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

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1answer
101 views

Solving right triangle given the area and one angle

Given right angle triangle $ACB$ (C is the right angle) has an area of 224 $mm^2$, what is the length of leg b if angle A equals 31.7deg? Here's the scenario: I have one right triangle completely ...
1
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1answer
123 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
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2answers
64 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
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2answers
50 views

Trig Question, Please help.

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $\cot \theta = −4$, $\sin \theta > 0$ Then it asks me to find: $ \sin(\theta) $, ...
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0answers
33 views

Looking for a method to solve simple trigonometric equations

Assume that $N$, $q$, $k_1$ and $k_2$ are integers such that $N>1$, $0\leq k_1<k_2<N$ and $q\geq 3$. Is there a method to solve the following trigonometric equation: $$ \sin\big(\frac{\pi ...
2
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1answer
179 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
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0answers
48 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
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4answers
967 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
2
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3answers
265 views

Using the Chain Rule to prove trig derivatives

I'm having trouble with this problem, I'm not sure how to tackle it and I was wondering if somebody could set me on the right path. The problem is as follows: Use the Chain Rule to show that if ...
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4answers
91 views

Prove trigonometry identity for $secx\quad -sinx$

I'm trying to prove this equality but I' stuck at the second step. Please give me some hints or other ways to proceed. $\frac { { tan }^{ 2 }x\quad +\quad { cos }^{ 2 }x }{ sinx\quad +\quad secx } ...
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1answer
174 views

Evaluate: $I = \int^{\pi/2}_0 (\sqrt{\sin x}+\sqrt{\cos x})^{-4}dx$

Evaluate : $$I = \int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx$$ Attempt : \begin{align} I&=\int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx\\ ...
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2answers
100 views

Integration practice of $\int \frac{\sqrt{25-y^2}}{y}dy$

I need to solve $\int \frac{\sqrt{25-y^2}}{y}dy$. I originally thought IBP, but that led to a very large and confusing algebra problem. Then I started to look at the $\sqrt{25-y^2}$ and started to ...
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1answer
74 views

How to solve $\sin{x}=c \,x$?

I'm trying to find $x$ in the following equation, where $c$ is a known constant: $\sin{x}= c \,x$ Any help is appreciated.
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2answers
68 views

Prove trigonometry identity for $\sin A+\cos A$

I’ve been struggling in proving this identity for hours (yes, shame on me), but I can’t see any light. $\frac { \cos(A) }{ 1-\tan(A) } +\frac { \sin(A) }{ 1-\cot(A) } =\sin(A)+\cos(A)$ I've been ...
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2answers
31 views

Applying the cosine even identity to the cosine difference identity

I'm slightly confused over what happens when you're applying cosine's "even identities" to the difference identity. Here's how I go about, please tell correct me as I feel i'm going wrong somewhere. ...
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0answers
72 views

How to solve this problem 4

Question $1$: Is $\frac{1}{\pi}\arccos\left(\frac{{\sqrt{2*\sqrt{2*\sqrt{2}*...n}}}}{2}\right)$ always a rational number when each$*$ is either $+$ or $-$ and $n$ may or may not be infinite? ...
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1answer
61 views

How to solve this differential equation sinusoidal?

I can't find how to separate variables. $$y= \sin(xy')$$
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2answers
73 views

Simple Trigonometric Equation

I am asked to solve the trigonometric equation $2cos \theta = \sqrt 3$ I rearrange it to $cos\theta = \frac{\sqrt3}{2}$ Now, at this point I am not sure what to do? Can someone describe to me the ...
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3answers
72 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
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1answer
240 views

Calculate length of radial intersecting a rectangle

In a rectangle like below, I need to calculate the length of any radial, from the center of the rectangle to where it intersects with the edge of the rectangle. Further, the angle of the radial is ...
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1answer
96 views

How Can I figure out when cosine = $\frac{2}{\pi}$?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ So I am trying to find $c$ for $f(x)=\sin x$ over the interval $[0,\frac{\pi}{2}]$. So using the Mean Value ...
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3answers
85 views

solve $3\sin\theta+4\cos\theta=0$

Solve for $0 < \theta < 360$ Question $3 \sin \theta + 4 \cos \theta = 0$ Please help. I really can't figure this out Thanks :) What I have tried I tried using the a $\sin \theta + b ...
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5answers
49 views

$12\frac{\sin 45^\circ}{\sin 60^\circ}$ Need help breaking this down.

