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

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Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
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33 views

A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy $$ where $B(x,t)$ is the ball with center ...
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98 views

Solving an equation with $\arccos(x)$ and $\sin(\arccos(x))$

I want to solve this equation, determining y (all others letters are constants) : $$2 \arccos(3+(1.6y-80)/R) - \sin(2\arccos(3+(1.6y-80)/R)) = 2π(1-P)$$ I've try to use some automatic solvers but ...
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34 views

Given only this graph of a certain cosine function, how can I determine what it's “c” is? (click to see graph)

Although my answer key says $c$ is supposedly $\frac{2\pi}{3}$, I actually get $c = \frac{4\pi}{3}$ Thus, $y$ should equal $\frac{5}{2}cos(\frac{\pi(x)}{2}-\frac{4\pi}{3})-0.5$ , right?
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44 views

What is the number of x-intercepts in this graph of sine?

The function : $y=3-4\sin(2\pi x-3\pi)$ .. how many $x$-intercepts over the interval $[0,2]$? I am confused if they're 3 or 5 because there are 3 $x$-intercepts that are really intercepting ...
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179 views

arrange div elements in circle and square

I n number of divs which are arranged in a circle using javascript. Right now i set the dimension of each div to 40*40. Below is what i am able to achieve so far. This is how i find X & Y of each ...
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93 views

proving $\tan^{-1}(x)+\tan^{-1}(y) = \pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\;,$ when $x>0,y>0,xy>1$

(1) How ca we prove $\displaystyle \tan^{-1}(x)+\tan^{-1}(y) = \pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\;,$ when $x>0,y>0,xy>1$ (2) How ca we prove $\displaystyle \tan^{-1}(x)+\tan^{-1}(y) ...
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57 views

How to I solve the inverse $(e^x – je^x)/(e^x+e^x)$?

I have tried using this method https://www.youtube.com/watch?v=V-LJWfuoCDs. But I am getting zero on one side thus cancelling $e^x$, which means that the the answer I get will not be in form of ln. Is ...
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54 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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47 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
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45 views

A right triangle with sides

Imagine a right triangle with sides: Long side C is $4n$, sides $b$ and $a$ are $2n$ and $n$, where $n$ is an integer. How many right triangles are of this form? My attempt: $$16n^2 = 4n^2 + n^2$$ ...
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41 views

Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
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116 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...
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60 views

Multiple Waves all in phase (Wave packets)

Suppose there are seven waves of slightly different wavelengths and amplitudes and we superimpose them (textbook is talking about wave packets). The wavelengths range from $\lambda _9 = 1/9$ to ...
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286 views

Shadow angle calculation for solar tracking application

Shadow Length Dear all, *I am looking for relationship Between Lmin and solar radiation angle.I know Here in above link they provided relation. But i don't know how to calculate it. x- modules ...
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28 views

Trigonometry integration with a bound

So, I want to integrate $\int_\gamma sinz\; dz$ where $\gamma$ is any curve joining $i\to \pi$. Can I say that it is beacause $\int sinz=-cosz$, and $-cosz$ is analytic on the domain containing ...
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26 views

Rotating two objects

I have two lines. Both created in this format: Line 1 $$line1 = \left\{ \begin{array}{c} startX, startY \\ endX, endY \end{array} \right\}$$ $$line2 = \left\{ \begin{array}{c} startX, startY \\ endX, ...
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65 views

what does the sine function tell you about an input

Intuitively, I'm trying to understand the significance of the input in a sine function. I'm currently, trying to develop intuition behind sinusoids and what the input tells you about the output and ...
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23 views

A method of calculation coordinates in order to implement it to a code language!

lets say that we have three points A(xa,ya,za), B(xb,yb,zv), C(xc,yc,zc) with known coordinates in 3d space. Is there a method to calculate the coordinates (x,y,z) of another point D for which the ...
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45 views

Which figure provides the greatest change in angle per change in distance? (trigonometry)

I have been having a lively discussion with others about the following: We (myself and others) are using triangulation to measure distance to an object with a linear image sensor (CCD) and a ...
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61 views

Determining the location of a point in a triangle under the given constraints

ABC is a triangle with AC = 1, AB = c/b and BC = a/b. Q is a variable point on AC such that CQ = x and QA = 1 – x. The perpendiculars from A and C to BQ (extended if necessary) are $d_2$ and $d_1$ ...
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48 views

Ratios of right triangle integer multiples to PI

It is known that in a right triangle with angles 30 and 60 degrees the cathetus at the 60 angle is equal to the 0.5 of hypotenuse. In other words an angle with cosine 0.5 is equal to PI/3. Is there ...
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68 views

independent/dependent values at different frequencies and phases

I am curious about the following problem. I would like to ask for help solving it. Consider the following $m$ sinusoidal functions $\sin(\omega_{1}⋅t+\phi_1),\sin(\omega_{2}⋅t+\phi_2),..., ...
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116 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
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127 views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...
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1k views

3D Game: Pitch Yaw Roll of a point

I have a flat elliptical plane and I'm trying to figure out how to represent it based on its direction. So I basically need to calculate its pitch, yaw, and roll. I have a camera at $C$, and a point ...
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47 views

Question on trigonometry (acute angle case)

Here is the question of "MOSP (The Mathematical Olympiad Summer Program) $2000$", and it takes me so much time to solve, but I can't. Let $ABC$ be an acute-angle triangle. The question is to prove ...
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80 views

$3$ equations - > $2$ unknowns and also trigonometry

I have a big problem with solving this one - I have $3$ equations, and need to find $2$ unknowns: $$\cos(-55.82) = (0.6893\cos(-70) + 0.3381\sin(-70)) \cdot (\cos b\sin c_1 + \sin b\cos c_1) - ...
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164 views

