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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0answers
17 views

What happens with the argument of arccos when it's multiplied with i (imaginary number)?

I'm studying for my calculus exam and stumbled upon this particular exercise where I should compute the polar form of the complex number \begin{equation} z=-4-2i \end{equation} so the sample solution ...
2
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1answer
32 views

Algebraic expression for the period of $\cos (\log (x))$?

This question relates only to $x \in \mathbb{R}^+$. The function $f(x) = \cos (\log (x))$ is clearly defined on the positive reals, with a monotonic decreasing period $p(x)$ which is defined at the ...
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2answers
18 views

How to find intersection of 3 sin waves of musical note frequencies

I am trying to look at how musical notes interact with each other. I've graphed the sine waves of multiple musical chords with the individual note's frequencies, both as three sine waves and as one. ...
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1answer
38 views

What do I do from here??? [closed]

Okay so, I have the opposite and adjacent legs, but not the hypotenuse. How would i put that in simplest form? Like what do I do from here? Update: I clearly didn't know what I was doing, and I feel ...
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1answer
25 views

Prove that three segments which intersect a circle pass through the same point

In a scalene triangle $\triangle ABC$ with $AB\ne AC$, I state we have $Y$ which is the point of intersection of the bisector of $\angle A$ with $BC$ and $D$ is the point where the perpendicular line ...
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0answers
5 views

Haversine formula on rotated poles sphere

I'm working with climate data from the CORDEX domain WAS-22. The data are on rotated poles grid defined as: ...
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0answers
19 views

Cross product, but with angles

I have two unit-length vectors $\vec{v_1}$ and $\vec{v_2}$ and I would like to find a unit-length vector that's perpendicular to them, so basically $\vec{v_3} = \vec{v_1} \times \vec{v_2}$. However ...
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3answers
39 views

Solution of a trigonometric linear equation with the method of the added angle

This morning I have done for my students of an high school an exercise with the method of the added angle. I not write all the steps but the principals. The equation is: $$\bbox[5px,border:3px solid #...
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1answer
15 views

Proof of a formula for the period of a $\sin$ function subtracting distances

I would appreciate it if someone could explain this demonstration of the formula for the period of the sine function: $f(x)=\sin ax$. It is as follows: We want to find the period for this function: $...
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0answers
21 views

Finding the analytic root of a multivariate function that involves trigonometric functions

I am currently stuck in a problem, where I need to find an analytic expression $x_0=f(\alpha,\beta)$. Let me first try to explain what $x_0$ is: Let $x$ be a real-valued variable, which must obey the ...
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0answers
32 views

Volume of a solid in the shaded region $y=\frac {2}{\sqrt{x^2+1}}, x=(-1,1)$, rotated about the $x$ axis

Find the exact volume of the solid formed when the shaded region $y=\dfrac {2}{\sqrt{x^2+1}}, x=(-1,1)$ is rotated completely about the $x$ axis. This is the question with the diagram I tried ...
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33 views

About circles and MVT

We know that the MVT proves the existence of some "c" inside the range $(a, b)$ such that $\frac{f(b)-f(a)}{b-a}=f'(c)$ for some well defined function f. We also know that the function $y=\...
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1answer
20 views

Can somebody solve this trigonometry problem by taking LHS? [closed]

If cosA - sinA = √2sinA prove that cosA + sinA =√2cosA I was able to find a solution to this by using cosA - sinA = √2*sin, but can somebody solve this by taking LHS? I tried using the first ...
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1answer
27 views

Computing Slopes Created by a Regular Polygon [closed]

Given a regular polygon with an even number, say $n\in\mathbb{N}$ of vertices, we know that these vertices form the total of $n$ slopes. However, is there a way of determining the exact values of ...
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0answers
25 views

Smallest regular pentagon covering a regular heptagon

A friend gave me a puzzle that seems very very tricky to me, and that we couldn't solve so far. It asks for the minimum size of a pentagon that covers a regular heptagon with sides of length $1$. It ...
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1answer
36 views

