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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2
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2answers
39 views

For angles $A$ and $B$ in a triangle, is $\cos\frac B2-\cos \frac A2=\cos B-\cos A$ enough to conclude that $A=B$?

Brief enquiry: $$\cos\frac B2-\cos \frac A2=\cos B-\cos A$$ Optionally $$\sqrt\frac{1+\cos B}{2}-\cos B=\sqrt\frac{1+\cos A}{2}-\cos A$$ Is above equality sufficient to prove that it ...
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1answer
34 views

Solving for side lengths of a triangle given specific conditions.

For a given constant $b > 10,$ there are two possible triangles $ABC$ satisfying $AB = 10,$ $AC = b,$ and $\sin B = \frac{3}{5}.$ Find the positive difference between the lengths of side $\overline{...
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4answers
37 views

How to solve $1-\tan(2x+\frac{\pi}{2})=0$? $0 \le x \le 2\pi$

I did not quite get how it is related to the period of the equation, I would really appreciate if anyone could please shed some light on this topic.
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2answers
41 views

Can we express circle with trigonometry function

As we know a unit circle equation is $y^{2}+x^{2}=1$. And we also know that in a unit circle adjacent(or x) is $\cos θ$ and opposite(or y) is $\sin θ$. So this means we can also express a unit ...
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1answer
41 views

I am having trouble with computing area of rose curve.

I computed the area of rose curve, $r = \cos (3\theta)$ by computing one petal first and then multiply it by $3$. then I evaluated the same integral again over the interval $[0,2 \pi]$. Why aren't ...
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2answers
27 views

Convert linear distance to steering angle

I need to calculate the angle of the front steering wheel using a collapsible piston(linear sensor). 'x' is used to represent the length in inches of the movable part of the sensor and is the ...
0
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1answer
40 views

How would I show the result below using contour integration? [duplicate]

How would I show the result below using contour integration? $$\int_{-\infty}^{\infty} \frac{\cos bx - \cos ax}{x^2} dx = \pi (a-b)$$ where a>b>0 using contour integration. Any help would be greatly ...
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2answers
79 views

Finding range of $\sin^{20}(\theta)+\cos^{30}(\theta)$.

We have to find the range of $$\sin^{20}(\theta)+\cos^{30}(\theta)$$ I have found the upper limit which is $1$. I am a high school student and we were taught to convert functions into a simpler ...
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0answers
26 views

A trigonometry and proving question

Given that that $$\tan x = \frac{\sin r-\cos r}{\cos r+\sin r}$$ where $x$ is an acute angle, prove that $$\sin x=\frac1{\sqrt{2}}(\sin r-\cos r).$$
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1answer
47 views

Question from PRMO 2019.

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more ...
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1answer
37 views

First trigonometric differential equation

Show that: $$\tan(x) \frac {dy}{dx}-y=\sin^2(x)+2\sec(x)$$ where $y=\sin^2(x)-2\cos(x)$ I get: $\frac {dy}{dx}=2sin(x)cos(x)+2sin(x)$ =$tan(x)(2sin(x)cos(x)+2sin(x))-sin^2(x)-2cos(x)$ From ...
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2answers
60 views

Determine $\arctan{e^i}$

In this answer, the quantity $\arctan(e^i)$ must be determined. It's obviously $e^i = \cos(1) + i\sin(1)$, but there is no formula for $\arctan(x + y)$ like $\sin(x + y)$, for example, and I'm stuck. ...
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0answers
42 views

If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ ; is $f$ zero a.e?

If $f \in L^1[0,1]$ satisfies $\int_{0}^1f(x)e^{-2\pi i n x}dx = 0$ for $n \in \mathbb{N}$ ; is $f$ zero a.e? Here $L^1[0,1]$ means the set of measurable functions $g : \mathbb{R} \rightarrow \mathbb{...
2
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1answer
23 views

Basic trigonometry + polygon geometry Where am I going wrong?

