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

learn more… | top users | synonyms (2)

0
votes
0answers
4 views

Find all the equilibrium solutions of $x' = \cos(x^2)$ and determine stability.

One can determine equilibrium solutions of autonomous EDO's setting the derivative equal to zero. So, in this case, all the equilibrium solutions take the form $x^* = \pm \sqrt{\frac{\pi}{2} + k\pi}$, ...
1
vote
4answers
57 views

How to solve $a = \cos x - b\sin x$ where $a$ and $b$ are real numbers?

I found this equation when solving a physics problem related to finding an angle when entering a river, that has a known current, and trying to get to a specific point on the other side. I'm not sure ...
1
vote
3answers
69 views

Evaluating the inverse trigonometric limit $\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}}$

$$ \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$ I was doing some questions on limits, I saw one in which there is $\arccos x$. I am stuck ...
0
votes
0answers
24 views

Trigonometric integral $\int_{[-\pi,\pi]^2}{\frac{1-e^{-in\cdot\theta_1}}{1-\cos(\theta_1)\cos(\theta_2)}\,d\theta_1\,d\theta_2}$

I am trying to compute the following integral (see here). Since it seems to be the wrong approach, I am trying to calculate another one which I hope it will give me what I am looking for. My point is ...
2
votes
1answer
414 views

how to solve $a\sin x+b\cos x$

Let's solve: $\sqrt{3}\sin x - \cos x=2$ The left hand side may be expressed as $R\sin(x+ \phi)$ We know that $R=\sqrt{3+1}=2$ We also know that $\tan \phi= \frac{-1}{\sqrt{3}}$ The solution to ...
0
votes
2answers
36 views

Determining exact value of $\cos (A+B)$ in a specific quadrant

The question reads: Angles $A$ and $B$ are obtuse angles in quadrant 2 (II). If $\csc A = 3$ and $\tan B$ = -1/3, determine the exact value of $\cos (A+B)$. How would I take on this question? ...
1
vote
2answers
806 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
1answer
14 views

Ration of sum and Product of Trigonometric expression.

If $A,B,C \in \mathbb{R}$ and $\displaystyle\cos(A-B)+\cos(B-C)+\cos (C-A)=-\frac{3}{2}\;,$ Then $\displaystyle \frac{\sum \cos^3(\theta+A)}{\prod\cos(\theta+A)}\;, $Where $\theta \in ...
1
vote
2answers
22 views

Maximum Height is giving me negative

Hey guys for this parametric equations its giving me negative Question is: A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. ...
1
vote
6answers
74 views

How would you verify the following trig identity $\frac{\sin(x)}{\sin(x)+\cos(x)}=\frac{\sec(x)}{\sec(x)+\cos(x)}$

How to verify the following trig identity. $$\frac{\sin(x)}{\sin(x)+\cos(x)}=\frac{\sec(x)}{\sec(x)+\cos(x)}$$ I started with the right side and multiplied the numerator and denominator by ...
3
votes
3answers
48 views

What is the fallacy of this trigonometrical proof that $1=-1?$

I have this equation which I solved- $$\sin^4x-\cos^4x=1$$ $$\implies -\cos^4x=1-\sin^4x$$ $$\implies-\cos^4x=(1+\sin^2x)(1-\sin^2x)$$ $$\implies-\cos^4x=(1+\sin^2x)\cos^2x$$ ...
1
vote
2answers
126 views

New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?

