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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2answers
160 views

Graph of an inverse trig function.

Which of the following is equivalent to the graph of $arcsin(x)$ ? (a) Reflecting $arccos(x)$ about the y-axis, then shift down by $\pi /2$ units. (b) Reflecting $arccos(x)$ about the x-axis, then ...
6
votes
3answers
217 views

Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $

I'm trying to find $$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$ I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using ...
1
vote
4answers
85 views

Solving a set of 3 Nonlinear Equations

In the following 3 equations: $$ k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1 $$ $$ 2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2 $$ $$ k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3 $$ $c_1$, $c_2$ and $c_3$ are ...
0
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2answers
854 views

The minimum and the maximum of $y=\sin^2x/(1+\cos^2x)$

I was asked to find the minimum and maximum values ​​of the functions: $y=\sin^2x/(1+\cos^2x)$; $y=\sin^2x-\cos^4x$. What I did so far: $y' = 2\sin(2x)/(1+\cos^2x)^2$ How do I check if ...
1
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2answers
43 views

Solve the trigonometrical product

solve this equation : $(2-\sec^21)(2-\sec^22)(2-\sec^23)........(2-\sec^288)(2-\sec^289)$ If tried from tangent approach with$(1+1-sec^21)......(1+1-\sec^289)$ and i do (1,89) ; (2,88);and........ ...
1
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0answers
33 views

Why is there a difference which way you rotate in different systems?

At school I've always rotate clockwise, with $90^0$ being straight up. But I see some system operates with $0$ being straight up and clockwise rotation, does someone know why this is?
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2answers
50 views

Can't solve this equation to $\phi$

It's the end of a Physics Problem. 2 Forces are equal, one is proportional to $\sin\phi$ the other to $$\frac{\cos\phi}{\text{distance}^2}$$ distance is proportional to $\sin\phi$ $$$$ I'm stuck at ...
1
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1answer
83 views

If $A+B+C=\pi$ does it imply that $\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$ [duplicate]

If $A+B+C=\pi$ does it imply that $$\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$$ If yes, how?
0
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2answers
105 views

Prove that $\tan^{-1}\frac{yz}{xr} + \tan^{-1}\frac{zx}{yr} + \tan^{-1}\frac{xy}{zr} = \frac{\pi}{2} $ where $x^2 + y^2+z^2=r^2$

Prove that $$ \tan^{-1}\frac{yz}{xr} + \tan^{-1}\frac{zx}{yr} + \tan^{-1}\frac{xy}{zr} = \frac{\pi}{2} $$ where $x^2 + y^2+z^2=r^2$ You have to use the formula : $$\tan^{-1}x_1 + \tan^{-1} ...
1
vote
1answer
51 views

Trig problem, finding angles and ranges

I have what may well be a simple problem, but it's been too long since I've done this type of problem. From a fixed point (intersection of all the lines), the angles to 3 other fixed points $a,b,c$ ...
1
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1answer
2k views

Differentiation of a natural log with fraction of trigonometric functions

I am starting with differenciation and I stumbled upon the following exercise: Find $\frac{df(x)}{dx}$, where $$f(x)=\ln \left({\cos x + \sin x \over \cos x - \sin x}\right).$$ So I applied the chain ...
2
votes
1answer
215 views

A sufficient condition to ensure $\alpha=\beta$

Let $\alpha$, $\beta$ be acute angles satisfying $$ \frac{\sin 2\alpha}{\sin(2\alpha+\beta)}=\frac{\sin2\beta}{\sin(2\beta+\alpha)} $$ Show then $\alpha=\beta$.
2
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3answers
174 views

Confusion regarding the derivation of $\cos(x)$ when differentiating $\sin(x)$

The textbook I'm reading derives it like this: $\eqalign{ & y = \sin x \Rightarrow \left( 1 \right) \cr & y + \delta y = \sin (x + \delta x) \Rightarrow (2) \cr} $ Subtracting equation ...
7
votes
3answers
462 views

How to find the eigenvalues and Jordan canonical form of this matrix

Question: let $a_{i,j}\in R,A=(a_{i,j})_{n\times n} $,and $a_{i,j}=\begin{cases} 1&i+j\in\{n,n+1\}\\ 0&i+j\notin\{n,n+1\} \end{cases}$ that's meaning: $$A=\begin{bmatrix} ...
6
votes
2answers
673 views

