(\dfrac{1}{x})$. How do I go about proving this?
0
votes
1answer
33 views

Trigonometric eliminations

These are a few problems which I wasn't able to do. I am new to these trigonometric eliminations. I don't really know how to start these problems. I couldn't get pass the first step in some of them.....
6
votes
1answer
115 views

Find the sum to n terms of the series

Find the sum to n terms of the series $$\frac {\sin x}{\cos x+\cos2x} + \frac {\sin2x}{\cos x+\cos4x} + \frac {\sin3x}{\cos x + \cos6x} +\dotsb $$ How can I solve this? Here is what I did for the ...
2
votes
0answers
71 views

Can one generate all possible binary strings by sampling a trig function at regular intervals?

I'm using a trigonometric function to generate binary strings by sampling the function at regular intervals and mapping each sample value to a binary bit. As a simple example: if the function is $g(x)...
2
votes
4answers
157 views

How to solve the trigonometric equation $\cos17x=20\cos x$?

How to solve the following trigonometric equation? $$\cos17x=20\cos x$$ I'm really awful in trigonometry. I tried division of both sides by $20$. Thanks.
3
votes
3answers
378 views

how to solve equation with cos

I have this equation $\cos2x +5 \cos x + 3=0$. To solve it I rewrite $\cos2x$ to $2 \cos^{2} x- 1$ and set $\cos = t$. I get the following equation $2t^2 - 1 +5t +3 = 0$ with that and then divide the ...
0
votes
3answers
35 views

When are you able to reduce equations such as $\tan(\pi/2-2x)=\tan3x$ to simply $\pi/2-2x=3x$?

as the title says, I am unsure when I can do this. Does this only apply to specific trigonometric functions? Any help clarifying this would be appreciated.
2
votes
3answers
55 views

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$

Show $|\sin(y)y - \sin(x)x| \leq C|y - x|$ for some $C > 0$. This is one of the steps in a bigger problem I'm trying to solve, and while it first appeared it would be entirely straightforward, I ...
0
votes
2answers
39 views

roughly estimate the angle between two lines that are really close to each other

Say, for example, what's the angle, theta, between y=10000x and y=10001x ? In terms of calculator-independent estimation, I tried: calculate tan(theta), then use taylor expansion of arctan(theta). ...
0
votes
2answers
40 views

Limit of a Trigonometric Function by Substitution

How to solve this limit: $$\lim_{t\rightarrow \pi} \frac{1+\cos(7t)}{\sin^{2}(4t)}$$ by using the substitution $\varphi=\pi - t$. I tried using trigonometric identities but it just gets messier. ...
0
votes
2answers
272 views

Calculate the angle from the given points coordinates.

I'm trying to figure out the way to calculate the a angle value from given coordinates of three points as showed on the illustration below: I know how to ...
2
votes
1answer
101 views

$2\cos(x)+x=0$? Advanced trig. question.

Title says it all: $$2\cos(\theta)+(\theta)=0$$ the interval should be between $0$ to $2\pi$. Been trying to figure this out for quite a while, still no luck. I'm trying to find if the solution ...
1
vote
0answers
217 views

Calculating point on sphere surface where sun reflection to a target point occurs

Imagine a mirror sphere at position O with radius R, and a target point at position P, at distance d from the sphere origin. There is an unknown point X on the surface of the sphere, where the light ...
1
vote
2answers
52 views

Parametrised curves.

I've been working through the following question: Q1= What points on the parameterised curve $x(\theta)=\cos^2{\theta}, y(\theta)=\sin{\theta}\cos{\theta}$ correspond to the parameter values ${\...
0
votes
1answer
109 views

Using Leibniz' formula to show the $(2n)$th derivative of $(2x^2 + 3x +1)\sin x$ is $(-1)^n(2x^2+3x-8n^2+4n+1)\sin x+(-1)^{n+1}(8nx+6n)\cos x$ wrt $x$

If I let $f=f(x)=\sin x$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\...
0
votes
2answers
93 views

In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$

Here is what I did, tell me whether I did correct or not: \begin{align*} y &= \cos A + \cos B + \cos C\\ y &= \cos A + 2\cos\left(\frac{B+C}2\right)\cos\left(\frac {B-C}2\right)\\ y &= \...
1
vote
5answers
206 views

Missing root of equation $\tan(2x)=2\sin(x)$

I tried to solve the equation $\tan(2x)=2\sin x$ and got the roots $x=n2\pi$ and $x=\pm \frac {2\pi}3 +n2\pi$. It seems that $x=n\pi$ is also a root but for some reason I didn't get that one out of my ...
10
votes
7answers
280 views