The below Golden Ratio Construction results in a ratio of PHI (1.6180...) between the blue line and red line, as found in Geogebra. This seems like a simple construction of the golden ratio, yet so ...
1
vote
3answers
33 views

Pentagon circumscribes a circle. Prove that its area is $5r^2\tan(36^\circ)$

Suppose that a regular pentagon circumscribes a circle of radius r. We are supposed to show proof, using the trigonometric area of the triangle (1/2)bhsin(36°) that the area of the pentagon is ...
56
votes
7answers
4k views

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
1
vote
1answer
24 views

Parametric Problem: Throwing a Dart <Test Review>

Yup it's me ... Parametrics, who would have thought xD! Anyways, again ... I am doing review and I really need this grade to get an A in math class; that's why I am asking questions here. And you guys ...
1
vote
0answers
19 views

Solve the equation $7\sin\theta+3\cos\theta=4$ for all solutions in the interval $0°\leq\theta\leq360°$, giving $\theta$ to the nearest $0.1°$

Using the Addition Formulae \begin{align} 7\sin\theta+3\cos\theta & = 4 \\ \Rightarrow \sin(\theta+23.2°) & = \frac{4}{\sqrt{58}} \\ \end{align} \begin{align} \sin(\theta+23.2°) & = ...
2
votes
2answers
83 views

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} - e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.
0
votes
1answer
26 views

How to deal with identities in which one expression, but not the other, evaluates to “undefined” in particular instances

I haven't touched trigonometry for awhile and whilst flicking through an old set of notes I came across the following expression: $$\csc(x) \cdot \sec(x) \cdot \sin^2(x)$$ I'm aware that this can ...
0
votes
3answers
56 views

Solving a trig equation that is quadratic?

I have to solve for $x$ given $$\tan^2 x = 2 + \tan x\;\;\;\;\;\;0≤x≤2\pi$$ I brought it all to one side and set it all equal to zero like: $$\tan^2 x - \tan x - 2 = 0$$ What am i supposed to do ...
0
votes
1answer
42 views

Calculating the absolute value of sum of rational numbers [on hold]

If $\sqrt{9-8\cos40}=a+b\sec40$, and $a$ and $b$ are rational numbers, then $\lvert a+b\rvert =\,{}$?
0
votes
0answers
12 views

Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
0
votes
0answers
21 views

Did I interpret this wiki article on spherical interpolation correctly?

In Lua pseudocode, I believe the wikipedia article here is saying that the formula is used in the following way: ...
0
votes
3answers
35 views

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$ Since evolute of a curve is the envelope of the normals of that curve.I ...
0
votes
0answers
14 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
0
votes
0answers
12 views

Converting a Parametric with trig and inverse trig functions to Rectangular form

I came up with a parametric equation for rotating a function $f(t)$ on a graph in three dimensions $$y=\sqrt{f(t)^2+t^2}\sin{\left(\beta+\arctan{\frac{f(t)}{t}}\right)}\cos{\alpha}$$ ...
3
votes
2answers
5k views

Is it possible to calculate sine by hand?

Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?
-1
votes
2answers
41 views

Getting $\sin^2$ and $\cos^2$ values from $\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$

How can you get from $$\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$$ to $$\sin^2 \alpha = \frac{1}{5} \\or \\ \cos^2 \alpha = \frac{4}{5}.$$ Sorry, but I can't see ...
8
votes
2answers
120 views

New very simple golden ratio construction incorporating a triangle, square, and pentagon all with sides of equal length. Is there any prior art?

Consider three regular polygons with 3, 4, and 5 sides wherein all the polygons have sides of equal length X throughout, as illustrated below. The ratio of the red line segment a to the blue line ...
-1
votes
0answers
27 views

Perimeter and Circumference of n-sided polygon

Given the sidelength, number of vertices and vertex angle in the polygon, how can one calculates the perimeter of an n-sided polygon that circumscribes a circle of radius r. And how can they use that ...
0
votes
0answers
21 views

Differential equations with inverse trigonometrical functions

$(x^2+y^2)^(1/2)$=$e^(asin(y/(x^2+y^2)^1/2))$ Prove that $\frac{d^2 y}{dx^2}$=$2((x^2+y^2))/(x-y)^3$, x>0 I started with taking the natural log of the given equation and differentiating it, I ended ...
0
votes
2answers
36 views

How to find area of isosceles triangle when given two heights? [on hold]

So I know the sine and cosine theorem and I tried using them but I got nowhere. (I got to an equation which I can't solve and I know there must be an easier method since we have not studied how to ...
1
vote
3answers
64 views

New very simple Golden Number Ratio PHI construction with Circle and Two Equal Segments of Circle Diameter. Is there prior art? Proofs?

