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

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2
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98 views

Application of integrating $\cos^4 x$?

A student asked a colleague the other day for a practical application that involved needing to integrate the fourth power of cosine, but no one here could think of one off-hand other than some volume ...
2
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2answers
55 views

Simplify a trigonometric equation

UPDATE **** AGGGH, I am embarrassed, but I made an error in deriving the equation in this question. Please disregard this question, and I will start a new one if I get stuck on the corrected version. ...
1
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0answers
206 views

The geometry of a spiral made of adjacent right triangles

In the above figure (not sure if you can see it clearly or not), while using the old standard technique of plotting irrational numbers on number line, I saw this property. If we go on plotting ...
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1answer
32 views

how we compute partial derivative for w respect to x?

If we have $$ f(x,y) = \arctan\Big(\frac{y}{x}\Big), $$ how we compute partial derivative for w respect to x, that is $\frac{\partial f}{\partial x}$? Thank you!
2
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5answers
97 views

Find range of $\sin x+\cos^{2}x$

By differentiating and equating to $0$ I know that the maximum must be $5/4$. The minimum is where I am confused. $-1$ would be logical but I'm not sure if this function can ever be equal to $0$. ...
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7answers
1k views

Can someone explain how $\frac{\tan x}{\sec x}=\sin x$?

What identities are used to get $\sin x$ from $\tan x \operatorname{/} \sec x$? I was looking at an example in my textbook and the problem went from $\tan x \operatorname{/} \sec x$ to $\sin x$. I don'...
1
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2answers
149 views

Books on Trigonometry; Specifically Trigonometric Equations

I'm just wondering if anyone knows of any good books that focus on trigonometric equations and solving them. I'm thinking of using Trigonometry by Saul and Gelfand.
3
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2answers
59 views

Solve for $x$ when $\sin 2x = \cos x$ where $ x$ is in the domain $ [0, 2\pi]$

Quick question on trig (which I haven't dealt with in a long time): since $\sin 2x = 2\sin x\cos x $ $2\sin x\cos x = \cos x$ $2\sin x\cos x/\cos x = 1$ $\sin x = 1/2$ since $\sin x = 1/2$ in ...
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0answers
128 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
0
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1answer
53 views

simplyfing trigonometric expression using phase shift identities

I need to simplify the following trigonometric expression: $$sin(a-\frac {3\pi}{2})cos(a-\frac {3\pi}{2})tg^{-1}(a-\frac {3\pi}{2})$$ Using the phase shift identities, I calculate the first factor ...
1
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3answers
57 views

Any shorter way to solve trigonometric problem?

If $10 \sin^4\theta + 15 \cos^4 \theta=6$, then find value of $27 \csc^2 \theta + 8\sec^2 \theta$ I know the normal method o solve this problem in which we need to multiply L.H.S. of $10 \sin^4\theta ...
3
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1answer
48 views

Number of solutions of this trigonometric equation.

Q. Find the number of solutions of the equation $\sin(x) + 2\sin(2x) - \sin(3x) = 3$, in the interval $x\in (0,\pi)$. I tried clubbing the $\sin(x)$ and $\sin(3x)$ terms together but got nothing. ...
8
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6answers
198 views

Prove that $ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x $

In a related question the following integral was evaluated $$ \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{\mathrm{d}x/2}{1 + \cos x \sin x} =\...
1
vote
1answer
27 views

Sum of cosines in a transmitter

Is there a closed form solution to this sum? $$ \max \sum_{k=1, 2, ...}^n \cos (k+m(k) \pi /4), m(k)= 0, 1, 2, or 3$$ The application where this arises is calculating the peak voltage of a radio ...
0
votes
3answers
50 views

How would you get to $\tan(\theta/2)$ if you are given $\sin\theta/(1+\cos\theta)$?

According to wolfram alpha, ${\sin{\theta}\over{1+\cos{\beta}}}={\tan{\theta\over2}}$. But how would you get to ${\tan{\theta\over2}}$ if you're given ${\sin{\theta}\over{1+\cos{\beta}}}$?
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2answers
28 views

Simplification of a function

I have to simplify the following function : $g(x)$ = ${\sin x\over \sin 1} \cos (x-1)-{\sin x\over \sin 1} \cos 1 + {\sin (x-1)\over \sin 1} \cos 1-{\sin (x-1) \over \sin 1} \cos x.$ My attempt: $\...
3
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1answer
110 views

How to solve $x\left(\sin x+\cos x\right)=1$?

Could you please give me some hint how to solve this trigonometric equation: $$ x\left(\sin x+\cos x\right)=1$$ Since $\sin x+ \cos x= \sin x+ \sin\left(\frac {\pi} 2-x\right)=2\sin\frac{\pi}...
5
votes
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
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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
74 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 }...
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2answers
83 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
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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
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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
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2answers
78 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
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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
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3answers
627 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 $...
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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
383 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
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1answer
300 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 = ...
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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
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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
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2answers
444 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
76 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.
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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
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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
113 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
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0answers
69 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
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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
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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
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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
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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 ...
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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). ...
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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
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2answers
260 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
211 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
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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
91 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 &= \...