Symmedian and orthic triangle properties

I am trying to prove the following lemma that may be useful for junior-level international contests: Given $\Delta ABC$ an acute triangle and $(BE)$ and $(CF)$ its altitudes. Let's consider $(AM)$ ...
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38 views

Proving that $\frac{1 + 3\sin A- 4\sin^3A}{1 - \sin A} = ( 1 + 2\sin A)^2$ [closed]

Can somebody solve this trigonometry problem? Prove that $$\frac{1 + 3\sin A- 4\sin^3A}{1 - \sin A} = ( 1 + 2\sin A)^2$$ I tried various methods but could not solve this.
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3answers
58 views

If $\sin^4a + \sin^2a = 1$, prove $\cot^4a + \cot^2a = 1$

Can somebody help me solve this trigonometry problem? If $\sin^4a + \sin^2a = 1$, prove $\cot^4a + \cot^2a = 1$ I tried various things but could not solve it.
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0answers
25 views

Difficult equations with fractions and variables

i tried but i'm stuck it seems too difficult for me why is this even true? please. help me. For any $\alpha \in \mathbb{R}$ and $n \in \mathbb{N}$ $$ \sum_{p=1}^n \frac{\sin(\alpha+ 2\pi p / n) }{x^2-...
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0answers
39 views

Proving an Identity involving sums related to the $Z(N)$-Ising model

Background: I am trying to construct meromorphic functions satisfying a number of axioms, so-called form factors which are important objects in integrable quantum models, following this paper. ...
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2answers
114 views

Evaluate $\frac{4}{\sin^2 20^\circ} - \frac{4}{\sin^2 40^\circ} + 64\sin^2 20^\circ$

Evaluate the following expression:$$\frac{4}{\sin^2 20^\circ} - \frac{4}{\sin^2 40^\circ} + 64\sin^2 20^\circ$$ I tried combining the whole into a single fraction and using double-angle identity, ...
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1answer
34 views

Showing $(\tan^2A+\tan^2B+\tan^2C)^3 \geq 27(\tan A+\tan B+\tan C)^2$ for non-right $\triangle ABC$ [closed]

I am trying to prove this: Show that if triangle $ABC$ is not a right triangle, then $$(\tan^2(A)+\tan^2(B)+\tan^2(C))^3 \geq 27(\tan(A)+\tan(B)+\tan(C))^2$$
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23 views

Sum and Difference Of Identities [closed]

Let sin A= -2/3 and sin B= 1/3, where A lies in Quadrant IV and B lies in Quadrant II FIND cos (A-B) csc (A+B) tan (A-B)
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1answer
35 views

Given a scalene triangle and its orthocenter, prove that a point on its Euler circle, has a constant line segment

Given a scalene triangle $\triangle ABC$ and $H$ the orthocenter of the triangle. $P$ is a point on the Euler circle of the triangle $\triangle ABC$. The segments $BH, CH$ intesect the opposite sides $...
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1answer
32 views

Given a scalene triangle $ABC$ with $H$ orthocenter, prove that two lines are parallel

Given a scalene triangle $\triangle ABC$ with $H$ the orthocenter of the triangle. The internal bisector of the angle $\angle BAC$ intersects the lines $BH$ and $CH$ at the points $Λ$ and $Θ$ ...
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0answers
30 views

Integral properties question

I have this exercise I have been working on the entire weekend and I just can seem to crack it, pls help. let $f: [0,\mathbb{\pi}]$ $\rightarrow$ $\mathbb{R}$ $$\int_0^{\mathbb{\pi}}{\sin(t)f(t)dt} = ...
2
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1answer
31 views

How to find the second time when a piezoelectric crystal vibrates given a cubic trigonometric equation?

