Questions tagged [trigonometry]

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

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Why are $\cosh$ and $\cos$ related by the imaginary unit?

I'm interested, why are $\cosh$ and $\cos$ related by the imaginary unit like$$\cos (x)=\cosh (ix)?$$Is there any visual proof? How can be circle related to hyperbola by complex unit?
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3 votes
1 answer
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Cosine of very large integers, different calculators giving different results

I want to calculate cosine of 452175521116192774 radians (it is around $4.52\cdot10^{17}$) Here is what different calculators say: Wolframalpha Desmos Geogebra Python 3.9 (standard math module) ...
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2 answers
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Calculate $C=\sin3\alpha\cos\alpha$ if $\tan2\alpha=2$ and $\alpha\in(0^\circ;45^\circ)$.

Calculate $$C=\sin3\alpha\cos\alpha$$ if $\tan2\alpha=2$ and $\alpha\in(0^\circ;45^\circ)$. My idea was to find $\sin\alpha$ and $\cos\alpha$. Then we have $\sin3\alpha=3\sin\alpha-4\sin^3\alpha$. So $...
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Find $B=\cos\frac{\alpha}{2}\cos\frac{3\alpha}{2}$ if $\tan\alpha=\sqrt2$ and $\alpha\in(0^\circ;90^\circ).$

Find $B=\cos\dfrac{\alpha}{2}\cos\dfrac{3\alpha}{2}$ if $\tan\alpha=\sqrt2$ and $\alpha\in(0^\circ;90^\circ).$ My try: $$\tan\alpha=\sqrt2=\dfrac{2\tan\dfrac{\alpha}{2}}{1-\tan^2\dfrac{\alpha}{2}}\\\...
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Does $\cos{ix}$ really equals to $\cosh{x}$ ? (graphic representation)

It is proved that $$\cosh{ix} = \cos{x}$$ so it also works for $u := ix$ such as: $$\cosh{u} = \cos{-iu} = \cos{iu}$$ Thus, I assumed that $\cosh{x} = \cos{-ix}$... Yet, when I draw both of those ...
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Evaluation of $\int\sqrt{1+2\sin(\theta)^2\cos(\theta)^2}\mathrm{d}\theta~~\text{where}~~\theta\in\left[0,{\pi\over4}\right]$

$$\begin{align} I&:=\int \sqrt{1+2\sin(\theta)^2\cos(\theta)^2} \mathrm{d} \theta ~~~\text{where}~~~~\theta\in\left[0,{\pi\over4}\right]\\ \cos(\theta)^2&=1-\sin(\theta)^2\\ I&=\int\sqrt{1+...
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1 answer
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Finding the side lengths of a 45 degree triangle with shared hypotenuse.

Right triangle A with hypotenuse 1 and sides x and y is graphed at the origin. Right triangle B is graphed so that it shares a hypotenuse with triangle A and it's sides w and z are parallel to the ...
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What is the significance of the tangent equation $f(x) = A \tan\left(\frac{x+B}{C}\right) + D$?

I stumbled upon it myself while working with computer vision. Does anybody know any other references to this equation? $$f(x) = A \tan\left(\frac{x+B}{C}\right) + D$$
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How to make calculations for a complex 3-D conduit shape?

I'm an electrician running conduit horizontally along a wall at a height of 6 feet off the floor, when I come to a corner of the room. Normally, I would bend a 90-degree elbow and continue level along ...
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Calculating a building's shade with building height

I work in GIS and to create a shade layer of a building, I need to "translate" the geometry/building or permanenently move it. But I just need some help with my formula based on this video. ...
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2 votes
3 answers
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Evaluate: $\prod_{n=1}^{n=9}(\sin(20n-10^\circ))$

The given answer is $2^{-8}$ My attempt $$=\sin(10^\circ) \sin(30^\circ) \sin(50^\circ)\cdots\sin(170^\circ)$$ after this I multiply and divide to fill in the even multiples of $10^\circ$ $\dfrac{\sin(...
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2 votes
2 answers
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How to find the closed form of $\int_{-\infty}^{\infty} \frac{x^{2 n+1} \sin x}{\left(1+x^{2}\right)^{n+1}} d x, \textrm{ where }n=0,1,2,3,…?$

