Questions tagged [trigonometry]

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

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Find the value of trigonometric expression: [closed]

Given, $$\frac{27\sin^3(9) + 9\sin^3(27) + 3\sin^3(81) + \sin^3(243))}{\sin (9)}$$
32 views

Shortest side of triangle

What is the shortest side of a triangle whose internal angle measures are in the ratio 2 : 5 : 8, and the perimeter 48 cm? I calculated the angle measures, a + b + c = 48 and sine law but i don't know ...
1 vote
43 views

How does one calculate the amount of bodyweight pressed in a push-up at varying levels of incline?

I came across the following chart: Push-Up Variation by Incline Level It was sought out because, well, quite frankly, I am very out of shape and limited to resources around the house. In other words, ...
1 vote
102 views

$\sum_{k=0}^{n-1}\cos^m\frac{2\pi k}{n}$ [closed]

Let $n$ be a positive integer and let $m$ be a positive even number such that $n>2$ and $n>m$. Show that $$\sum_{k=0}^{n-1}\cos^m\frac{2\pi k}{n}=\frac{(m-1)!!}{m!!}n.$$
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Why, fundamentally, does adding sin graphs together always produce another sin graph?

Consider if you want to graph: $$6\cos x\ +\ 3\sin x$$ It produces another sin graph: The green is the new graph. I understand the auxiliary angle explanation. We can transform $6\cos x\ +\ 3\sin x$ ...
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1 vote
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Help in calculating the following infinite product [duplicate]

$[ \lim_{{n \to \infty}} \cos \frac{\pi}{2^2} \cos \frac{\pi}{2^3} \cdots \cos \frac{\pi}{2^n} ]$ I know the answer should be $2/π$ but stuck otherwise. Thought about using series expansion and ...
1 vote
67 views

How to show that $\displaystyle\arccos{\alpha} = 2\arctan{\sqrt{\frac{1-\alpha}{1+\alpha}}}$ for $\alpha\in (-1,1]$? [duplicate]

I was working through a solution for an integral, and one of the last steps made use of this identity, which I have never encountered and struggled to find anywhere. I've tried messing around with ...
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Determining a tetrahedron from 2 faces and a 1 dihedral angle [closed]

If I know 2 faces of a tetrahedron (that is, 5 edge lengths in total as the 2 faces share 1 edge), and 1 dihedral angle, how can I determine the 6th edge? Is the 6th edge always unique? Does it matter ...
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Calculating the trapezoid angle needed to fill the space to produce an n-gon cone

The situation is a bit convoluted. But essentially I am trying to create a bowl using a table saw. This table saw can do angled cuts (we call them mitered cuts) as well as tilt the table saw. I am ...
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Showing $\sum_{cyc}\frac{\cos^2A}{1+\cos A} \geq \frac12$ for acute $\triangle ABC$ [duplicate]

If $A+B+C = \pi$ and $0< A,B,C < \frac{\pi}2$, then: $$\frac{\cos^2(A)}{1+\cos(A)} +\frac{\cos^2(B)}{1+\cos(B)} +\frac{\cos^2(C)}{1+\cos(C)} \geq \frac12$$ I used Lagrange's multipliers and ...
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Maximum value of $\cos(A)+\cos(B)+\cos(C)$ [duplicate]

How can we find the maximum value of $\cos(A)+\cos(B)+\cos(C)$ where $A+B+C=\pi$. Is there any beautiful way rather than to tediously solve the equation?
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Show that $\sum_{r=0}^{14}\cos(\frac{2\pi ir}{15})\cos(\frac{2\pi jr}{15})=0$ for all $0\leq i\neq j\leq 7$. [duplicate]

As an exercise I've been working through constructing the character table of $D_{15}$, and am getting caught up on one of the final steps. To show that the representations I constructed are ...
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How does $\delta \theta_1 = \frac{\Delta v \delta t}{\Delta x}$?

In matrix form, the rate-of-strain tensor is: \{e_{ij}\} = \begin{pmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \...
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Angle between two bisecting lines when given their perpendiculars

I am attempting to calculate the angle B in the attached image. Essentially, I am using a radiograph to calculate the angles of each vertebra and then computing the angles between them using the ...
1 vote
101 views

How do I calculate the points within in a polygon that changes height and width to create a hollow frame shape?

I am using JavaScript to create a shape using clip-path. I have no issues with the code but my issue lies with the math. I can get close to my goal but it fails once the shape's height or width is ...
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Finding angle from point a to point b given grid coordinates in MGRS format

SI mils to Military Mils and the distance formula I have a question about SI mils vs Nato Mils and this looks like a great place to ask for help. Im currently in the military and am a mortar, we use a ...
53 views

Verifying the Identity $-i \log(x + iy) = \tan^{-1}\left(\frac{y}{x}\right)$

I am trying to verify the following complex identity: $$-i \log(x + iy) = \tan^{-1}\left(\frac{y}{x}\right)$$ where $x$ and $y$ are real numbers. Steps Taken Define the complex number $z = x + iy$. ...
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How many roots does $\tan(z)-z^n$ have for $n \in \mathbb{N}$, $\frac{-\pi}{2}\le \Re(z)\le \frac{\pi}{2}$?

Now asked on MO here. I am investigating the number of roots of the equation $$\tan(z) - z^n = 0$$ within the vertical strip $|\text{Re}(z)| \leq \frac{\pi}{2}$ for positive integers $n$. Numerical ...
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Evaluating $\lim_{n\to\infty} \frac{2^n}{x_n}$, where $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$, without proving $x_n=\cot{\frac\pi{2^{n+1}}}$?

Let the sequence $(x_n)$ be such that $x_1 = 1$ and, for $n\geq 1$, $$x_{n+1} = x_n + \sqrt{x_n^2 + 1}$$ Find $$\lim_{n\to\infty} \frac{2^n}{x_n}$$ Is there any way to do this without proving by ...
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Given a quadratic bézier curve, how can I know if P1 is within $\vec{V_1}\bot\overrightarrow{PT\to P2}$ and $\vec{V_2}\bot\overrightarrow{PT\to P0}$

In this case, t = 0.5 so PT is the middle of the curve From what I understand, there are 2 possible solutions to this problem: ...
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Relationship between Pythagorean Triples and double/half angles

I just happened upon something that I had never noticed before. I was doing a problem given $\tan u = \frac34$ where $0<u<\frac\pi 2$, find the exact values of $\sin 2u$, $\cos 2u$, and $\tan 2u$...
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How to find the period of a trigonometric function out of its graphical representation [closed]

I don't know how to find period on this graph, from what I know it should be the distance between full cycle points. On this graph we don't see the other point so I'm confused. Hereby my graph:
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