Question You are given a regular polygon with $2⋅n$ vertices (it's convex and has equal sides and equal angles) and all its sides have length $1$. Let's name it as $2n-gon$. Your task is to find the ...
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2answers
36 views

Finding the hypotenuse of a triangle using angles and segment lengths. [closed]

In the diagram, $\angle CAB = 90^\circ.$ Let $D$ be a point on $\overline{AB},$ and let $E$ be a point on $\overline{AC},$ such that $AB = AC = DE,$ $BD = 9,$ and $CE = 8.$ Find $DE.$ [asy] unitsize(...
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1answer
14 views

Calculating Velocity from Speed, X,Y Bearing, and Z Bearing

I'm not entirely sure how to go about it, but I have a data structure that gives me ...
0
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4answers
67 views

How to calculate a point on the surface of a cone?

Consider a cone whose central line is located at $z=0$. We know the points on the upper and lower lines (red lines). Let's call them $x_1,y_1$ and $x_2,y_2$. How can we obtain the value of $z$ for ...
1
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1answer
19 views

A doubt regarding proof of values of trigonometric functions at allied angles

There are certain identities that help us to determine the values of trigonometric functions at $\dfrac{\pi}{2}+x \text{, } \pi-x$ etc. given the values of $\sin x, \cos x$. Now, when we prove such ...
3
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3answers
54 views

If $z+\frac{1}{z}=2\cos\theta,$ where $z\in\Bbb C$, show that $\left|\frac{z^{2 n}-1}{z^{2n}+1}\right|=|\tan n\theta|$

If $z+\frac{1}{z}=2 \cos \theta,$ where $z$ is a complex number, show that $$ \left|\frac{z^{2 n}-1}{z^{2 n}+1}\right|=|\tan n \theta| $$ My Approach: $$ \begin{array}{l}|\sin \theta|=\left|\sqrt{1-\...
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0answers
44 views

Is there another way of expressing $\sin^{-1}(\cot\theta)$?

Is there another way of expressing $\sin^{-1}(\cot\theta)$? Something similar to, for example, $\sin^{-1}(\cos\theta)=\frac{\pi}{2}-\theta$.
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2answers
44 views

Prove that the function $f :\Bbb R \to \Bbb R$ defined by $f(x) = e^{-\cos(x)^2}$, for all $x \in\Bbb R$, has a unique fixed point on $\Bbb R$.

Hint: some arguments might be simpler if you recall the trigonometric formula $2\sin(x)\cos(x) = \sin(2x)$. Remember also that $\cos$ and $\sin$ are $2\pi$-periodic functions. I am a bit lost with ...
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1answer
22 views

Trigonometric identities: harmonic form, negative statements

Find the min and max of these expressions and state smallest non-negative value of θ for which each occurs: 1) 10 - 2sinθ + cosθ 2) 1/((7 - 2cosθ + sqrt(5)sinθ)) 1) Harmonic form: R[sin(θ+α) = ...
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0answers
26 views

Check if the system: $\sin \left(\frac{2\pi n x}{b-a} \right), \cos \left(\frac{2\pi n x}{b-a} \right)$is orthogonal over the interval [a=12, b=14].

Check if the system: $\sin \left(\frac{2\pi n x}{b-a} \right), \cos \left(\frac{2\pi n x}{b-a} \right)$is orthogonal over the interval [a=12, b=14]. My work thus far: $$\begin{align} \int^{14}_{12} \...
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2answers
32 views

What is the proof for the factor formula? [duplicate]

This is known as the factor formula. It is used for the addition of sin functions. I don't understand how the two are equal though. How would you get to the right side of the equation using the left? ...
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1answer
30 views

Drone camera view frustum

I am trying to derive formula (2) from here. Formula describes the size of an area seen from UAV camera: Length $W$ is given in the paper as $D\frac{\sin(α)}{cos(θ)cos(α)}$. Camera view $α$ is same ...
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2answers
30 views

$\operatorname{tg}2x=-1$ find $x$ [closed]

$\operatorname{tg}2x=-1$ $x∈[\pi/2,\pi]$ I tried expressing $\operatorname{tg}2x=\sin2x/\cos2x$ but is there any elegant other method?
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0answers
59 views

History of $\sin(nx) = 2^{n-1} \prod_{0}^{n-1} \sin\left(x + \frac{\pi k}{n}\right)$

What is the name of this identity? Who discovered this identity? What is the history behind this? I have looked up on Wikipedia with little documentation of the identity see finite product of ...
2
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2answers
38 views

Lebesgue integral, Is the solution right?