In my post, I had found the exact value of the integral $$\displaystyle I:=\int_{-\infty}^{\infty} \frac{x^{3} \sin x}{\left(1+x^{2}\right)^{2}}dx= \frac{\pi}{2 e}\tag*{} $$ by differentiating $J(a)$w....
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If $2\arccos(\frac45)-\arcsin(\frac45)=\arctan(y)$ then find the value of $y$

If $2\arccos(\frac45)-\arcsin(\frac45)=\arctan(y)$ then find the value of $y$ My Attempt: Using $2\arccos(x)=\arccos(2x^2-1)$, I get $$2\arccos(\frac45)=\arccos(2\times\frac{16}{25}-1)=\arccos(\frac7{...
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2 votes
4 answers
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Is $\tan^{-1}(0)=\pi$ or is $\tan^{-1}(0)=0$?

This question came in the Dhaka University admission exam 2006-7 Q) The value of $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3$ is - (a) $0$ (b) $\frac{\pi}{2}$ (c) $\pi$ (d) $2\pi$ My attempt: $$\tan^{-1}1+\tan^{...
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1 vote
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Why is the sine of x squared equivalent to sine squared of x?

Why is the sine of x squared equivalent to sine squared of x? How can we verify this or what is the intuition behind it?
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2 answers
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Find n evenly spaced points on circle with radius r

I have a circle that has a given radius $r$. I want to generate $n$ evenly-spaced points along this circle, something like this diagram (with $r=5$ and $n=4$):
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Number of solutions of equation $\tan^{-1}(2\sin x)=\cot^{-1}(\cos x)\;$ in $[0,10\pi]$.

Number of solutions of equation $\tan^{-1}(2\sin x)=\cot^{-1}(\cos x)\;$ in $[0,10\pi]$. My Approach: Using the formula $\tan^{-1}(x)\;+\;\cot^{-1}(x)=\dfrac{\pi}{2}$ our equation get converted to $\...
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4 votes
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A navigation problem: Is the path of the ship straight or curved?

My first post here: I’m looking for some guidance with a maths problem. A ship sets sail from England (A) to France (B) covering a distance of 20 miles at an average speed of 5mph. If the ship sails ...
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Proof of equality given by wolfram alpha

By 'playing' with wolfram alpha and entering this sum $$\sum_{n=0}^{+\infty}(-1)^n\left(\frac{1}{1+nx}+\frac{1}{x-1+nx}\right)$$ I found this alternative form that interest me: (since I don't know the ...
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1 answer
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Why am I getting different angles between the vectors in these two different processes?

This question came in the Dhaka University admission exam 2007-08 Question: Two vectors $\vec{P}=\hat{i}+2\hat{j}-2\hat{k}$ and $\vec{Q}=3\hat{i}+2\hat{j}+2\sqrt{3}\hat{k}$ are acting at a point at ...
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1 answer
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The graphs of $y=ax$ and $y=\arctan(bx)$ intersect at three distinct points if? [closed]

I have a question relating to calculus and inverse trigonometric functions. Any help is appreciated. The question is: The graphs of $y=ax$ and $y=\arctan(bx)$ intersect at three distinct points if? A:...
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Find solutions to $2\cos^2\frac x2=\cos^2x$ in range $[0,2\pi]$

I've been stuck on this problem for a few hours now. I have tried half-angle formulas but it seems like it doesn't work. A little help (or hint) would be really nice.
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How to prove cosine law using the power series expansion of cosine and sine?

How to prove the equation $\cos(x + y) = \cos(x) \cos(y) -\sin(x) \sin(y)$ using the power series expansions \begin{equation*} \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, \qquad \sin(x)...
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How does cos(4000pi/3) have the same value as cos(4pi/3)?

Why does the magnitude not matter in a trigonometric function like this? Please give as detailed an answer as possible thank you.
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4 votes
1 answer
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Is there a way to derive sin/cos series from multiple angle formulas?

