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# 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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### Multiple angle formulas for tangent

There are direct multiple angle formulas for sine and cosine, in terms of Chebyshev polynomials. For example, the cosine of $n\theta$ is $$\cos{n\theta}=T_n(\cos\theta)$$ Where $T_n$ is the n-th ...
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### I need help with mathematical induction on trigonometry [closed]

I need to prove that $\cos x \cos 2x \cos 4x \ldots \cos 2^{n}x= \sin 2^{n+1}x/2^{n+1}\sin x$ while $x \in \mathbb{R}$.
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### How to see the graph of this? $n=\sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$ [closed]

$$n \text{ (constant)} = \sin(x)\cos(y)+\sin(y)\cos(z)+\sin(z)\cos(x)$$ Code is this, but I can't emulate this. Why? Even WolframAlpha can't make graph too. ...
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### Inverse trigonometry , find x [closed]

What is the value of x ? There are many different ways to do . Is my method correct? If so can anyone help me to go further. After this part , i have no idea to go with !!
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### In concentric circles Triangle formed from intersection of a line making 45 degree with x axis where inner circle meets x axis to outer circle

I have two concentric circles one of radius 5 cm and outer one of 10 cm, their centers being 0,0 I want to calculate P B and H of the triangle formed by intersection of a line on outer circle making ...
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### $\sin A+\cos A+\sec A+\operatorname{cosec}A+\tan A+\cot A=7$, find the value of $\sin(2A)$. [closed]

How can I find the value of $\sin(2A)$, when $$\sin A+\cos A+\sec A+\operatorname{cosec}A+\tan A+\cot A=7?$$
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### Convert pair of parametric trig equations to $y=f(x)$ form

My apologies if this already has an answer, I've spent some time looking but haven't found anything that (to me) looked directly applicable. I have a set of parametric equations describing a periodic ...
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### Identify and describe the surface described by the equation: φ = c where π/2 < c < π.

In this context, φ refers to the polar angle between the z-axis and the radial line ρ as described by the spherical coordinate system with the following equations: x=ρcos(θ)sin(φ) y=ρsin(θ)sin(φ) z=...
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### Find the exact value of $\DeclareMathOperator{\cosec}{cosec} \cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$ [duplicate]

Find the exact value of $\cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$. The equation can be written as $$\cosec(x) + \cosec(60^\circ-x) - \cosec(60^\circ+x),$$ where $x = 10^\circ$. I ...
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### Find $X$ and $\theta$ in this quadrilateral

$ABCD$ is a convex quadrilateral, with $\angle A = 120^\circ$. $\angle B = \angle C =\theta$. $AB = 10$, $AD = BC = X$ and $CD = 2 X$. Find $X$ and $\theta$. My attempt: From the law of ...
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### Determining concavity of a certain function

Let $t\in(0,1)$ be a fixed number. Consider the function $f(x)=2\cos(t)\arccos(\cot(t)(\csc(x)-\cot(x)))+\arccos(\csc^2(t)\cos(x)-\cot^2(t))$. I believe that this function is concave down for $x>0$,...
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### Solving a trigonometric equation from a geometry problem

Question statement: In triangle $ABC$, angle $A = 54^\circ$ and angle $C = 24^\circ$. $P$ is a point on $AC$ such that $AP = BC$. Find angle $CBP$. (HK IMO Prelim 2015) I am looking into using ...
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### Looking for similar trigonometric functions [closed]

Has anyone seen such a function before? I am not looking for the exact formula, but rather the type or structure of such a function. I.e. I would be happy to see the explicit formula of a function ...
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### Evaluate $x^2 = \sin(2x), x≠0$ [closed]

