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

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72 views

solving a trigonometric equation 1-cos(180°-x)+[(sin180°+x)/2]=0 .

$1-\cos(180^{\circ}-x)+\left(\frac{sin(180°+x)}{2}\right)=0$ Can someone help me on solvinf it. I did $1-(-\cos x) + \left(\frac{sin(180°+x)}{2}\right)=0 $
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1answer
49 views

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...
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1answer
588 views

Coordinate of intersection between line and square

TL;DR given a square and a point $p$, I need the intersection between the perimeter of the square and a ray cast from the center of the square through point $p$. This is my approach so far, but I will ...
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4answers
194 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
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5answers
230 views

Is $\cos(x^2)$ the same as $\cos^2(x)$?

I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
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5answers
83 views

Solving a trigonometric equation: $2 \sin(3a)=\sqrt{2}$

I have the following equation : $2 \sin(3a)=\sqrt{2}$ Not sure how to solve it (Because it's a transformed sin function, meaning 6 solution with 3 cycles in $2\pi$) after a moment I finally found ...
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6answers
476 views

How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
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195 views

Prove that $\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is an increasing function

Could anybody show that $f(x)=\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is increasing on the interval $x\in[0, 0.25]$? Of course, $\csc{(x)}=\frac{1}{\sin{(x)}}$. Here is ...
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59 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
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351 views

integrate $ \frac {(x^3 + 36)} {(x^2 + 36)}$

I know I have to use long division first, but I don't really know how to do it in this case $$\int \frac{x^3 + 36}{x^2 + 36}dx$$
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218 views

Tan Binomial formulas from a set S and its k-subsets

Working around, I found some Tan Binomial formulas. Let's $S$ be a set such that: $$ S=\left\{\text{ }\tan ^2\left(\frac{1\pi }{n}\right), \tan^2\left(\frac{2\pi }{n}\right), ...
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1answer
387 views

Is sine of angles greater than 90 degrees a convention?

The sine function is defined as the opposite side of the angle in question over the hypotenuse of the $90^\circ$ triangle. $$\sin(â) = \frac{opposite}{hypotenuse} \tag{$0^\circ<â<90^\circ$}$$ ...
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2answers
98 views

Finding $\cos(A+B)$ from $\sin(A+B)$

The question states to start with the identity: $$ \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$$ It then asks to use the above to prove that: $$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$ All I can think ...
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2answers
103 views

Find the magnitdue of a vector along a specific heading

I've gotten some seemingly conflicting answers about this, so I'm hoping someone can help straighten this out for me. Given a vector of components East/West (u) and North/South (v), how do I find the ...
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4answers
1k views

Integral of $\sin^2 \pi x$

Evaluate $$\int_0^{1/4} \sin^2 \pi x \; dx$$ Can someone please explain what to do if theres a power and how to do it in general thanks
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0answers
264 views

Determining similarity between paths (sets of ordered coordinates).

With limited knowledge of mathematics, I am not sure what tags to use for this question. I have a path on a 2D surface called $(p1)$. A path consists of a set of ordered $(x,y)$ coordinates. By ...
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1answer
88 views

For the slope of the line at a point, why am I getting a different result by using the calculus method?

I am evaluating the slope of the secant as it approaches $f(30)$ for the function $f(x) = 2\sin(x) - 2$. Using calculus I can easily find that the derivative is $f'(x) = 2\cos(x)$. If I sub in $30$ ...
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2answers
368 views

If $\dfrac{\cos^4\theta}{\cos^2\phi}+\dfrac{\sin^4\theta}{\sin^2\phi}=1$, show $\dfrac{\cos^4\phi}{\cos^2\theta} +\dfrac{\sin^4\phi}{\sin^2\theta}=1$

If $\dfrac{\cos^4\theta}{\cos^2\phi}+\dfrac{\sin^4\theta}{\sin^2\phi}=1$, prove that $\dfrac{\cos^4\phi}{\cos^2\theta} +\dfrac{\sin^4\phi}{\sin^2\theta}=1$. Unable to move further ...request you ...
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2answers
761 views

Figuring the volume of a partially filled cone without the radius of the material inside the cone

Say I have a cone. For simplicities sake, this cone is at the bottom of a storage silo. Is has a flat bottom, flat top and angled sides. I know the height of the material inside the cone but don't ...
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3answers
138 views

How to find $\cos A$ from $\sec$ and $\tan$?

It's given that $$\sec A + \tan A = 4$$ How would you find $\cos A$ from this?
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2answers
157 views

Why is this solution to $2\sin^2(x) + \sin(x) - 1 = 0$ incorrect?

$$2\sin^2(x) + \sin(x) - 1 = 0$$ $$\sin(x) = \cos (2x)$$ $$ \sin(x) = \sin (2x + \dfrac{1}{2} \pi)$$ $$ x = -\dfrac{1}{2}\pi - k \cdot 2\pi$$ or $$ x = \dfrac{1}{6} \pi + k \cdot \dfrac{2}{3} ...
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4answers
85 views

Solving a set of 3 Nonlinear Equations

In the following 3 equations: $$ k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1 $$ $$ 2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2 $$ $$ k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3 $$ $c_1$, $c_2$ and $c_3$ are ...
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52 views

inverse trigonometry derivation

Prove that : $sin^{-1}x+sin^{-1}y = sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$ If -1 $\leq x \leq 1; -1 \leq y \leq 1 $ and $x^2+y^2\leq 1$ or if $xy <0 $ and $x^2+y^2 > 1$ solution : Let ...
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183 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
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3answers
276 views

Order of operations in rotation matrix notation.

