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

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5 views

where will all cosines with frequency multiples of harmonic frequency be zero?

Let every cosine be of form $x_i(t) = a_i\cos(2\pi f_i t)$ with $a,t \in \mathbb{R}$ and $f$ is labelled as frequency. Suppose we have cosines with frequencies $f_1,f_2,..,f_n$ and every frequency is ...
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3answers
27 views

Alternative form of a Trigonometrics Expression

Express $8\sin\theta \cos\theta - 6 \sin^2 \theta$ in the form $R \sin(2\theta + \alpha) + k$ Edit: I am sorry, I thought it was a somewhat interesting question. I shall let you know of the progress ...
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2answers
60 views

How to show whether this limit exists or not?

It's my intuition that $$\lim_{x\to+\infty} \frac{\sin(x+\frac1x)}{\sin(x)}$$ does not exist. And I have been working on proving it. I have tried Heine's Theorem but for this moment I get stuck and ...
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4answers
69 views

What's the easiest proof for $\left|\frac{\sin x}x\right|\le\frac 12$ for all $|x|\ge 2$?

As asked in the title: What's the easiest proof for $$\left|\frac{\sin x}x\right|\le\frac 12\;\;\;\text{for all }|x|\ge 2$$
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3answers
41 views

Limit of two variable

Suppose that I have equation: $$\tan(a) = \dfrac{b-c}{bc + 1},\;\text{ where }\,a, b, c\, \text{ are variables.}$$ How can I show that, as $a \to \pi/2, \; bc+1\to 0\;$ mathematically, not ...
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1answer
39 views

The expression 2sin2 A sin4 A + cos4 A is simplified to (A) 2 (B) 1 (C) sin2 A (D) cos2 A [on hold]

The expression $$2\sin(2 A) \sin(4 A) + \cos(4 A)$$ is simplified to $$(A) 2, (B) 1, (C) \sin(2 A), (D) \cos(2 A)$$
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3answers
37 views

Differentiability of $f(x)=sin(x)/x$ if $x\ne0$ and $1$ if $x=0$

I am trying to see if $$f(x)= \begin{cases} \frac{\sin(x)}x &\text{ if x}\neq0\\ 1 &\text{ if x}=0. \end{cases} $$ is differentiable more than once. This is what I did: $$f'(0)= \begin{cases} ...
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1answer
27 views

noob question on arithmetic with proving identities

how does $$\sin^2x-\cos^2x+\cos^4x$$ simplify to $$\sin^2x\times\sin^2x=\sin^4x \,\,\,\,\ ?$$ I would appreciate how the steps are done to arrive to the final answer. Thanks!
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1answer
31 views

Determining Exact Values of Trignometric Equations

Use the special triangles to give exact solutions where possible. Find all values of $x$ such that $0\le x \le 2\pi$ . (a) $\tan^2 x=1$ $\,$ (b)$\, \, 2\cos x + \sqrt{3}=0 \, \,$ (c) $\, \, ...
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0answers
24 views

Construct triangle from three points on base and difference in distances to third vertex

Imagine such a triangle: We know the differences in distances: $\overline{OA} - \overline{BO}$ and $\overline{CO} - \overline{BO}$, as well as the distances between the points on the base: ...
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2answers
24 views

Verifying identities - trigonometry [on hold]

$$\frac{1 - \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 - \cos \theta} = 2 \csc \theta$$ I've taken the first step into combining using LCD... and I get royally stuck..
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1answer
22 views

Proving an identity, cos and sin, two variables

$$\frac{\cos(2x)+\cos(2y)}{\sin(x)+\cos(y)} = 2\cos(y)-2\sin(x)$$ The question asks to prove the identity. I tried simplifying the first half, thought maybe I could expand and simplify with the ...
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1answer
12 views

Finding intervals using local min and max (in interval notation form)

I am having some trouble with the following question: Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum ...
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0answers
24 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
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1answer
43 views

Is there any meaningful way in which the tangent function relate to e?