There should be another solution ($x \approx0.967$), but can it be done without doing graphs? I tried drawing a triangle to solve but it failed miserably.
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From Apostol's Calculus Volume 1 2nd ed., 6.22 #46, the task is to integrate $$\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$$ Method 1: The provided hint is to use the substitution $x-a=(b-a)\sin^2(u)$. Thus $b-... -2 votes 0 answers 24 views ### prove the addition identity cos(x+y) using the law of cosines [closed] I’ve drawn a triangle with sides a b and c and written down two equations for side c using the law of cosines with angle x+y and using the sine of angle x and sine of angle y. I don’t know how to ... 3 votes 2 answers 83 views ###$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$$$\int\frac{\sin 2x\sin 3x}{\sin 2x+\sin 3x}\,\mathrm dx$$ My attempt: Rewriting the numerator: We can use the double angle identity for sine: $$\sin(A)\sin(B) = \frac12\left(\cos(A - B) - \cos(A + B)\... 0 votes 3 answers 85 views ### When solving \sin{x}+\cos{x}=\sin{2x}+\cos{2x}, where does the extra solution come from? Background This question is from the 1907 Victorian Universities and Schools Examination Board Trigonometry (Senior) Examination. My solution is given below, but apparently there is an extra solution ... • 2,671 -2 votes 2 answers 89 views ### Is there a way to simplify \cos^{-1}\left(\frac{\cos x}{A}\right)? Is there a way to simplify the following:$$\cos^{-1}\left(\frac{\cos x}{A}\right)$$I'm thinking something along the line of pulling A out and then getting \cos^{-1}(\cos x) which would be x ... • 329 7 votes 2 answers 97 views ### Solving the integral \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx My textbook has the following problem:$$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} \ dx$$Using the trigonometric identity:$$\sin 2x = 2 \sin x \cos x$$the ... • 296 0 votes 0 answers 40 views ### Using the Maclaurin series of sin(x) to find Si(x) The Maclaurin series for \sin\left(x\right) is given by:$$\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}x^{2n+1}}{\left(2n+1\right)!}$$So the series for \frac{\sin\left(x\right)}{x} would be$$\... 0 votes 2 answers 186 views ### summation of$\cos{x} + \cos{2x} + \cos{3x}$and so on is$-\frac{1}{2}$$$\cos{x} + \cos{2x} + \cos{3x} \ldots = y$$ $$2\cos{x} + 2\cos{2x} + 2\cos{3x} \ldots = 2y$$ By grouping every alternate term by$\cos{A} + \cos{B} = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$$$\... • 11 1 vote 3 answers 82 views ### Solving the integral \int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x} [duplicate] My textbook has the following problem: \int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\ dx}{\cos^2 x + 4\sin^2 x}. This website solves the problem in a very convoluted way and I was wondering if there's an ... • 296 1 vote 3 answers 43 views ### 3D trigonometry bearing question - finding the bearing of a point to a tower I'm currently struggling with 3D trigonometry, particularly with drawing a proper diagram and interpreting bearings. The question is below: The angle of elevation of a tower QR of height 100m at a ... • 11 -1 votes 0 answers 23 views ### Find the most efficient path, time wise, to get from points S to Q at varying speeds [closed] Both me and my friend are a bit stumped on this question and keep getting different answers. I think the sailor should land 4km up shore (1km from market) but am not entirely sure. Q: A fisherman can ... 2 votes 2 answers 55 views ### Alternative proof g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0 for u\in\left[-\frac \pi 2, \frac \pi 2\right] The given inequality$$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$for u\in\left[-\frac \pi 2, \frac \pi 2\right], comes out from an answer given to this other recent question. The ... • 1 0 votes 1 answer 25 views ### Range of values of θ that satisy these values. The question:The value of θ for which$$x+y(\sin θ)=1, x(\sin θ)+4y=2$$satisfy$$x>=\frac{4}{5}, y>=\frac{1}{3}$$Note here θ must belong from$$(-\frac{\pi}{2},\frac{\pi}{2})$$This is a ... • 128 0 votes 1 answer 59 views ### Maximum length of a boat able to turn at a junction of two orthogonal canals [closed] Let's say you are making a boat, and the goal is to navigate down a canal of width a. But not only that, you want to make a turn into another canal that is perpendicular to the one you navigate of ... • 339 0 votes 1 answer 86 views ### Prove that the average of the numbers n\sin(n^{\circ}),n=2^{\circ},4^{\circ},6^{\circ},\ldots,180^{\circ} is \cot(1). Prove that the average of the numbers n\sin(n^{\circ}),n=2,4,6,\ldots,180^{\circ} is \cot(1^{\circ}). I am trying to use complex numbers to solve the given problem. We can write$$\sin(n^{\circ})=... • 624 1 vote 0 answers 68 views ### Middle School Narrative Math Textbook A couple decades ago, I remember teaching myself trig and calculus from some textbooks that I got from the library. They were set in a fantasy kingdom and concepts were introduced to solve various ... • 11 0 votes 1 answer 81 views ### Is it possible to calculate the distance in red? [closed] In the context of a physics experiment preparation I am trying to figure out if I can calculate theoretically the distance in red on the below schema in order to determine the waist of a laser beam. ... 0 votes 0 answers 20 views ### Calculate offset from corner coordinate [closed] Illustration Given 4 corner coordinates. Compared to leftBottom or leftTop, I want to determine the displacement of a point P in the area bounded by the 4 coordinates. The picture is looking north. ... 0 votes 2 answers 107 views ### Solving the system$\cos(x)-\cos(x+y)=0$and$\cos(y)-\cos(x+y)=0\$

At some point when dealing with a problem I had to solve the following system $$\begin{cases} \cos(x)-\cos(x+y)=0\\ \cos(y)-\cos(x+y)=0 \end{cases}$$ I seem to have problems with doing so, or maybe ...
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