Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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1answer
1k views

Finding force for trajectory

Using the trajectory equations provided on hyperphysics, I have developed some code to plot the trajectory of an object. I now need to work out the x and y forces to apply to make the object follow ...
3
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1answer
464 views

Functional inverse of $\sin\theta\sqrt{\tan\theta}$

What is the functional inverse of $f(\theta) = \sin\theta\sqrt{\tan\theta}$? Or, equivalently, what is the inverse of $$f(\theta)=\sin^2\,\theta\tan\,\theta=\frac{\sin^3\,\theta}{\cos\,\theta}$$ It ...
3
votes
1answer
138 views

Graphically display $a^3+b^3+c^3 = d^3$

As a kind of 'addition' to Fermats Last Theorem, a friend of mine has come up with a different idea. Let $$a^3+b^3+c^3 = d^3$$ with $(a,b,c,d): a,b,c,d \in \mathbb{N}$. We were discussing Pythagoras: ...
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4answers
5k views

How to remember the trigonometric identities

I have a test tomorrow and I am having trouble remembering those pesky trigonometrical identities (such as $1-\cos x=2\sin^2(\frac{x}{2})$ ) Do you guys have any tips on how I can remember these? ...
4
votes
4answers
382 views

Why is $\lim\limits_{x \space \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$?

As part of this problem, after substitution I need to calculate the new limits. However, I do not understand why this is so: $$\lim_{x \to \infty}\space{\arctan(x)} = \frac{\pi}{2}$$ I tried ...
6
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1answer
261 views

Solving $\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$

I'm trying to solve $$\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$$ By substituting $x=2\tan{t}$. I get as far as: $$\int_{x \space = -\infty}^{x \space = ...
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0answers
3k views

A complicated trigonometric equation

I have the following trigonometric equation $$f(\theta)=100(A_2 B_3 - A_3 B_2)^2 - (c_1B_3 - c_2 B_2)^2 - (c_2A_2 - c_1 A_3)^2=0,$$ where: $ A_2 = 3\cos(\theta)-5$ $B_2 = 3\sin(\theta)$ $A_3 = ...
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1answer
565 views

Proving formulas for $\cos(nx)$ and $\sin(nx)$

How do I prove the following formulas? Let $n \in \mathbb{N}, x \in \mathbb{R}$. Prove that: $$\cos(nx)=\sum_{j=0}^{[n/2]} (-1)^j {n \choose 2j} (\cos x)^{n-2j} (\sin x)^{2j}$$ ...
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6answers
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Relationship between $\tanh x$ and $\arctan x$

The functions $\tanh x$ and $\arctan x$ have a similar graph. Is there a formula to transform $\tanh x$ to $\arctan x$?
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2answers
206 views

Losing information when solving trig problem

I was doing a simple trig question when it turned out I was missing several answers. I have read somewhere that it is possible to lose information about the signs when dealing with squares and square ...
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4answers
150 views

How do I find the solutions of this equation?

How do I find the solutions of this equation: $$\tan^2 (x)=-1$$
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7answers
251 views

Unconventional way to solve trig question

I was working on a trig question and got stuck, but then I noticed a possible way to solve the problem. However, this way seemed to be slightly unconventional and possibly not what the book was ...
2
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6answers
559 views

Intuitive Explanation of the graph $y = \sin x$ [duplicate]

Possible Duplicate: Intuition for graphing Sine/Cosine We've all seen the graph of $y = \sin x$ (I can't post an image because of reputation so I posted a link to a graph) Sine Graph ...
3
votes
2answers
330 views

Simplifying trig expression

I was working through some trig exercises when I stumbled upon the following problem: Prove that: $ \cos(A+B) \cdot \cos(A-B)=\cos^2A- \sin^2B$. I started out by expanding it such that $$ \cos(A+B) ...
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2answers
176 views

Solving trig equation with $\sin A$ and $\sin 2A$

I am stuck trying to solve for $A$ in $$3 = 11\sin^2 A - 2\sin 2A$$ I cannot see a way to manipulate to get like terms and hence factor it. Thanks!
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2answers
498 views

Intuition around why domain of x of arcsine and arccosine is [-1;1] for “real result” & domain for arctangent is all real numbers

Context I'm working my way through basic trig (this question has a focus on inverse trig functions, specifically arcsine, arccosine and arctangent ), using Khan Academy, wikipedia and some of "trig ...
6
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2answers
2k views

Calculating $\pi$ using two formulas: $\pi/4 = \arctan(1/2) + \arctan(1/3)$ and $\pi/4 = 4\arctan(1/5) - \arctan(1/239)$