I'am trying to understand Lebesgue integration Compute $\int_{0}^{\pi}$ f(x)dx Where $f(x) = \begin{cases} sin x & \text{ if } x \in \mathbb{I} \\ cosx & \text{ if } x \in \...
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0answers
44 views

Integrate $ \ln(49\sin^2\theta +81\cos^2\theta) $ [duplicate]

$$\int_0^{\frac{\pi}2} \ln(49\sin^2\theta +81\cos^2\theta) d\theta$$ The only thing that I observed that : $$49\sin^2\theta +81\cos^2\theta =(7\sin\theta)^2+(9\cos\theta)^2$$ And we can also use the ...
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2answers
36 views

Deriving polar coordinate form of ellipse. Issue with length of a distance to a foci.

I am reading through in Spivak on how to obtain the polar coordinate form of the ellipse. I'm given the following diagram: All I'm trying to do is establish that the distance between $(x,y)$ and $(-...
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1answer
59 views

Is it possible to define trignometric functions over a finite field?

Is it possible to define trigonometric functions such as sine, cosine, etc modulo a prime p?
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0answers
19 views

Trigonometric identity for $\cos^3 x \cos y$? [closed]

Is there any identity for $\cos^3x\cos y$? If $\cos x \cos y = \frac{1}{2}\left[cos(x-y)+\cos(x+y)\right]$, then how could I rewrite the this product as a sum?
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3answers
37 views

Finding the $\lim \limits_{x \to 0} {1 - \cos(x)\over \sin(x) \ln(1+x)}$ using Taylor's series.

I am a bit stuck. This is what I have so far and I am not sure how to simplify it further: $${{x^2\over 2} - o(x^4)\over (x - {x^3 \over 6} + o(x^5))(x - {x^2 \over 2} +o(x^3))} $$ How do I proceed ...
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3answers
60 views

Solve the equation $6\cos^4 y+\sin^2 y =5$

Solve the equation $$6\cos^4 y+\sin^2 y =5 \qquad (0\leq y \leq 360^\circ).$$ I used quadratics and got the answers $y= 18.0, 162.0, 198.0,$ and $342.0$ degrees, but it differs from the answer given ...
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0answers
24 views

Using trigonometric functions for equation simplification

Assume I have a matrix $D$ whose its entries are as below : Where $A$ and $B$ can be written using using the trigonometric functions for (1) as: My question, Is it possible to simplify (1) more? Or ...
0
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1answer
74 views

How to figure out $\sin(1°)$? [closed]

Can $\sin(1°)$ be found manually so that I can know other measurements of sine without memorizing? As I am a student of class 8, please give the answer in a way that I can understand.
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0answers
20 views

Range of arctan(x) in terms of anticlockwise angles [closed]

Why is the range of $\arctan (x)$ from $-\pi/2$ to $\pi/2$ ? It could be from $0$ to $\pi$ as well since $\pi/2<x<\pi$ gives the negative values of $\arctan(x)$ like $-\pi/2<x<0$.
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3answers
78 views

If $\cos 17x = f(\cos x)$, then show that $\sin 17 x=f(\sin x)$

If $f$ denotes the function which gives $\cos(17x)$ in terms of $\cos x$, that is $\cos(17 x) = f (\cos x)$, then, prove that it is the same function $f$ which gives $\sin(17x)$ in terms of $\sin x$. ...
3
votes
1answer
44 views