Is there any meaningful value for which $\tan(x)$ relates to $e$? What I mean to say is $\tan(1.21828\ldots)=e$, but is there any significance to this number? i.e. could it be expressed as some ...
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1answer
39 views

Are these trigonometry problems correct?

I have to prove the following trigonometric identities. However, since I can't prove them, I am starting to think they are not correctly stated. Problem 1: ...
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2answers
62 views

Integral of $\big((1+\cos(x))\sin(x)\big)^2$

What is $$\int \big((1+\cos(x))\sin(x)\big)^2dx$$ ?
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2answers
45 views

$\sin(\pi - a) = \sin (a)$. How/why? [on hold]

Can someone please explain how and why $\sin(\pi - a) = \sin (a)$? The handout from class doesn't really explain it. I tried asking the teach but there's a bit of a language barrier and I'm not ...
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0answers
31 views

what is the shortcut approach to find the maximum value of $ \sin^8 \theta + \cos ^{17} \theta $ [on hold]

what is the shortcut way to find the MAXIMUM value of $ \sin^8 \theta + \cos ^{17} \theta $ Bacially I need the approach to solve it quicker.
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2answers
52 views

Help evaluating an integral involving tan(x) [on hold]

Find the simplest way of evaluating $$ \int \tan^{4}(x) \,dx $$
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4answers
84 views

Simplifying An Inverse Tan Function

I would like to know how this equality holds. $$ \tan^{-1} \frac{(2n+1) - (2n-1)}{1 + (2n+1)(2n-1)} = \tan^{-1} \frac{1}{2n-1} - \tan^{-1} \frac{1}{2n+1}.$$ I was told to use the double angle ...
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1answer
20 views

Absolute Maximum and Minimum of cos function

I am having a little trouble trying to figure out the following problem: Find the absolute maximum and minimum values of the function $f(x) = x-2\cos x$ on the interval $[0, 2\pi]$. I have taken the ...
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2answers
62 views

Why cant I solve the equation $4\sin(2x)\cdot \cos(2x)=2\cos(2x)$ by dividing both sides by $\cos(2x)$??

This assignment assumes the domain is $[0, \pi]$. Why can't I solve the following equation by dividing both sides by $\cos(2x)$? $$4\sin(2x)\cdot \cos(2x)=2\cos(2x)$$ If I would continue to do so, ...
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3answers
55 views

Trigonometric substitution needed for integral [on hold]

I got stuck with this problem. Any idea to solve this problem please? Any trig. rule may help? $$\int \frac{2\,dx}{x^3\sqrt{x^2-1}}$$
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1answer
35 views

Trigonometric integration

I could not figure out how to solve this problem. Can anyone give me a hint how to solve this: integral $\sqrt{1-\sin2x}dx$
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3answers
43 views

trigonometry of interfering waves

I have a book which makes the following claim: $$ A \cos( \omega t + \phi ) = A_1 \cos( \omega t + \phi_1 ) + A_2 \cos( \omega t + \phi_2 ) $$ ...where: $$ A^2 = A_1^2 + A_2^2 + 2 A_1 A_2 \cos( ...
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2answers
35 views

Proving a trigonometric identity (I'm a newbie)

Why does $$-\frac{\cos^2 x}{\sin x}= -\cos x\cot x?$$ Sorry for the dumb question.
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0answers
37 views

Can trigonometric equations be graphed?

I was solving various trigonometric equations. I was confused that how are they solved easily by using methods that are useful to solve algebraic equations. Do the trigonometric functions in ...
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1answer
15 views

Weighed Euclidean Distance to compare players

Good Day, I am writing some code to compare how similar two players in a sport are. Right now I am using three features: age (between 17.0 and 17.99), points/game (normally between 0.0 and 3.0) and ...
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2answers
48 views

If $\sum (2009-r)\cos(\frac{2\pi r}{2009})=-n/2;\quad 1\leq r\leq 2008,$ then the digits in the unit's …

$$\text{If}\;\sum (2009-r)\cos\left(\frac{2\pi r}{2009}\right)=-n/2;\quad 1\leq r\leq 2008$$ then the digits in the unit's place of $(9417709487)^n$ must be equal to? Well how do I proceed? Hints ...
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0answers
23 views