How can I use these two formulas to come up with two infinite series, each of which is used to calculate $\pi$?: $$\begin{align*} \frac{\pi}4 &= \arctan(1/2) + \arctan(1/3)\\ \frac{\pi}4 &= ...
10
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4answers
837 views

Finding angles in a parallelogram without trigonometry

I'm wondering whether it's possible to solve for $x^{\circ}$ in terms of $a^{\circ}$ and $b^{\circ}$ given that $ABCD$ is a parallelogram. In particular, I'm wondering if it's possible to solve it ...
2
votes
1answer
345 views

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, ...
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1answer
239 views

How do you integrate $\cos(x^n)$, specifically for $n=-1$?

How does one integrate $\cos(x^{-1})$? I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put ...
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votes
1answer
239 views

How to calculate the coordinates of the middle point of a given arc?

Does anybody know how to solve this problem? I am trying to calculate the green sides of this triangle: I know / have : the arc length, the arch base, the radius, and the h (distance from the red ...
2
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1answer
386 views

How to calculate the coordinates of the middle point of a given arc? [duplicate]

Possible Duplicate: How to calculate the coordinates of the middle point of a given arc? I am trying to calculate the green sides of this triangle: I know/have: the arc length, the ...
6
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1answer
198 views

Integral of difference of sin functions squared is $2\pi$?

Can someone explain why this is true? $$\int_{0}^{2\pi }\left(\sin nx-\sin kx\right)^2dx=2\pi$$ for natural numbers $n,k$ with $n<k $
3
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1answer
454 views

Trigonometric Factorization

How can I factorize this expression: $\sin(x) + \cos(y)$? I've entered this input in Mathemetica: TrigFactor[Sin[x] + Cos[y]] and the output was: $$ ...
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3answers
215 views

Intuition around why Sine of X angle always equals same result.

My understanding so far. Sine represents a ratio of two sides of an interior angle within a right angle triangle. So given the three lengths of a triangle you can find the sine of any of the 3 ...
8
votes
3answers
236 views

Is my trig result unique?

I recently determined that for all integers $a$ and $b$ such that $a\neq b$ and $b\neq 0$, $$ \arctan\left(\frac{a}{b}\right) + \frac{\pi}{4} = \arctan\left(\frac{b+a}{b-a}\right) $$ This implies ...
7
votes
4answers
438 views

Intuition for graphing Sine/Cosine

So I'm working my way through some basic trig (Khan Academy) - I'm trying to get a better intuition for what graphs of sine and cosine represent. I've seen the nice unit circle animations that do ...
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0answers
177 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...
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3answers
323 views

How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$

I want to prove that$$2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}, x\geq 0$$ I have started from the arcsin part and I tried to end to the arctan one but I failed. Can anyone help ...
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1answer
507 views

When is a nail in a rotating wheel below a stated height, given its height as a displaced sinusoidal function of time?

I have this question from my textbook, however I keep getting "unidentified", which is not the answer at the back. I was wondering what I'm doing wrong. The question is: Given that in a water wheel ...
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2answers
875 views

Why is arctan of $-\frac{\sqrt{3}}{3} = -\frac{1}{6}\pi$?

I've been studying the unit circle and inverse trig functions on Khan Academy. One of the questions asked, is, what is the arctan of $-\frac{\sqrt{3}}{3}$. The solution is $-\frac{1}{6}\pi$, I don't ...
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1answer
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How would I approach this identity: $\cos^3x+\sin^3x=(\cos x+\sin x)\cdot(1-\sin x\cdot\cos x)$?

Prove the identity: $$ \cos^3x+\sin^3x=(\cos x+\sin x)\cdot(1-\sin x\cdot\cos x) $$ I was able to change both sides to $\cos x-\cos x\cdot\sin^{2}x+\sin x-\sin x\cdot\cos^{2}x$, which is kind of ...
0
votes
1answer
189 views

Solution for following trigonometric equation?