Proving the orthogonality of $\sin\frac{2\pi x}{\pi-e}$ and $\cos\frac{2\pi x}{\pi-e}$

I want to prove the orthogonality of the functions: $\sin\left(\dfrac{2\pi x}{b-a}\right)$ and $\cos\left(\dfrac{2\pi x}{b-a}\right)$, where $b=\pi$ and $a = e$ My work: $$\begin{align} \int^{\pi}_{...
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0answers
63 views

What are the possible approaches to try to prove $(\sin x)^{\sin x}<(\cos x)^{\cos x}$ for $x\in\left(0;\frac{\pi}{4}\right)$ [closed]

The original question is here and it states: prove the inequality $(\sin \theta)^{\sin \theta} < (\cos \theta)^{\cos \theta}$ for all $\theta \in (0, \pi/4)$ I'm not enuogh with $+45$ points ...
1
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1answer
32 views

I have a face in 3D space that I would like to move and place (calculate rotation between vectors)

Say I have a plane in 3-dimensional space: I have a normal for the face in the form <x,y,z> I have a rotate function, ...
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0answers
28 views

tan^2 3x-4tan3xSin2x+16Sin^2x, find the range. [closed]

enter image description heretan^23xCos^2x-4tan3xSin2x+16Sin^2x find the range.
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0answers
36 views

How to solve trigonometric product like $\prod_{k=0}^{k=6} \sin\left(\frac{ (2k+1) \pi}{14}\right)$ in fastest way possible?

I considered taking two terms at a time and applying on each pair $$\sin(A)\sin(B) = \frac{1}{2} [\cos(A+B) - \cos(A-B)]$$ But that seems too long, is there a faster way to do this?
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0answers
59 views

A proof for : Let $0<x\leq\frac{\pi}{4}$ then $\sin(x)^{\sin(x)}\leq \cos(x)^{\cos(x)}$

Hi yesterday I see a question (closed) about this inequality : Let $0<x\leq\frac{\pi}{4}$ then we have : $$\sin(x)^{\sin(x)}\leq \cos(x)^{\cos(x)}$$ I found that beautiful so I ...
1
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1answer
79 views

Product $\left[\sin(x)\cos\left(\frac{x}2\right)\right]^{1/2}\cdot\left[\sin\left(\frac{x}{2}\right) \cos \left(\frac{x}4\right)\right]^{1/4}\ \cdots$

I came across this question in the following form: Compute the following infinite product $$\left[\sin (x)\cos \left(\frac{x}{2}\right)\right]^{1/2}\cdot \left[\sin \left(\frac{x}{2}\right) \cos \...
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0answers
24 views

simple trigonometry solving and find the possible answers if [closed]

mpenter image description here here its a multi anser type question
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2answers
41 views

If $C$ is the smallest angle of a triangle, SHOW $\sin(C/2)\leq 1/2$. What's the significance of $C$ being smallest?

Let $A$, $B$, and $C$ be the angles of a triangle, with angle $C$ as the smallest of them. Show that (i) $\sin \left(\frac{C}{2}\right) \leq \frac{1}{2}$ (ii) Hence, or otherwise, show that $...
-3
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1answer
45 views

What are the best books for studying Algebra 1,Algebra 2,Geometry and trigonometry? [closed]

Background :ALgebra I.Non-eucledian geometry.And currently began trigonometry and ALgebra II. What the answer should be:Good Bokks on algebra I(elementary algebra),Algebra II(INTERMEdiate),Geometry ...
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0answers
25 views

Find the rotation of a box to project a corner to a point on a line.

Here are my elements: A camera positioned at [4, 0] with a 45-degree field of view. A box that is 2 units wide, centered at [0, 0], where a particular corner is at [1, -1]. A line segment that is ...
0
votes
1answer
35 views

Right Triangle within another right triangle inside a square

As you can see in the below picture, We have a right triangle inside a big square, and within the triangle there is another small right triangle. The question as follows: Find the length of ...

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