How long will it take me to learn math as a beginner

am 18 years old. I won't like to say i don't know maths, but i have a very poor foundation in maths. I am very weak in maths, even some 8 year olds are better than me. I can answer about 4 out of 20 ...
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165 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
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343 views

Solving system if equations containing trigonometric functions with Ti-Nspire

In trying to solve the following system of equation: $20000\times9.81+a\cos b=0$ $a\sin b=6.17\times20000$ Find $a$ and $b$ . It gives me something containing "n2" in bold and I don't know why? ...
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62 views

Find angle inside of isosceles triangle

The figure explains it best. http://imgur.com/V9tX22Z We have $ABC$ isosceles triangle. We know a few angles as follows: $ACB = 20°$ $PAB = 50°$ $ABQ = 60°$ Find $BQP$ angle
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103 views

Law of Cosines Manipulation

I'm supposed to use law of cosines on $S_1S_2P$ in the following diagram: To arrive at the following equation: $$ \frac{r_2}{r_1} = [1 - 2(\frac{a}{r_1})sin(\theta) + ...
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55 views

Solving a System of Equations with Cosine

How do I solve a system of equations when there is a cosine. Here is the system: $$ \left\{ \begin{array}{c} a+b=77° \\ \cos(a)=\frac{y}{3.5} \\ \cos(a)=\frac{y+1}{3.5+x} \\ ...
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333 views

Definite integral involving exponential, powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} e^{-qx^{ak}(x^2+d^2+2 \, dx \cos[t])^{-a/2}} dt $$ I am not able to find any related formula. Note that this integral follows ...
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312 views

Definite integral involving powers and trigonometric functions

Is it possible to evaluate the following integral? $$ \int_{-\pi}^{\pi} {m \over m + x^{ak}\left[\,x^{2} + d^{2} + 2dx\cos\left(t\right)\,\right]^{-a/2}} \,{\rm d}t $$ I am not able to find any ...
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44 views

Trigonometry problem gives a wrong answer

A. Find the BC. Information: <) BAC = 65 (the base), AD = 10 I could not get it to work. What I did is, I took ADB and then I divided ...
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812 views

Matrix with trig functions and Cramer's rule

Using Cramer's rule solve for $x'$ and $y'$ in term of $x$ and $y$ $x = x'\cos\theta - y'\sin\theta\\ y = x'\sin\theta + y'\cos\theta$ So what I have is this $\det\begin{bmatrix} \cos\theta& ...
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228 views

Problems with Circles and Lines on a Cartesian Plane

(a) Find the equations of the two circles each of which touches both coordinate axes and passes through the point $(9,2)$. (b) Find the coordinates of the second point of intersection of the two ...
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64 views

Relative angles on unknown surface

I'm hoping someone here could help me with a problem I am having. I have an electronic sensor that measures tilt, and I put this sensor on a 10 degree ramp. The problem is that this portable ramp is ...
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maximize $\csc{(\pi b)}\sin{(\pi ab)}+\csc{(\pi (\frac{1}{a}-b))}\sin{(\pi a(\frac{1}{a}-b))}$

Let $x\in [0, \frac{1}{a}]$ for a positive integer $a$. $f(x)=\csc{(\pi x)}\sin{(\pi ax)}=\frac{\sin{(\pi ax)}}{\sin{(\pi x)}}$. When $b\in[0, \frac{1}{2a}]$, I want to prove that $b$ maximizing ...
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82 views

Solutions of the equation: $x\cos(\pi y)=y\cos(\pi x)$

I have to find the solutions $y=f(x)$ of the equation: $$x\cos(\pi y)=y\cos(\pi x)$$ in $x\in[0,\frac{1}{2})$. Is it possible to solve it analytically or I should use only numerical methods? Thanks
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213 views

What is the phase shift of a sinusoidal function?

Given the function f: $$ f(x) = \sqrt{3} \cos(2x) - \sin(2x) $$ Question: What is its amplitude and phase shift? My attempt: Let c be the hypothenuse of a triangle with the sides from the ...
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67 views

Can this trigonometric equation be solved exactly or approximately?

I need to find a solution for $\alpha$ that satisfies the following equation for $i=1,2,...,N$ $$\frac{2i}{K} = \sin(2 \alpha i) \left(\frac{\frac{\cos(k-1)\alpha}{2}-\frac{\cos(k+1)\alpha}{2}}{1- ...
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283 views

Proving Question (Complex Numbers, De Moivre's)

Prove that $$(1+\cos \theta + i \sin \theta)^n + (1+\cos \theta - i \sin \theta)^n=2^{n+1}\cos^n\frac{\theta}{2}\cos\frac{n\theta}{2}$$ I want to avoid using the $e^{i\theta}$ form since I haven't ...
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148 views

Splitting parametric curve equation into two ranges

I am examining the speed of motion on curves and in the textbook i am reading , the example was showing that using a parametric equation for an ellipse will result in a regular motion (as expected) , ...
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51 views

Coefficients making expression independent of variables NOTE: updated

I have the following expression: $f_S(\lambda,\tau,\beta)=\displaystyle\sum_{n=1,\ \text{odd}}^{2S-1}\ ...
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450 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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136 views

Bound on the angle between a vector and a subspace

Suppose you have three complex vectors $x_1$, $x_2$, and $x_3$. Define $a = \angle(x_1,x_2)$, $b = \angle(x_1,x_3)$. My question is about $c = \angle(x_1, span(x_2,x_3))$, the angle between the vector ...
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249 views

Gps Coordinate 1 mile away - Haversine

I've come across the need for doing GPS calculations. I've implemented the haversineformula for gps distances but I need the opposite, a point 1 mile away in any direction. The GPS coordinates are ...