The problem is as follows: In an electronics factory a quartz crystal is analyzed to get its vibration so this frequency can be used to adjust their components. The length that it expands from a ...
26
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5answers
324 views

Teacher claims this proof for $\frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}$ is wrong. Why?

My son's high school teacher says his solution to this proof is wrong because it is not "the right way" and that you have to "start with one side of the equation and prove it is equal ...
2
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2answers
72 views

How to approach $\int\limits_{-\infty}^{a}\frac{\sin^{-1}e^x+\sec^{-1}e^{-x}}{(\tan^{-1}e^a+ \tan^{-1}e^x)(e^x+e^{-x})}\mathrm{d}x$

How to approach this integral? $$\int\limits_{-\infty}^{a}\frac{\sin^{-1}e^x+\sec^{-1}e^{-x}}{(\tan^{-1}e^a+ \tan^{-1}e^x)(e^x+e^{-x})}\mathrm{d}x$$
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38 views

Interval for $x$ where the equation holds

Find the interval for $x$ where equation holds.
1
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1answer
54 views

$f(x) = \int\limits_{a(x)}^{b(x)} \sin{\sqrt t}\,\mathrm{d}t,$ compute $f^{'}(0)$.

Let $a(x)=\frac{\pi^2}{4} + \cos(3x+\frac{\pi}{2})$ and $b(x)=\frac{25 \pi^2}{4} + 2x^2$. $$f(x) = \int\limits_{a(x)}^{b(x)} \sin{\sqrt t}\,\mathrm{d}t,$$ compute $f^{'}(0)$. My thoughts: To evaluate $...
2
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2answers
23 views

How to find the number of times a weather station registered a certain temperature?

The problem is as follows: The temperature in the city of Daegu on March 1st, 2020 is given by: $15+5\sin\left(\frac{\pi t}{12}+\frac{\pi}{2}\right)$ in celcius, where $t\in [0,24]$. Assume $t$ is ...
1
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2answers
29 views

How to find the number of solutions in $2\cot 4x=3-3\cot 2x$?

The problem is as follows: Find the number of solutions from the equation from below in the given range. $2\cot 4x=3-3\cot 2x$ Assume $x\in \left[0,\frac{3\pi}{4}\right]$ I am not sure what to do ...
1
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2answers
49 views

How to evaluate from the number of solutions for $\sin 2x +\cos 3x - \sin 4x = 0$?

The problem is as follows: First find the number of solutions for the equation from below: Assume $x \in [0,2\pi]$ $\sin 2x +\cos 3x - \sin 4x = 0$ Let $n$ be the number of solutions. Using this $n$ ...
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2answers
47 views

How to find the solution set for $\tan ^2 x= (\sqrt{2}-1)^2$?

The problem is as follows: Find the set solution for the equation from below: $\sqrt{2}-\tan (2x)=\cot \left(\frac{\pi}{4}+x\right)$ Well after doing all the necessary algebra I'm getting the ...
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2answers
33 views

Struggling with the Missing Side Length with One Length

In class, we have been learning about using proportions to solve for missing lengths of triangles, however, this problem has been leaving me confused. Both of the hypotenuses of the two right ...
3
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2answers
68 views

“Summing” the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-…$

"Summing" the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$ Pose $$S=\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$$ $$C=\...
1
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1answer
54 views

Why does $\sin(x)=\sin(y)$ not equal $\sin^{-1}\left(\sin\left(x\right)\right)=y$?

The equation $\sin(x) = \sin(y)$ does not produce the same output as $y=\sin^{-1}(\sin(x))$. Can anyone explain why this is?
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2answers
37 views

$0 = \sqrt{2}\cos(x) - 2\sin(x)\cos(x)$, why do I have to factor out $\cos(x)$, not to divide by $\cos(x)$?