For all real $\theta$ prove that $ \cos(\sin\theta) \gt \sin(\cos\theta)$

How do I prove this? Im not able to even start it. Help please!
2
votes
3answers
734 views

Finding the exact value of $\tan(\pi/5)$

Hi, I realise there has been a question already asked regarding this particular exact value, but this question requires for it to be done under different conditions, which is the part I require help ...
1
vote
1answer
47 views

Finding Solutions to Trigonometric Equation

Find all $x$ in the interval (0, $\frac{\pi}{2}$) such that $$\frac{\sqrt{3}-1}{\sin x} + \frac{\sqrt{3}+1}{\cos x} = 4\sqrt{2}.$$
1
vote
2answers
55 views

Proving a trigonometric relation using circle properties

Hi, I've been having trouble with this question, and would really like some help. What I've done so far is applied the cosine rule in the triangle PQR to find that $PR^2=a^2+c^2+2ac\cos\theta$. What ...
1
vote
3answers
98 views

How is $\sin x$ considered a function

Yeah, it's a simple question, I forgot the exact reasoning behind it. also would a one-to-one function be considered a function (obviously yes)? If it is then how come $\sin x$ is a function. I'm ...
0
votes
1answer
599 views

Calculation of Nutation and Rotation from Pitch and roll (yaw is fixed to 0)

i am stuck attempting to convert two angles (pitch and roll) to represent tilt within a circle. Take a plane $(Z\ \text{(Vertical)}, X\ \text{(Roll)}, Y\ \text{(Pitch)})$ If i have $45$ degrees ...
0
votes
3answers
371 views

How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?

I have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity cheat ...
3
votes
3answers
844 views

Trigonometry Difficult Question

If $\cos x + \cos y = a$ and $\sin x + \sin y = b$. Find $\cos(x+y)$ and $\sin(x+y)$. I only need some hints to start as I am not able to get any way to go forward to.
0
votes
2answers
42 views

2 equations for 2 variables with trigonometry

2 equations for 2 variables $x$ and $t$ are unknowns $a$, $b$, $c$, $d$, $e$, $f$, $g$ are constants $$a+\left(8(t+b)+\frac{b^2}2-32\right)\sin(c)=d+et\sin(x)$$ $$f+\left(8(t+b)+\frac{b^2}2-32\...
1
vote
2answers
46 views

The maximum possible value of $ (xv - yu)^2 $ over the surface …

The maximum possible value of $ (xv - yu)^2 $ over the surface given by the equations $ x^2 + y^2 = 4 $ and $ u^2 + v^2 = 9 $ is : I solved it and my answer comes out to be $9$ but the correct ...
1
vote
1answer
191 views

How to calculate civil twilight timings?

I am writing a program that requires calculations of civil twilight timings. I know how to calculate sunrise and sunset timings(actually I just searched online for a formula and copied it into my code ...
1
vote
1answer
62 views

Trigonometry curiosity [duplicate]

How prove that: $$-\tan[(\frac{10\pi}{41}+4[\sin[(\frac{2\pi}{41}+\sin[(\frac{4\pi}{41}+\sin[(\frac{12\pi}{41}+\sin[(\frac{20\pi}{41}-\sin[(\frac{26\pi}{41})]-\sin[(\frac{30\pi}{41})])])])])]])]=\cot[(...
2
votes
0answers
146 views

Trigonometric curiosity

How prove this $$-\tan\frac{10\pi}{41}+4\left(\sin\frac{2\pi}{41}+\sin\frac{4\pi}{41}+\sin\frac{12\pi}{41}+\sin\frac{20\pi}{41}-\sin\frac{26\pi}{41}-\sin \frac{30\pi}{41}\right)= \\\cot\frac{2\pi}{41}-...
1
vote
1answer
23 views

Evaluating Inverse Trigonometry Functions

Say we are given a problem such as $\arctan(\frac{\sqrt2}{2})=\theta$ Theta evaluates to $\frac{\pi}{4}\text{rad}$. I know this only because I have it memorised. What is an intuitive and procedural ...
0
votes
0answers
50 views

Construction of the equation from a given graph

The above diagram shows a sine-like graph is plotted in the range $0° \le x \le 360°$. It passes through 5 distinct points, namely A(0°, 0), B(90°, 2), C(180°, 0), D(270°, -2) and E(360°, 0). In which,...
3
votes
1answer
54 views

Simplifying a summation involving “cos”.

$\sum_{r=1}^{n-1}\cos ^{2}\left ( \frac{r\pi }{n} \right )$ How can I simplify this summation when I do not know whether "n" is odd or even?