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)

2
votes
1answer
68 views

Sum of fractions of squared sines

I'm trying to prove the following approximate identity for $p$ integer: $$ \sum_{l=1}^m\frac{\sin^2\left(\frac{\pi l}{p}\right)}{\sin^2\left(\frac{\pi l}{mp}\right)}\sim \frac{m^2(p-1)}{2}+O(m) $$ ...
0
votes
1answer
31 views

Simple algebraic question mixed up

I know it is very simple but do not know why I am mixed up in it $(.5)(r^2)\cfrac{20-2r}r$ how is this equal to $10r-r^2$ Sorry if it is too easy, thanks for the help.
0
votes
1answer
28 views

Trigonometric equation problem.

Simply and shortly how do I show that this $33 = 33 + 5 \cos(720\cdot t)$ is equal to this $720 \cdot t = 90.$ Thank you for your help.
2
votes
0answers
51 views

How find the range value $a^2+b^2$ if $\cos{(a\sin{x})}=\sin{(b\cos{x})}$ have no solution

if the equation $$\cos{(a\sin{x})}=\sin{(b\cos{x})}$$ have no zero solution,then $a^2+b^2$ range of value $A:[0,\dfrac{\pi}{4})$,$B: [0,\dfrac{\pi^2}{2})$,$C: ...
2
votes
4answers
70 views

Minimum value Of trigonometry expression

FIND THE MIN VALUE OF 4 cosec^2 x + 9 sin^2 x ? Please explain by both calculus and non-calculus methods ?
1
vote
2answers
221 views

What is the minimum value of $\csc x - \sin x$?

What is the minimum value of $\csc x - \sin x$? Differentiating and setting it to zero yields nothing meaningful. How can I find the minimum value?
1
vote
0answers
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?
2
votes
6answers
119 views

explicit expression sought

Consider the equation $$ \cos^2\phi + \alpha\sin\phi\cos\phi-\beta=0\;, $$ where $\alpha,\beta\in\mathbb{R}$. I need to find an explicit expression for $\phi$. I have tried completing the square, but ...
3
votes
2answers
81 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
2
votes
2answers
77 views

Complex Numbers and Hyperbolic Functions

How would you evaluate: $\mathfrak{R}\left[(1+i)\sin\left(\dfrac{(2+i)\pi}{4}\right)\right]$? I know that $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$ and $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$. I have also ...
0
votes
1answer
154 views

Discontinuity of principal argument in nonpositive real axis

Let $\operatorname{Arg}(z)$ be principal argument function defined in branch $(-\pi, \pi]$. Prove that $\operatorname{Arg}(z)$ is discontinuous in every point in nonpositive real axis. "Solution": ...
2
votes
1answer
60 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
2
votes
2answers
132 views

How to simplify this trignometric expression: $4( 3 \sin \theta)( 3 \cos \theta)$?

I was given a circle with a radius of $3$ and in it was a rectangle and an angle $\theta$ extending from the $x$ axis to up with coordinates of $(3 \cos \theta, 3 \sin \theta)$ and the question asks ...
3
votes
3answers
123 views

Ordinary differential equation $y'(t)=\sin(f(t,y))$

One whose solution never makes me happy is the following: $$y'(t)=\sin(y+t)\text{.}$$ I would start by substituting $z(t)=y(t)+t$ to get an ODE in $z(t)$, but then I'm not sure about how to substitute ...
0
votes
1answer
57 views

How do I multiple these matrices together?

As a personal brain exercise, I've recently been trying to work out the math involved with rotating vertices around an arbitrary axis in 3D space. To do so, I've been relying very heavily on the ...
0
votes
1answer
37 views

Compound Angles

I was working on compound angles formula problems, when I encountered this problem: $\sin\left(\frac{5\pi}{9}\right)$ $\cos\left(\frac{7\pi}{18}\right)$ I know how to use the formula, but I'm not ...
0
votes
2answers
79 views

Is 1 rad important?

Of course radians generally come in ratios of π. So is 1 rad important/useful/special? Or, for that matter, is any integer radian measure important? Besides being approximately 57°, I can't seem to ...
2
votes
1answer
32 views

How do I simplify this difference of angles expression using conjugates?