Expressing $\cos\theta - \sqrt{3}\sin\theta = r\sin(\theta - \alpha)$

My book explains that $a\cos\theta + b\sin\theta$ is a sine (or cosine) graph with a particular amplitude/shift (i.e. $r\sin(\theta + \alpha)$) and shows me some steps to solve for $r$ and $\alpha$: ...
1
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0answers
72 views

Pure Phase Number

I am read a solution (4.9) Here say: ... both $a, d$ are pure phases, so that it is always possible to find (non unique) real numbers $\alpha, \beta, \delta$ such that $a = ...
3
votes
2answers
959 views

Trouble visualizing sin and cos

I'm working on building tetris now in Java and am at the point of rotations... I originally hardcoded all of the rotations, but found that linear algebra was the better way to go. I'm trying to use ...
1
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2answers
596 views

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
0
votes
2answers
86 views

Taylor Series of $\sin 2x$ finding $f^{(n)} (a)$ where $a = 0$

ok so i get; f (x) = sin 2x f ' = 2cos 2x f '' = -4sin 2x f ''' = -8cos 2x f '''' = 16sin 2x f ''''' = 32cos 2x f (0) = 0 f '(0) = 2 f ''(0) = 0 f '''(0) = -8 f ''''(0) = 0 f '''''(0) = 32 ...
1
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3answers
882 views

Show that $ \frac{\tan x}{1+\sec x}+\frac{1+\sec x}{\tan x}= 2 \csc x$

Verify the following identity: $$ \frac{\tan x}{1+\sec x}+\frac{1+\sec x}{\tan x}= 2 \csc x$$
1
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4answers
71 views

What is $g'(x)$ if $g(x) =x^2 \int_{x-2}^{\sin x} \cos^2t dt $?

What is $g'(x)$ if $$g(x)= x^2 \int_{x-2}^{\sin x} \cos^2t dt?$$ So i get $g'(x) = 2x(\int_{x-2}^{sinx} cos^2t dt ) + x^2(cos^2(sinx)-cos^2(x-2))$ as my final answer. Is this right?, thanks. Use ...
0
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1answer
55 views

Solution of two equivalent quadratic equation solutions

I have two solutions to quadratic equations, based on the quadratic formula. The solutions are equivalent. Additionally, one of the variables (Tx, ...
0
votes
1answer
189 views

De Moivre's theorem question.

State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$ I know De Moivre's theorem, how to prove it, and converting to ...
1
vote
1answer
113 views

$\cos$ not a contraction on $\mathbb R$

I know that $\cos$ is a contraction mapping on $[0, a]$ with $a<\pi/2$. I also know that the proof of this uses the mean value theorem and it fails on $\Bbb R$. However, this is not a proof to the ...
6
votes
3answers
2k views

Prove that the Tangent of 75 degrees equals 2 plus the square-root of 3

My (very simple) question to a friend was how do I prove the following using basic trig principles: $\tan75^\circ = 2 + \sqrt{3}$ He gave this proof (via a text message!) $1. \tan75^\circ$ $2. = ...
10
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2answers
144 views

How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?

I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?
0
votes
1answer
94 views

Maximum and minimum of $y = 4x-8*(\cos(x))$ between $-\pi$ and $\pi$

I have found that the maximum of this function is at $\pi$, where the function will equal $$4\pi+8,$$ which is approximately $20$. However, I tried to get the minimum value, and it was incorrect. The ...
1
vote
4answers
173 views

$\lim_{x\rightarrow\infty}\sin(x)$?

In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function. Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
6
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4answers
1k views

Finding the limit of $\frac{1-\cos(2x)}{1-\cos(3x)}$ for $x \to 0$

As $x$ goes to $0$, what is the limit of $$\frac{1-\cos(2x)}{1-\cos(3x)}$$ Thanks.
2
votes
0answers
112 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
5
votes
1answer
208 views

Power series for $\cos(n\theta)$ in terms of $\sin^{2i}(\theta/2)$?

