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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0
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0answers
45 views

$\sum_{n=1}^{\infty} \frac{\sin(n^2)}{n^2}$

Question: $$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{n^2}=\,?$$ Previously I calculated a similar summation but it was more luck than wisdom, and insight led me to believe my methods were super incorrect (...
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0answers
22 views

What properties can I deduce about an $f(i)$ that satisfies $\sum\limits_{i=1}^{x} \cos(f(i)) = x\cos(\ln(x)) $?

I have an $f(i): \mathbb{N} \rightarrow \mathbb{N}$ that satisfies $\sum\limits_{i=1}^{x} \cos(f(i)) = x\cos(\ln(x)), \,\,x \in \mathbb{N}$ What general properties does $f(i)$ satisfy? Can I deduce ...
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2answers
75 views

Proving $\lim_{x\rightarrow {\frac{\pi}{2}}^{-}}\tan(x)=+\infty$ and $\lim_{x\rightarrow {-\frac{\pi}{2}}^{+}}\tan(x)=-\infty$ by definition

I have to prove \begin{eqnarray} \lim_{x\rightarrow {\frac{\pi}{2}}^{-}}\tan(x)=+\infty \hspace{1cm} \lim_{x\rightarrow {-\frac{\pi}{2}}^{+}}\tan(x)=-\infty \end{eqnarray} by definition. I don't find ...
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0answers
75 views

Solving $ c\sin(a(x-b))+d=\frac{c+d}{\frac{\pi}{2a} + b}x$ with $ a,b,c,d \in\Bbb{R} $

Help me Solve this Equation $$ c\sin(a(x-b))+d=\frac{c+d}{\dfrac{\pi}{2a} + b}x \quad\to\quad x=\dfrac{\pi}{2a} + b $$ $$ a,b,c,d \in\Bbb{R} $$ Are you expected to find "closed form" ...
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0answers
17 views

How to proof that $\cos x <( \frac{\sin x}{x})^3$ when $0<|x|<\pi\over 2 $ [duplicate]

How to proof: $0<|x|<$ $\pi\over 2 $ $\quad$ $\Rightarrow \quad$ $\cos x < (\frac{\sin x}{x})^3$
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1answer
218 views
+50

Five Porismatic Equations.

Here is a really tough problem. If $$\boldsymbol{a\cos\alpha\cos\beta+b\sin\alpha\sin\beta+c=0}$$ $$\boldsymbol{a\cos\gamma\cos\delta+b\sin\gamma\sin\delta+c=0}$$ $$\boldsymbol{a\cos\beta\cos\gamma+b\...
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1answer
44 views

Simplifying $9\sin^2(x)\csc^2(x) − 9\sin^2(x)$ [closed]

I'm new to trigonometry, and I have tried and tried to solve this... would someone be willing to explain how to solve this? Factor the expression and use the fundamental identities to simplify. There ...
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2answers
61 views

Finding $\log_{2b – a}(2a – b)$, where $a=\sum_{r=1}^{11}\tan^2(\frac{r\pi}{24})$ and $b=\sum_{r=1}^{11}(-1)^{r-1}\tan^2(\frac{r\pi}{24})$

Let $a = \sum\limits_{r = 1}^{11} {{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \right)} $ and $b = \sum\limits_{r = 1}^{11} {{{\left( { - 1} \right)}^{r - 1}}{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \...
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1answer
71 views

Problem in defining a trigonometric equation (ellipse)

I have an updated problem of my question from: Problem in defining a trigonometric equation @David K gave me a very nice solution here. But now the problem is that since I do not have a circle but a ...
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0answers
59 views

Limit without l'Hospital for a function with denominator $x$

I have to find the limit $$\lim_{x\to 0} \frac{\sqrt{x^2+1}-\sin x+\cos x-2}x$$ without using L'Hospital. I know I have to find $-1$, indicated by plotting the graph of the function and by ...
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4answers
57 views

Differentiability of $f(|x|)$

What are the rules for the differentiability of $f(|x|)$? In hindsight, and upon inspecting $\sin(|x|)$ and $\cos(|x|)$, the only rule I can deduce is that $f(x)$ should not be zero at $x=0$. But I ...
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4answers
198 views

Number of triangles $\Delta ABC$ with $\angle{ACB} = 30^o$ and $AC=9\sqrt{3}$ and $AB=9$?

I came across the following question just now, A triangle $\Delta ABC$ is drawn such that $\angle{ACB} = 30^o$ and side length $AC$ = $9*\sqrt{3}$ If side length $AB = 9$, how many possible triangles ...
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3answers
48 views

Number of solution of ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$

Number of solution of the equation ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$ in the interval $[0,2\pi]$ is equal ...
3
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1answer
449 views

Have you seen this golden ratio construction before with three squares (or just two) and circle ?Geogebra gives PHI or 1.6180.. exactly

Geogebra gives PHI or 1.6180.. exactly Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a ...
3
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1answer
108 views

Finding the $n$th derivative of $f(x) = e^{\sin(x)}$

I have a function $$f(x) = e^{\sin(x)}$$ I want to expand it in a infinte series using Maclaurin's theorem and for that I need to know if the remainder term $$R_n = \frac{x^n}{n!}f^{(n)}(\theta x),\...
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1answer
63 views

How do you simplify expressions like $\tan^{-1}(\cos(x))$ or $\cot^{-1}(\sec^{-1}(x))$? [closed]

In my tutorial, we went over practice problems for inverse trig functions, but I still don't really understand how to arrive at solutions for these types of problems. Is there any methodology/logic I ...
9
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1answer
2k views

Is a triangle with two equal angles always isosceles?