Otherwise known as $12\dfrac{\left(\frac{1}{\sqrt2}\right)}{\left(\frac{\sqrt3}{2}\right)}$ How do you simplify this multi level fractional radical expression into $4\sqrt{6}$.
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3answers
193 views

A few questions regarding the cosine difference identity

I've a few questions that stem from the proof given in my textbook regarding the cosine difference identity. The proof goes like this: Let $\alpha$ and $\beta$ be angles plotted in standard ...
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2answers
88 views

Find the derivative of $y=\cos(x) - 2\sin(x),$ when the gradient is $1$

I need to find the smallest positive value of $x$ for which the gradient of the curve has value 1. For this equation: $$ y =\cos(x)-2\sin(x) $$ The answer is 2.5c grad. The following is my ...
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1answer
60 views

Not understanding where the ratio comes from

I'm completely stumped at how they come up with the ratio AB : AC. Why not AC : AB? Where does this ratio come from? How can I get to this ratio myself? Please help.
2
votes
2answers
65 views

What is an algorithm for making text form a circle

Ok it's beyond the scope of this programming exercise, but I want to create a loop that will allow me to input any number of characters and the loop gets each character in the string and places it at ...
0
votes
1answer
36 views

What is the effect of taking the sine of inverse cosine?

How can I evaluate the sine of an inverse cosine? for example: sin(arccos((x)^1/2))
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4answers
49 views

Finding the $\cot\left(\sin^{-1}\left(-\frac12\right)\right)$

How can I calculate this value? $$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$
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6answers
156 views

Solving a trigonometric limit

First off, please excuse my n00bishness I have only just begun learning about algebraic manipulation of limits so this is probably a really dumb or obvious question. I'm trying to solve the following ...
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1answer
37 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
2
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2answers
62 views

How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $?

How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x? $x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $ for $\sin 2\theta ...
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2answers
46 views

Solving equations with powers without logarithms

Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms. I was asked to find, for example, $27^x = 3$. (without the use ...
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2answers
70 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
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1answer
47 views

If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is…

Problem : If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is a) $\frac{\sin\alpha +\sin\beta}{2}$ b) $|\sin\alpha -\sin\beta|$ c) $\frac{\cos\alpha ...
2
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4answers
273 views

Solve The Triangle

I am having a tough time trying to solve this problem. I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its ...
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2answers
100 views

Trigonometric solution of a 6 degree polynomial

How do i prove that $\sin^2 \frac{\pi}{13}$ is a root of the equation $$2^{12}.x^6-13(2^{10}.x^5-5.2^8.x^4+3.2^8.x^3-7.2^5.x^2+7.2^2x-1)$$? Any hints/answers would be appreciated.
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2answers
244 views

How do you find this product?

Is there a way to find the exact value of the product $$P=\displaystyle\prod_{n=1}^{1007} \sin {\left(\dfrac{n\pi}{2015}\right)}$$
1
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3answers
123 views

Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
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1answer
503 views

Easier way to solve this problem of trigonometry.

Prove that $\sin x \sin y \sin(x-y) + \sin y \sin z \sin(y-z) + \sin z \sin x \sin(z-x) + \sin(x-y)\sin(y-z)\sin(z-x) = 0$ . When I expanded them ,it became horrendous. Is there any easy way or trick ...
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2answers
100 views

Finding the derivative of sinus and cosinus. Trigonometric identities

How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$ How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$ Do ...
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2answers
318 views

Finding a 3rd coordinate of the rectangle points in 3d

I have a 4 3-D-points, each of them has only 2 of 3 known coordinates, as follow (? is unknown here): P5 (P5x, P5y?, P5z) P6 (P6x, P6y?, P6z) P3 (P3x, P3y, P3z?) P4 (P4x, P4y, P4z?) They build ...
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2answers
136 views

Trigonometry sum of solutions question

Problem: For which $a$ will the sum of solutions be equal to $100$, in $\sin(\sqrt{ax-x^2})=0$. The attempt at a solution: For $\sin(x)=0$, $x$ must be equal to $0$, so we get ...
1
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1answer
81 views

Calculating amount of rotation to straighten an imaginary line created by 2 points.

I am trying to build a small app where my users can straighten up a tilted face with just 2 clicks I ask my users to click on the middle of the nose and the middle of the eyebrows of the face ...
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5answers
475 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
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1answer
47 views

Trigonometric identity proof problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 5: d) $\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha}$ I would appreciate some hints on how to ...
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2answers
66 views

Trigonometry - simplifying a given equation [duplicate]

Question: $$\tan 9 - \tan 27 - \tan 63 + \tan 81$$ Answer I'm getting : 0 What I did: Well I clubbed together $\tan 9$ and $\tan 81$ and $\tan 27$ and $\tan 63$ (took out negative as common). Then ...
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2answers
163 views

Clarification on the domain of $\arcsin(\sqrt{1-x^2})$

As the title says, I don't understand how to find the domain of $\arcsin(\sqrt{1-x^2})$. I kinda understand how it would equate to it would be -1 < x < 1 (inclusive of 1 and -1) by definition of ...
3
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1answer
109 views

For which integer $n$, $\sin\left(\frac{\pi}{n}\right)$ can be a rational?

When I studying the trigonometric functions, I sow that most of the values of $\sin\left(\dfrac{\pi}{n}\right)$ and $\cos\left(\dfrac{\pi}{n}\right)$ where $n\in\mathbb{N}$ are irrational. How can we ...
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2answers
19 views

A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...