I'm trying to fill in the gaps in my knowledge of simplifying rational expressions using conjugates, but this one stumps me. Given $\tan(\frac{\pi}{4}-\frac{\pi}{6})$, I can work the formula down to: ...
1
vote
2answers
35 views

Show this function can be defined as the limit function

Let f: $ \mathbb{R} \rightarrow \mathbb{R} $ be defined by f(x) = 1 for x $\in \mathbb{Q} $, f(x) = 0 otherwise. We can see f is not regulated. Show that f may be obtained as a limit function: f(x) = ...
0
votes
2answers
36 views

Find extrema on the interval

Problem Find the extrema of the function $$f(x) = cos^2(x)$$ on the interval $ [-4,4]$ I can differentiate and get $$f'(x) = -2 \sin(x) \cos(x)$$ And set that to zero, but I'm pretty sure that's ...
1
vote
2answers
25 views

Multiply segment

Suppose that I have the segment between the points (2, 2) and (3, 4). Empirically, drawing on a piece of paper, I can say that "doubling" the segment leads me to the segment (2, 2), (4, 6) and making ...
0
votes
4answers
100 views

How to solve trigonometric equations of the form $\tan(x)=m$ where $m$ is a real number?

How to solve trigonometric equations of the form $\tan(x)=m$ where $m$ is a number? I know how to solve $\sin(x)=m$ where $m$ is a real number and $\cos(x)=m$ where $m$ is a real number but I don't ...
0
votes
0answers
71 views

Given a non-negative integer $m$ and a positive integer $n$, calculate $\lfloor \frac{m}{n} \rfloor$

Here is the problem: I have a non-negative integer $m$ and a positive integer $n$ I would like to calculate $\lfloor \frac{m}{n} \rfloor$, $\lceil \frac{m}{n} \rceil$ and $m \bmod n$ But I want to ...
6
votes
2answers
154 views

Show that $\sin 10^\circ$ is irrational

So, this is the problem I am working on. Show that $\sin 10^\circ$ is irrational. The solution to the problem is $$1/2 = \sin 30^\circ = 3 \sin 10^\circ - 4\sin^3 10^\circ .$$ Let $$x = 2\sin ...
0
votes
1answer
50 views

How do I prove this Triginomerty equasion?

I got this in a math preparation course I'm taking for college. Can you please help me proving this? $$\frac{\sin2a\cos a-\cos3a\sin a}{2\sin a}=\cos a$$ Thanks!
4
votes
3answers
204 views

How prove this $\cos{x}+\cos{y}+\cos{z}=1$

Question: let $x,y,z\in R$ and such $x+y+z=\pi$,and such $$\tan{\dfrac{y+z-x}{4}}+\tan{\dfrac{x+z-y}{4}}+\tan{\dfrac{x+y-z}{4}}=1$$ show that $$\cos{x}+\cos{y}+\cos{z}=1$$ My idea: let ...
2
votes
1answer
67 views

Solving this equation for $\theta$

$$ 0 = x\cos\theta\cos\phi+y\sin\theta\cos\phi+z\sin\phi$$ Here's what I've tried doing. $$\begin{align} x\cos\theta\cos\phi+y\sin\theta\cos\phi & = -z\sin\phi \\ x\cos\theta+y\sin\theta ...
0
votes
0answers
47 views

Phase vocoder equation

I know I can adjust the frequency of a waveform using a modified version of the sine wave equation $$\mathrm{amplitude}\times\cos(2\pi\times\mathrm{frequency}\times\mathrm{time}+\mathrm{phase})$$ ...
2
votes
2answers
48 views

area of ​​a quadrilateral

Get the area of ​​a quadrilateral? $‎\angle ‎A‎‎‎_{1}‎+‎\angle ‎C‎_{3}‎=30‎^{‎\circ‎}‎‎‎‎‎$‎ $\angle ‎A‎‎‎_{2}‎+‎\angle ‎C‎_{4}‎=90‎^{‎\circ‎}‎‎‎$ $CD=9, DA=5, BC=8 , AB=4$
0
votes
2answers
41 views

Condition of periodic function for |sin πx|

Period of |sin πx| = 1 Wolfram alpha : So why this condition for Periodic function is not true? f(x) = f(x + T) Wolfram alpha :
0
votes
1answer
191 views

Why do I get a domain error for the following computation using law of cosines

Following are two points. lat1, lon1 represent one point and lat2, lon2 represent another point. ...
15
votes
2answers
467 views

Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$

Calculate $$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 ...
0
votes
1answer
23 views

Obtaining consistent triangle surface normals.