An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles, ...
6
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2answers
307 views

How is it that $\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$ for all angles, even though the derivation holds only for $\cos A\cos B\neq 0$?

How is it that $$\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$$ for any value of $A$, $B$? I have doubts about this since we arrive at this by dividing the numerator and denominator of $$\...
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2answers
241 views

Finding the maximum and minimum values of $a^2\sin^2\theta + b^2\csc^2\theta$

How do I find the maximum and minimum values of the following? $$a^2\sin^2\theta + b^2\csc^2\theta$$ Is the max value $\infty$? I tried to find the minimum value by using A.M$\geq$G.M. inequality (is ...
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0answers
29 views

What is the length of X? [closed]

I need to find the length of X in the diagram. How do I do it? Thanks, Ryan
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2answers
1k views

An alternative to integration by trigonometric substitution?

When I was a Calculus II student many (about 4800) moons ago, our professor taught us an alternative to trig sub. For example, if we have $$ \int \frac{dx}{x^2\sqrt{x^2 - 9}}, $$ we would evaluate ...
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4answers
853 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since $...
2
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0answers
53 views

A circle and two perpendicular lines enclose four regions. Can the regions have distinct rational areas?

A circle and two perpendicular lines enclose four regions. Can the regions have distinct rational areas? (I stipulate distinct to eliminate trivial cases in which one of both of the perpendicular ...
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0answers
37 views

Sum of powers of Cos and Sin [closed]

Does anyone know how to work out the formula for like the infinite sum of cos^2n(x) and sin^2n(x) or maybe even like how to re write the terms in a simpler way. Because the problem that I am stuck ...
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2answers
118 views

Solve for $y$, $\sin(x)=y+\cos(y)$

I’m helping a friend of mine solve an equation: Solve for $y$, $\sin(x)=y+\cos(y)$ Substituting $y$ multiple times isn’t what he wants, he wants an exact explicit function. I tried doing $\sin(x)-y=\...
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0answers
32 views

Solving Simultaneous Equations with Nested Trigonometric and Inverse Trigonometric functions.

Deriving an equation for the geodesic on a sphere between two arbitrary points A and B given in spherical coordinates, I obtained $$\theta = -\arcsin{\left(c \cot{\phi}\right)} - d,$$ where $c$ and $d$...
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0answers
37 views

Desmos sin wave sum shows different to code

I have followed The Coding Train's Discrete Fourier Transformation series, and added a feature which spits out the coefficients of the equation. I wrote test in the program, which looked nice. However,...
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3answers
31 views

How to calculate the offset distance of triangle

I need to find the offset distance of 2 points. I start with my basic shape The dashed lines are help lines Now I will offset the line and need to find point X and Y Since I only know the offset and ...
2
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1answer
362 views

Generalizing a Trigonometric Infinite Product of Vieta

The second exercise in "Statistical Independence in Probability, Analysis and Number Theory," by Mark Kac is to prove that $$ {\sin x\over x}=\prod_{k=1}^{\infty}\frac13\left(1+2\cos{2x\over3^k}\right)...
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1answer
26 views

How to solve this spherical trigonometry situation with missing information?

Suppose $b$ and $c$ were given constants. If $C = \theta - y$ (where $\theta$ is given) and $a = 90 \deg - y $, is it possible to solve for $y$? It seems like there must be a solution, since $y$ can't ...
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2answers
49 views

How does $\tan^2(x) \sec(x) + \sec^3(x)$ turn in to $2\sec^3(x) - \sec(x)$

Can someone explain how $\tan^2 $ disappeared and $\sec^3$ turn into $2\sec^3$ ??? The derivatives of the function $2\sec(x)\tan(x)$ is apparently $2(-\sec(x) + 2\sec^3(x))$
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0answers
17 views

Calculating the center of a rotated 2D rectangle given it's bottom left point, angle of rotation and dimensions

I have a rectangle that can be rotated around its center point. I would like to know what the coordinates of the center are, but I'm not sure how to get that with the information available to me. I ...
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2answers
2k views

When solving $\tan(3x) - \cot(4x)$, how to formulate the answer?

when I solve the following equation: $\tan(3x) = \cot(4x)$ I get the following solution: $x = \frac{\pi}{14} + \frac{\pi n}{7}, n \in \mathbb{Z}$ But as x must be $\neq \frac{\pi}{6} + \frac{\pi k}...
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2answers
84 views