I was reading about multiple angle formulas to expand $\sin{(nx)}$ or $\cos{(nx)}$ to be in terms of $\sin{x}$ and $\cos{x}$ on Wolfram MathWorld, and came across these formulas: These formulas look ...
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3 votes
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If $\arcsin(x^2+y^2)+\arctan(4y^2-1)+\operatorname{arcsec}(x)=a$ then $a$ belongs to

If $\arcsin(x^2+y^2)+\arctan(4y^2-1)+\operatorname{arcsec}(x)=a$ then $a$ belongs to p) $\{0\}$ q) $(\frac\pi2,\frac{3\pi}2)$ r) $[0,\frac\pi2)$ s) $\mathrm R$ t) $\mathrm R-\{0\}$ My Attempt: $\...
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1 vote
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Derivative of $\arcsin(x)=\frac{1}{\cos(x)}$ .

My intuition says that since $\sin(x)$ and $\arcsin(x)$ are inverse of each other, their derivatives must be reciprocal. I have previously proved the fact that $\arcsin(x)'=\frac{1}{\sqrt{1-x^2}}$ but ...
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-3 votes
2 answers
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How to express $(2\cos^2{x}-1)\sin{x}\cos{x}$ as $ \frac14 (\sin{4x})$?

I was wondering how I could do this problem. I haven't seen many questions that ask me to express one expression as another in such a format. I know the the two are equal, but I'm not really sure how ...
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Rotate a object to face a point

I have an object (point with rotation) and a point. I want to get the degree of rotation that will allow the object to point towards the point. The following equation gets me correct behavior in ...
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1 vote
4 answers
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How to show that $f(x)=\frac{\arcsin x}x$ is increasing when $x\ge0$?

How to show that $f(x)=\frac{\arcsin x}x$ is increasing when $x\ge0$? My Attempt: $f'(x)=\frac{\frac x{\sqrt{1-x^2}}-\arcsin x}{x^2}$ Since $0\le x\le1\implies0\le\sqrt{1-x^2}\le1$ And $0\le\arcsin x\...
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0 votes
1 answer
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Closed form of $\int_{0}^{\infty} \frac{x^{a} \ln^n x}{1+x^{b}} d x$ for natural number $n$?

I am going to evaluate the integral $$\displaystyle I=\int_{0}^{\infty} \frac{x^{a} \ln x}{1+x^{b}} d x, \tag*{} $$ where $b>a+1>0$, by its partner integral $$\displaystyle J(a)=\int_{0}^{\infty}...
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Find the value of $\tan ^{-1}(\frac 12 \tan 2A)+\tan^{-1} ( \cot A) +\tan ^{-1}(\cot ^3 A)$

$\tan ^{-1}(\frac 12 \tan 2A)+\tan^{-1} ( \cot A) +\tan ^{-1}(\cot ^3 A)=\begin{cases} 0 & \frac {\pi} 4\lt A \le \frac {\pi} 2\\ \pi & 0\le A \le \frac {\pi} 2 \end {cases} $ Try $\tan ^{-1}(\...
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4 votes
4 answers
134 views

Is there a shorter or more trivial way to prove that $ x > \cos (x)-\cos (2 x) $ holds for all $x>0$?

I want to prove that the inequality $$ x > \cos (x)-\cos (2 x) $$ holds for all $x>0$. My attempt: Since the function on the RHS is periodic, we can find the position of extrema (on first ...
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What would be the number of solution for eqn $\tan x \sin x -1 = \tan x - \sin x$ for all $x\in [0,2\pi]$

number of solution for eqn $\tan x\cdot\sin x-1=\tan x-\sin x$ for all $x$. I have tried solving this in two different methods. Form 1 st method I get 2 number of solutions possible and from 2 nd ...
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Derive the Pythagorean Theorem from the Taylor series for trig functions

Inspired by this recent question. Starting with the function definitions: $$C(x) := \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$ $$S(x) := \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$ ...
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0 votes
1 answer
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If $2 (a\cos x - \cos 2x) = 1$

If $2 (a \cos x - \cos 2x) = 1$ for $x\in \mathbb R$ then find all possible real values of $a$. I first expanded $\cos 2x$ then used quadratic equation inequalities for the domain $\cos x \in (-1,1)$ ...
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2 votes
0 answers
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How to find the intersections between a sine and cosine function when both have translations and dilations applied