I am given 3 points in a random order like so... calculateSurfaceNormal(point1, point2, point3); I have implemented the method by simply saying... ...
1
vote
1answer
171 views

How do I prove this trig identity without geometry?

I need to prove this: $$\cos(x) = \frac{1-\tan^2(\frac{x}{2})}{1+\tan^2(\frac{x}{2})}$$ using only $\sin(a-b)$ and $\cos(a-b)$ formulas wich I already proved. I also proven this: $$\cos^2(x) + ...
0
votes
1answer
44 views

Change in length of a right triangle

And so my question is how do I prove the ??
0
votes
1answer
94 views

How to solve $4\sin x\cos x+2\sqrt3\sin x-2\cos x-\sqrt3=0$?

How can I solve the trigonometric equation $$4\sin x\cos x+2\sqrt3\sin x-2\cos x-\sqrt3=0$$ I used to replace $\sin x$ by $\sqrt{1-\cos^2 x}$ but doesn't work very well ;° I just want a hint kiss:°
0
votes
2answers
78 views

Using the formulas de Moivre to deduce trigonometric identities.

Yesterday I made a test of complex variables, and this contained a question (in which I could not solve) that asked to use the de Moivre formulas to deduce the following trigonometric identities: ...
1
vote
2answers
97 views

How to prove: $1-\cos 2\alpha = 2\sin^2\alpha$ [closed]

I am taking a preperation course for college. I got this as homework. Prove that: $1-\cos2\alpha = 2\sin^2\alpha$ Thank you!
0
votes
1answer
134 views

Trigonometry for steradian angle

Polar coordinate system is very closely associated with trigonometry. For instance, given an angle in radian, we can find its corresponding 2-D cartesian coordinates using ...
0
votes
2answers
120 views

Establish a trigonometry-based $floor$ function

I have established the following function for calculating $floor$: $$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$ It works correctly for all real values in the range ...
0
votes
2answers
398 views

Is it possible to get the angle between two vectors in a single direction?

I'd like to compute the angle between two vectors but always in a anticlockwise manner. Is this possible? I know the formula is arc cos (dot product of vectors / product of magnitudes of vectors) but ...
0
votes
1answer
592 views

How do I simplify a radical within a radical in this half-angle problem?

I don't understand how to simplify the following radicals and arrive at the final answer below. I can make it to this point: $$\sin\left(-\frac{3\pi}{8}\right)=\pm\sqrt{1+\frac{\sqrt2}{2}\over2}$$ ...
0
votes
1answer
60 views

Proof of half angle identity

Remember one of the step in proving it is to replace theta with 1/2 A. I am wondering how could you replace a variable to prove the identity. A is not the same as theta. Plus aren't we trying to find ...
10
votes
2answers
393 views

Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
1
vote
3answers
349 views

Prove that $\sec^2{\theta}=(4xy)/(x+y)^2$ only when $x=y$

Show that the equation below is only possible when $x=y$ $$ \sec^2{\theta}=\frac{4xy}{(x+y)^2}$$ The only way I can think of doing this is by rewriting it as $$ ...
0
votes
3answers
162 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
0
votes
1answer
45 views

With which function I can replace $\arctan x^2$ in limit calculation?

I have read I can replace in limits $\arctan x$ with $x$. I conjectured that I can replace $\arctan x^2$ with $x^2$, however in following example it doesn't work: $$\frac{2x\left( 105x^4+150 x^2 ...
2
votes
2answers
64 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
0
votes
1answer
21 views

Question about graph of sin function

There is question I had on my mind for 3-4 years. If you have a function:- f(x) = (sin(x))^n If you increase the value of n slowly the Value of the function at all points, except where it is one, will ...
1
vote
2answers
97 views

Proof for value of sum of sine and cosine

I have come across those sums of sine and cosine, while trying to show that windmills dont move without external force. Although it is clear that should be the case, i'm stuck in proofing it. I wonder ...