Problem in defining a trigonometric equation

I want to define an equation and I already solved the probem for a special case. Here is the description: Given: $x_\phi$, $y_\phi, \alpha$ Unkown: $r_k$, $r$ I solved this specific case where the ...
4
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3answers
179 views

Find the minimum of $\frac{\sin x}{\cos y}+\frac {\cos x}{\sin y}+\frac{\sin y}{\cos x}+\frac{\cos y}{\sin x}$

Let $0<x,y<\frac {\pi}{2}$ such that $\sin (x+y)=\frac 23$, then find the minimum of $$\frac{\sin x}{\cos y}+\frac {\cos x}{\sin y}+\frac{\sin y}{\cos x}+\frac{\cos y}{\sin x}$$ A) $\frac 23$ B) ...
2
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5answers
6k views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ Then, what will be the set of $x$ for which this equation is true? I tried to solve it by putting $x = \sin a$ or $\cos a$ but got no ...
3
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1answer
210 views

If $3\sec^4\theta+8=10\sec^2\theta$, find the values of $\tan\theta$

If $3\sec^4\theta+8=10\sec^2\theta$, find the values of $\tan\theta$. $$3\sec^4\theta-10\sec^2\theta+8=0$$ $$3\sec^4\theta-6\sec^2\theta-4\sec^2\theta+8=0$$ $$(3\sec^2\theta-4)(\sec^2\theta-2)=0$$ $$\...
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2answers
40 views

Find the general value of $\theta$ for the inverse trigonometric function

$\text { Find the general value of } \theta, \text { when } 9 \sec ^{4} \theta=16$ My work- Given $\sec ^{4} \theta=\frac{16}{9}$ or, $\sec ^{4} \theta = \frac{(4)^{2}}{(3)^{2}}$ $\implies \sec ^{2} \...
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2answers
1k views

Determinant of a matrix with trigonometry functions.

Prove that the matrix is invertible for any value of $\beta$. I've done several exercises of this type. But I'm not sure with this one: $$\begin{bmatrix}\cos \beta & \sin \beta & 0\\ -\sin\...
19
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9answers
11k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: $\sin(...
9
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5answers
7k views

Proving that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$

Prove that $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$ This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < \sin(x) &...
0
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1answer
42 views

How do I calculate the new position of a given rectangle after subjecting it to rotation and scaling?

I want to calculate the new position of a modified version of a given rectangle. The given rectangle is shown in Yellow. The rectangle undergoing rotation and scaling is shown in Red. Step-1: The ...
5
votes
2answers
341 views

In calculus, how should I interpret the -1 superscript in trigonometric functions?

In calculus, and in the context of differentiating functions for practice, how should I interpret the following expression (i.e., what is the convention here) $$f(x) = \tan^{-1}(x)$$ Should I treat it ...
1
vote
1answer
61 views

Simplifying $\frac{\omega\sin(\omega t)\tan(\omega{t})+\omega(\cos(\omega t)+1)\sec^2(\omega t)}{(\cos(\omega t) +1)^2}$

How can I simply this : $$ \dfrac{{\omega}\sin \left( {\omega} {t}\right) \tan \left( {\omega}{t}\right) +\omega \left( \cos \left( \omega t\right) +1\right) \sec ^{2}\left( \omega t\right) }{\left( \...
1
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1answer
93 views

Is there a unit equal to 2pi radians?

We can cut up circles in whatever size chunks we choose -- we normally choose to cut them up so that the size of the angle of an entire circle is $2\pi$ or 360. Said differently, we choose units to be ...
0
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0answers
69 views

General solution of $a\sin(b+x)=x$

What is the general solution of the equation $a\sin(b+x)=x$ for $x$? I am interested in the case, when $0<a<1$. I tried to solve it the following way: calculate the left side when $x=0$. Then ...
-3
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1answer
43 views

Can we prove Trig functions as continuous in this way? [closed]

I am having a bit of confusion in proving some expressions involving Trig functions as continuous.Everywhere in my textbook they use the method where they substitute $c+h$ instead of $x$ and letting $...
2
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1answer
97 views

Why and when can I just peacefully substitute into $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ without checking range conditions?

This is an example question in my book: To solve for $x$: $$\tan^{-1}\frac{x-1}{x+2}+\tan^{-1}\frac{x+1}{x+2}=\frac{\pi}{4}$$ and it is solved by a direct formula given by $$\tan^{-1}x+\tan^{-1}y=\...
0
votes
0answers
58 views

Why can $-n\cdot \pi $ be changed to $n\cdot \pi $?

This task was just to solve this equation: $\cos2x=\cos4x$. I solved it correctly apart from one step. My book somehow changes $-n\cdot \pi $ to $n\cdot \pi $. How is it possible?
2
votes
1answer
65 views

A triangle has one vertex at a circle's center and two vertices on the circle. Can the three enclosed regions have rational areas?

A triangle has one vertex at a circle's center and two vertices on the circle. Can the three enclosed regions have rational areas? Let $r=$ radius of circle, $\theta=$ angle at vertex of triangle at ...

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