Solve for $x$. $$0 = \sqrt{2}\cos(x) - \sin(2x)$$ $$0 = \sqrt{2}\cos(x) - 2\sin(x)\cos(x)$$ Here, I thought I could divide all terms by $\cos(x)$ to get $$0 = \sqrt{2} - 2\sin(x)$$ But the solution ...
2
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2answers
81 views

$\int \sin( \pi x) \cos( \pi x)\ dx$

I would like some help computing this integral: $$\int \sin(\pi t)\cos(\pi t)dt$$ Apparently, the answer should be $\frac{\sin^2(\pi t)}{2\pi}+C$ but I get $\frac{-\cos(2 \pi t)}{4 \pi}$+C instead. ...
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0answers
50 views

Interesting sequences when computing nested square roots

In my previous post, I defined $$f(\color{blue}{m},\color{red}{n})= \sqrt{2\color{blue}{-}\sqrt{2\color{blue}{-}\cdots\color{blue}{-} \sqrt{2\color{red}{+}\sqrt{2\color{red}{+}\cdots\color{red}{+} \...
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1answer
36 views

Restricting Domain and Range in Inverse Trigonometric Function

After an explanation of the restricted domains and ranges of inverse trigonometric functions, I.M. Gelfand's Trigonometry gives the following exercise: Show that $$\sin(\arccos b) = \pm \sqrt{1-b^2}$$ ...
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0answers
54 views

How to calculate this product: $\sin 1^\circ\times\sin 2^\circ\times\cdots\times\sin90^\circ$ [duplicate]

How to calculate this product: $$\sin 1^\circ\times\sin 2^\circ\times\cdots\times\sin90^\circ$$
1
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1answer
31 views

trigonometric equation containing sin(x) and x

I have a problem solving the following trigonometric equation and would be happy for any help. $$\sin(2x) = 3,5x$$ What I tried is: $$2\sin(x)\cos(x) = 3,5x\\ ...
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0answers
62 views

Solving $3\sin (x)- \tan(x) = -\sqrt{3} \sin(x)\tan(x) - \tan(x)$. Why are $0$ and $\pi$ solutions?

For $0\leq x < 2\pi$, solve $$3\sin (x)- \tan(x) = -\sqrt{3} \sin(x)\tan(x) - \tan(x)$$ The solutions were $0, \frac{2}{3}\pi, \pi$ and $\frac{5}{3}\pi$, and I don't understand why $0$ and $\pi$ ...
1
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2answers
43 views

Proving that if $y=\operatorname{arcsec}(x)$ then $\frac{dy}{dx}=\frac{1}{x\sqrt{x^2-1}}$

I'm trying to prove this formula however, I cannot seem to figure out how to single out the $x$ and remove its power. I would very much appreciate your help towards this. $$y=\operatorname{arcsec}(x) \...
0
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1answer
57 views

Why is $\arctan(e^{i0})=\frac\pi4$? [closed]

Why is $\arctan(e^{i0})=\dfrac{\pi}{4}$? I am doing exactly what A Level-Student is doing here I wish to ask why $\arctan(e^{i0})=\dfrac{\pi}{4}$?
1
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0answers
7 views

Jacobian of curvilinear dynamics, simplifying to avoid division by 0

This comes as a follow-up to other question. The original was about discretizing a nonlinear curvilinear dynamics equation, this follow up is about taking the Jacobian and simplifying such as remove ...
0
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1answer
48 views

Prove that $\sum_{j=0}^{n} r_{j}e^{i\cdot j\theta} = 0$ for any rational $\cos\theta$

Prove that for any $\cos\theta$ that its value is a rational number, there are always some non-negative integers $r_j$, not all $0$, such that $$\sum_{j=0}^{n} r_{j}e^{i\cdot j\theta} = 0.$$ For ...
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0answers
46 views

Is the 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 1 + 𝑐𝑜𝑡2 𝜃 = 𝑐𝑠𝑐2 𝜃 𝑖𝑠 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 ??? [closed]

Is the 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 1 + 𝑐𝑜𝑡2 𝜃 = 𝑐𝑠𝑐2 𝜃 𝑖𝑠 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 ?

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