I have the following trigonometric equation in $\theta$: $$0=G_{\omega}(1/r^2)({\csc^2}\theta){(r\cos\theta-x)}^2+(\cot\theta)(r\cos\theta-x)+r\sin\theta-y.$$ Is there an analytical solution for ...
3
votes
1answer
206 views

Trigonometry basic question

I am training Trigonometry just for fun, so I am not in a hurry, but would like to know how to answer this question - not the result, but how to do it. Sorry because I understand this is too basic for ...
3
votes
1answer
457 views

Integration by Parts with Trigonometric Functions

Trying to evaluate this indefinite integral: $$ \int (x^2 + 1)\cos2xdx$$ So far I have the following: $u=x^2 + 1 \Rightarrow du = 2xdx$ and $dv=\cos2x \Rightarrow v = \frac {\sin2x}{2}$. So the ...
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2answers
844 views

Parallelogram trigonometry

(Sorry for the ambiguous title, couldn't think of a better one) While leafing through a highschool textbook, I found what looked like an interesting question in trigonometry. My trigonometry skills ...
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1answer
238 views

When are $\theta$ and $\sin\theta^\circ$ both rational? [duplicate]

Possible Duplicate: Sine values being rational I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it ...
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4answers
2k views

When is $\sin(x)$ rational?

Obviously, there are some points (like $\pi,30$) but I am unsure if there are more. How can it be proved that there are no more points, or what those points will be? EDIT: I largely meant to ask ...
4
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2answers
164 views

Trigonometric limit

In order to prove that $\displaystyle\lim_{x \to 0}\frac{1-\cos(ax)}{ax}=0$, with $a \ne 0$, I managed that $a=2$ and evaluated this limit: $$ \begin{align*} \quad \lim_{x \to ...
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1answer
134 views

Exact solutions to the equation $\sin(x) = n \cos(x)$

Let $n$ be a positive integer. Can we precisely solve the equation $$\sin(x) = n\cos(x)$$ in $x$? For $n=1$, we get $x=\pi/4$.
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2answers
278 views

Adding tangents of natural numbers and getting the tangent of another natural number

Is it possible to find a set of numbers $(a_i)_{i\leq n} \in \mathbb {(N^{\star})}^n$ and another natural number $b$ such that $$\sum_{i=0}^n \tan a_i = \tan b $$? In other words, given integral, ...
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1answer
259 views

Showing that $\sec z = \frac1{\cos z} = 1+ \sum\limits_{k=1}^{\infty} \frac{E_{2k}}{(2k)!}z^{2k}$

Show that there are complex numbers $E_2,E_4,E_6,\dotsc$ such that $\sec z = \frac{1}{\cos z} = 1+ \sum\limits_{k=1}^{\infty} \frac{E_{2k}}{(2k)!}z^{2k}$ in a neighborhood of $0$. What is the ...
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1answer
83 views

Trigonometric Equation

Given the function $2 \sin(2x+\frac{\pi}{2})$ find the amplitude, the period, the phase shift and the intersection points with the parent function, $\sin(x)$. I was able to find the ...
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0answers
325 views

trigonometry and formula

I asked the same question some time ago, but it is closed. This time I will be clearer (I hope). If I have a right triangle $c ^ 2 = a ^ 2 + b ^ 2$, and $b > a$ (so $a$ is the shorter side and ...
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1answer
380 views

Fitting a sine function to data

I have a sequence of $n$ points $(x_i,y_i)$, for $i=1,\dots,n$. I would like to find the function, of the form $y=V\sin(x+\phi)$, which best fits the points. Which numerical method could I use? I have ...
3
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1answer
147 views

How to solve this: $\int \frac{1}{(5-x-x^{2})^{5/2}}dx$ using a trigonometric substitution

I've completed the squares in order to get a fraction in the integrand of the form $\frac {1}{\sqrt{a^2-x^2}}$ that can be easily substituted by a trigonometric function (drawing the respective ...
0
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1answer
180 views

AP Calculus Homework Help Needed

I am having another problem with basic math in my homework. Help is much appreciated. The answer is: $\frac 1{\sqrt{4-x^2}}$ The problem is: Find $\frac {\mathrm{d}}{\mathrm dx} \sin^{-1} (x/2)$ ...
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1answer
103 views

Formula to convert $1$ to $90$, $2$ to $180$?

Really quick question about Trig! If I've got $\sin^{-1}(1)$ or $\sin^{-1}(2)$, how would I convert that to $90^\circ$ and $180^\circ$ ? I'm trying to convert a graphed version of $f(x) = \sin(x)$ ...
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3answers
69 views

How to get from $2^{99} \cdot (\cos{(99\times \frac{5\pi}{6}) + i\cdot \sin{(99\times \frac{5\pi}{6})}})$ to $0+2^{99}i$?

How do I get from 2nd last to last step? How did they simplify cos & sin $99\times \frac{5\pi}{6}$?
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1answer
106 views

Function to Generate Two Circles For a Color Gradient

I am working on a project creating a gauge package in software, but I am posting here because it is more of a mathematical question. In short, I need a function to create a color gradient to fill in ...