Geogebra gives me the golden number PHI to fifteen decimal places for this simple construction illustrated below wherein the ratio of the blue line i to the red line h is PHI or 1.6180.... The golden ...
0
votes
0answers
18 views

Increasing or decreasing theta based on direction of vector

I am a programmer, and I have some holes in my math knowledge that I am working on filling in. Right now I'm working with a simple process involving drawing curves and straight lines. The line that is ...
-1
votes
1answer
15 views

Calculate edge of right triangle, two edges given

Given two coordinates of a right triangle's leg/cathetus' edges (x0,y0),(x1,y1) and the length of the other leg/cathetus (L). How do I calculate the coordinates of the remaining edge depending on ...
0
votes
0answers
15 views

Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
1
vote
1answer
21 views

proof of $\sin(420º+\alpha) + \cos(60º+\alpha) = \sin(90º-\alpha)$?

I was trying to proof this using the right side, and I'm aware that $\cos (60 + \alpha) + \cos(60 + \alpha)$ it's what I'm really looking for but I can't find a way to proof it. \begin{align} \sin ...
0
votes
2answers
40 views

Does $f(x) = -\sin(2x)$ have two integrals?

I found $\cos^2(x)$ and $\sin^2(x)$ which happily differ by the constant of $1$ though I've also found $\frac 12 \cos(2x)$, which both of the former diverge from by a sinusoidal function, what's wrong ...
0
votes
1answer
66 views

Polynomials with roots having the same module and linear dependent arguments

Is it possible for a polynomial with integer coefficients to have some of its roots: $$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$ such that there exist nonzero integers ...
2
votes
4answers
127 views

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I've learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is ...
2
votes
4answers
48 views

how to verify $\frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x))}$? [on hold]

How would I verifty the following trig identity? $$ \frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x)} $$ I am not sure how to start.
1
vote
3answers
61 views

Angle between squares at which they just touch along the circumference of a circle

Say I have two squares whose centers fall along the circumference of a circle. The circle has radius $x$. The squares have the same height and width $y$. The height of one square is parallel to the ...
2
votes
4answers
10k views

How to get sine / cosine value out of tangens

I know that: $\tan(\alpha) = 1/2$. How can I get clean values for sine / cosine without the calculator? Is there a relationship? I know that $\sin(\arctan(1/2))$ is a way ... But I hope you get the ...
0
votes
0answers
21 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
2
votes
1answer
22 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
1
vote
0answers
38 views

Pythagorean triplet triangle associated with Golden Ratio Construction. [on hold]

MoreGoldenPythagorusTriangle Just recognised simple connection by Astrophysics Math with a particular Pythagorean triplet proportion leads to an easy method to construct $ \varphi$ or its reciprocal. ...
0
votes
0answers
38 views

Finding sides using trigonometry.

I am currently studying for a maths exam and I came upon a question with this diagram With the question asking to solve all of the variables. No other information was given. How is this possible? ...
1
vote
1answer
46 views

Why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$

The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve. I want to ...
0
votes
3answers
40 views

Value of the given expression …

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ? I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't ...
1
vote
1answer
67 views

Area of Pentagon question

Suppose that a regular pentagon circumscribes a circle of radius $r$. Show that the area of the pentagon is $5r^2\tan(36°)$. I know that the area of a triangle is $\frac{1}{2}bh$, where $b$ is the ...
0
votes
3answers
2k views

Harder Trigonometry Identity ($\sec A+\csc A$)

How do I prove: $\sin A (1 + \tan A) + \cos A (1 + \cot A) = \sec A + \csc A$ I've tried expanding the brackets by multiplying sin A and cos A to the left hand side but to no avail. Where should I ...