Does anybody know an expression for the weights in $$ \cos(n\theta) = \sum_{i=0}^n c_i \sin^{2i}(\theta/2) $$ I checked the standard sources (Abramowitz & Stegun, Gradshteyn & Rhyzik) and ...
1
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1answer
155 views

Solve the integral $\int_{x=0}^{\infty}\frac{1}{x}\int_{y=0}^{x}\frac{\cos{(x-y)}-\cos{x}}{y}dydx$

Find the value of: $$I=\int_{x=0}^{\infty}\dfrac{1}{x}\int_{y=0}^{x}\dfrac{\cos{(x-y)}-\cos{x}}{y}dy \ dx$$ I think we could take: ...
3
votes
3answers
237 views

What is -cos(t) equivalent to in terms of cos(t)

I want to know if, $-\cos(t) = \cos(t+180)$ or $-\cos(t) = \cos(t-180)$ Please guide me. Thanks
8
votes
3answers
317 views

Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$

It is known that$$\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$$ for exactly one set of positive integers $(a,b,c)$ where $0<c<90$ find the value $$\dfrac{b+c}{a}$$ my idea,$ \sin 50^\circ ...
0
votes
1answer
53 views

Inverse trigonometric question

Prove that $\tan^{-1}(\frac{x\sin\alpha}{1-x\cos\alpha})-tan^{-1}(\frac{x-\cos\alpha}{\sin\alpha})$ is independent of $x$ and is equal to $\frac{\pi}{2} - \alpha$ Please guide how to proceed in this ...
0
votes
1answer
28 views

What is the amplitude of this function

What is the amplitude if I have these two functions? $$y1: y=\cos(x)$$ $$y2: 2y=3\cos(3x/2)$$ I answered 3 but it was wrong
10
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2answers
1k views

Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$

$$ 2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x $$ How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
1
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1answer
54 views

inverse trigonometry derivation

Prove that : $sin^{-1}x+sin^{-1}y = sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$ If -1 $\leq x \leq 1; -1 \leq y \leq 1 $ and $x^2+y^2\leq 1$ or if $xy <0 $ and $x^2+y^2 > 1$ solution : Let ...
0
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1answer
129 views

looking for reference for 2 trig facts

Math people: I am looking for a reference for two trigonometry facts, one of which I proved myself, and another which a random person had posted on the Web. I have evidence to believe the second ...
0
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2answers
127 views

What is required to establish the law of cosines?

In my quantum computation course, we have been given nothing more than the basic axioms of a linear vector space, and and the properties of an inner product; but we have started referring to "the ...
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3answers
1k views

Solving the equation $\sin 40^{\circ}=\cos x$

How to solve this equation (finding $x$ ): $$\sin 40^\circ=\cos x$$
2
votes
2answers
2k views

Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$

How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
3
votes
1answer
183 views

Largest element of the set $\{ \sin{1}, \sin{2}, \sin{3}\}$

i have to find the largest element of the following set $\{ \sin{1}, \sin{2}, \sin{3}\}$. I converted every element to the first quadrant so i can use the monotony of cosine, the set becomes: ...
1
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2answers
113 views

How do I find the surface area of an angled conic base?

Thank you for viewing my question. I need help creating a formula for finding the surface area of a conic base. (eg. I install a flood light on my roof, I want to know how much surface area it will ...
1
vote
2answers
133 views

Finding an infinite trigonometric sum

Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
4
votes
1answer
258 views

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at Find area bounded by two chords and an arc in a disc . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 ...
0
votes
1answer
275 views

Upper and lower bound of $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$

Let define $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$ Please help me prove, that $f$ reaches its lower bound in only one point $x_1$ and reaches its upper bound $x_2$ also in only one ...
1
vote
1answer
189 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
0
votes
4answers
73 views

Inverse trigonometric function

Prove that $$\tan^{-1}x + \tan^{-1}\frac{2x}{1-x^2}=\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\;,\;\; |x| < \frac{1}{\sqrt{3}}$$ By taking R.H.S $\tan^{-1}(\frac{3x-x^3}{1-3x^2}) = ...
3
votes
3answers
1k views

How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?

Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$ My book says the answer is $(0,0),(a,0),(a,\pi)$. However I calculated $ (a,0),(a,\pi),(a,2\pi)$.