I'm trying to convert this equation to C# but I'm not a mathematician and I find math notation ambiguous: See the first matrix in this article: http://mathworld.wolfram.com/RotationMatrix.html ...
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101 views

Trigonometry Identities

Consider a collection of five points evenly spaced around a circle to form a regular pentagon. Assume the figure is scaled so that the sides of the pentagon have length 1. Question: Use Ptolemy’s ...
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322 views

How do I compute $\cos$ and $\sin$ in a given interval if I know $\tan$?

$$ \tan x = -\frac{2}{3} $$ when $\dfrac{5\pi}{2} < x < 3\pi$. I understand this, but I don't know how to calculate the two other functions' values, $\cos x$, $\sin x$, using $\tan x$
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135 views

Another trigonometric equation

Show that : $$31+8\sqrt{15}=16(1+\cos 6^{\circ})(1+\cos 42^{\circ})(1+\cos 66^{\circ})(1-\cos 78^{\circ})$$
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466 views

Solving a trig equation with inverse?

How would I solve the following trig equation? $\arctan(x)+\arcsin(x)=\frac \pi 2$ I am kind of confused on how to solve it.
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638 views

How do I solve $\int \sec^3 \theta d\theta$ [duplicate]

Possible Duplicate: Indefinite integral of secant cubed I guess I need to learn a new technique because those I know didn't help me here. Wolframalpha uses the reduction formula, which I ...
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4answers
461 views

Frequency of a trigonometric function - Where is my mistake?

I need to find the frequency of the following trigonometric function.$$y=\sin^4(x)+\cos^4(x)$$ The "answers" section says the answer is: $$F_y=\frac{\pi}{2}$$ This is what i did: Finding $\sin(x)^4$ ...
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2answers
450 views

Possible to evaluate definite integral of inverse trigonometric function as function of $Y$?

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. Now I have the ...
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203 views

How to solve $1/2 \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0$?

How to solve trigonomtry function involving $\sin x \cos x$ and $\sin 2x$: $$\frac{1}{2} \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0. $$
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3answers
507 views

Use equalities to derive important trigonometric functions

The trigonometric functions I must know: (A) $\sin(-x)=-\sin x$ (B) $\cos(-x)=\cos x$ (C) $\cos(x+y)=\cos x\cos y-\sin y\sin x$ (D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$ $\sin^2x+\cos^2x=1$ (Use ...
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4answers
1k views

Understanding the Unit Circle

I'm going to need to know this (the unit circle, English) for a test, but I'm so horrible with both charts and memorization, the whole thing is failing to stick into my brain. Using non-complicated ...
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1answer
1k views

Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think ...
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431 views

Determining location based on image of rectangle

Given an image of a rectangle (and assuming I could find the corners of that rectangle), what is the math I would need to determine where I am relative to the center of the rectangle, assuming that I ...
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2answers
3k views

How to use atan2?

I've looked at wikipedia and don't really understand what I'm reading or how to implement atan2. I was hoping someone could point me to a better resource or give me an example of how to use atan2. ...
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201 views

A boundary value problem over an infinite interval

This is the edited version of the original problem, hopefully presented in a clearer manner. (I have also renamed this post with a more befitting title) Problem: $$y'(x) = ...
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2answers
4k views

Calculating point around circumference of circle given distance travelled

Its the end of the day and my brain just cant cope anymore. Can anyone help with, what I hope, is a simple question.. Given a point on a circle (x, y coordinates) how can I calculate the coordinates ...
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940 views

Using the spherical law of cosines

Compute angular length c of the great-circle route between these two cities: Daytona Beach (location A): $29^\circ12'\ N, 81^\circ1' \ W$. Sidi Ifni (location B): $29^\circ23' \ N. 10^\circ10' \ ...
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959 views

Help with calculating the angle to turn towards a target in a coordinatesystem

I know the following: my own position my own facing (the angle im turned) my targets position What i would like some help with is how i calculate the shortes way to turn and the angle to turn. If ...
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3answers
340 views

Creating a sine function with specific parameters

Part1(Solved) Is it possible to create a function that maps x to y as follows: x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... y 0 1 2 3 4 5 4 3 2 1 0 1 2 3 4 5 ... So far, this is what I have ...
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46 views

The period of a non-linear pendulum

The period of a non-linear pendulum is $T = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \frac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. My problem: what will happened with the period $T$, ...
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37 views

Comparing the sum of squared trigonometric functions and the sum of functions of double argument

Let $D,E$ be two non-negative real numbers such that $D \neq E$. Do there exist real numbers $A,B,C$ such that: $$A \cos(2\theta)+B \sin(2\theta)+C = D \cos^2(\theta)+E \sin^2(\theta)?$$ I used ...
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62 views

If $\cos(A + B) < 0$, then why $-\pi \leq A + B \leq -\frac{\pi}{2}$?

The value of $A + B$ should be less than $\frac{-\pi}{2}$ for the cosine to be negative. Then relation should be $-\pi \leq A + B < -\frac{\pi}{2}$. But the question's relation is not this; that ...
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39 views

Geometric transformations with matrix multiplication

Say you have a matrix of position vectors, could someone please explain the intuition behind why a rotation by an angle $\theta$ about the origin can be represented by the matrix: $$ R= ...
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1answer
33 views

Calculate the convolution of the product of two identical sine functions. (5.6-7)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of ...
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3answers
66 views

Find the smallest interval for parametric equations

$\mathcal{G}$ is the graph of parametric equations $\begin{align*} x = \cos(4t), y = \sin(6t). \end{align*}$. Find the length of the smallest interval $I$ such that the graph of the parametric ...
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1answer
105 views

Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D

this is my first post.. I hope this good I have 1 triangle in space (3D)... and I know all data except the coordinates of 3er point(vertex)... for example this: then: ...