I'm asking for how to solve the general equation $ A \sin(x+B)+C=D \cos(x+E)+F$ essentially No matter which approach I take, I find that I cannot isolate the terms within the two functions to find x ...
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solve for the unknown in each triangle given and round it to the nearest tenth using the law of simes [closed]

this is the triangle one and two. find the missing or unknown in each triangle so that the triangle will be completed
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Place a point on a line, given only the two points (not the equation)

Suppose I have this line, but I do not know the equation to draw it. I only know that there are 2 points on this line : point 1 at (1, 2) and point 2 at ...
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6 votes
3 answers
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Analytic definition implies geometric definition of trigonometric functions

It is well known how we can arrive to the power series definition of trigonometric functions starting from their definition in terms of the unit circle. I'm trying to do the converse, i.e. start from ...
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0 votes
3 answers
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Solve complex equation $\cos(iz)$=$\cos(z)$

I'm not sure how to find all solutions. Of course, it's true for $z=0$. I also know that $\cos(iz)=\cosh(z)$. So far I've managed to transform it into such an equation: $\cos(y)\cosh(x)=\cos(x)\cosh(y)...
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3 votes
1 answer
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Find the period of the trigonometrics functions $\sin[\cos(x)]$ and $\cos[\sin(x)]$

So I need to find the period of these two trigonometric functions: $$\cos[\sin(x)]$$ and $$\sin[\cos(x)]$$ but algebraically and without using the graph of this functions. So to be more clear, my ...
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0 votes
0 answers
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analytic definition of trigonometric function by means of curve length

I'm trying to analytically define trigonometric functions following the geometric intuition. To this aim I define the length $\ell(u)\in[0,+\infty]$ of any curve $u$ in $\mathbb R^2$, consider the ...
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2 votes
1 answer
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Solve: $\sec(2x) \ge\sec(x) , x\in [0,\pi]$\ {$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$}

Here is the following question: Solve: $\sec(2x) \ge\sec(x) , x\in [0,\pi]$\ {$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$} Note: This is part b of a question where in part a, I was asked to solve: $...
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1 vote
1 answer
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How to do the following trigonometric simplification: $ \frac{1- \cos (3\alpha) }{1- \cos (\alpha)} = (1 + 2\cos (\alpha)^2) $

This is probably a trivial question but I don't understand it. Somehow I can't seem to understand how to simplify this expression: $$ \frac{1- \cos (3\alpha) }{1- \cos (\alpha)} = (1 + 2\cos (\alpha))^...
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0 votes
0 answers
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Prove the $\sin$ function is continuous at $0\in\mathbb{R}$

I need to test this and I don't understand how the first part is related to the second part. Claim Use the identity $|\sin(x)| \leq |x|$ when $0 < |x| < \pi/2$ to show that $\sin(x)$ is ...
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2 votes
1 answer
65 views

Finding angle in circle to produce equal areas

I have a circle that is divided into 4 quadrants with a vertical and a horizontal axis. The center of the circle (where the axes cross) is point b. The top of the vertical axis is point d. On the ...
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0 votes
1 answer
109 views

$\sin(x)$ is continuous at $x = 0$ [closed]

I need to test this and I don't understand how the first part is related to the second part. "Use the identity $|\sin(x)|<|x|$ when $0<|x|<\pi/2$, to show that $\sin(x)$ is continuous at ...
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4 votes
0 answers
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Why is $\int_{0}^{2\pi} \int_0^{2\pi} \frac{\ln(21-4(\cos x+\cos y+\cos(x+y)))}{2\ln(9/2)}\frac{dx}{2\pi} \frac{dy}{2\pi}$ almost $1$?

Consider the function $$ f(x,y) = \frac{\ln(21-4(\cos(x)+\cos(y)+\cos(x+y)))}{2\ln(9/2)} $$ Its average value is awfully close to unity: $$ \int_{0}^{2\pi} \int_0^{2\pi} f(x,y) \frac{\mathrm dx}{2\pi} ...
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1 vote
3 answers
76 views

Solution of $\theta$ when $\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$

I came across this trigonometry problem. If, $$\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$$ What is the value of $\theta$ I got the solution that $\theta$ will be $\frac{\pi}{3}$ by expanding the ...
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