$2^{199}\sin(π/199)\cdots\sin(198π/199)$ Sine Product Series [duplicate]

What will be the value of $2^{199}\sin(\pi/199)\cdots\sin(198\pi/199)$ ? I could have found in case the functions were cosine but what should i do in case of sine?
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3answers
34 views

Trigonometry-Complex Numbers Based Problem

If $2^7\cos^5x * \sin^3x$=$a\sin8x- b\sin 6x +c\sin 4x + d\sin 2x$ where $x$ is real then what will be the value of $a^4 + b^4 + c^4 + d^4$? Even a hint will suffice... I don't know how to proceed! I ...
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1answer
43 views

Solving several equations involving sine function

Recently, I asked for root of this equation $2x - \sin(2x) = \frac{\pi}{2}$, then i got $x = \frac{Dottie}{2} + \frac{\pi}{4}$. Thanks everyone. Now can i define a function like this: $f(n) = x$ to ...
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2answers
21 views

Simple harmonic motion~

I am stuck in a question which says that: A particle moves on the X- axis according to equation $x=A+Bsin(\omega t)$. The motion is simple harmonic. Find the amplitude of SHM. The answer of the above ...
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2answers
30 views

How do I find the period of the sine function $y = 20\sin\left[\frac{5 \pi}{2}\left(\frac{x -2}{5}\right)\right] + 100$

Using Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off.
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2answers
33 views

finding value of $\cos^{-1}$ unsure why it changes to negative in answer

$\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$ Another of the find the exact value of this questions. The answer it gives is: $\cos^{-1}(\frac{-\sqrt{3}}{2}) = \theta \Leftrightarrow \cos(a) = ...
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2answers
46 views

Finding the exact value of $\sin^{-1}$

I have a question that says to find the exact value of $\sin^{-1} (1/\sqrt{2})$ For the answer it shows $\frac{\pi}{4} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ Where does that last part in ...
1
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1answer
43 views

Approximation of radial displacement

I read, in a physics textbook, that a small displacement $\Delta\mathbf{s}$ of a body -from point $P_i$ to $P_f$ in the image whose bad quality I apologise for: it is the first time I use Geogebra- ...
0
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1answer
10 views

Trigonometry to radially spread points around a center

I'm having a basic Math problem. I have number of point collections, each with a different amount of points. Let's say one collection has five points and I want them to spread radially around a ...
0
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2answers
37 views

How to compute Dottie number accurately?

Dottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$. I wonder how can I compute it ? I have tried to do it with an approximating ...
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4answers
68 views

Solve $\tan\left(\arccos\left(\frac{-\sqrt{2}}{2}\right)\right)$ without calculator

This is a question from the practice exercises of Barron's AP Calculus. The directions state that I cannot use a calculator. (No trig tables I think, since those are not allowed in the exam) So, ...
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3answers
42 views

Trignometric Equation Solution

Question : On the interval $[0,2\Pi]$ there is one point on the curve $r = \Theta - 2cos\Theta$ whose x-coordinate is 2. Find the y-coordinate there. The solution simply states: Solving $(\Theta - ...
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2answers
43 views

Finding value of sin (15 degrees) with half angle identity

The answer I got when trying to solve it was $\sqrt{ [1 - \sqrt3]/2 }$ but the book says it's $\sqrt{ [2 - \sqrt3]/2}$ and I don't know how the two on the top half gets there.
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1answer
52 views

Simplifying identity cos*cos+sin*cos

$$\cos(3\pi/2 - a) = -\sin(a)$$ According to an answer to one of the questions in my book that's true, but come up with that? $$cos(\pi/2 - a) = \sin(a),$$ but this is $3\pi/2$. do you just ignore ...
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2answers
30 views

Implicit Differentiation involving trigonometric functions.

We are given the following condition: $$\tan(x^3y^2)=6x^2+y^2$$ Find the derivative of $y$ w.r.t. $x$, i.e., find $y'=\dfrac{\textrm{d}y}{\textrm{d}x}$ I am having trouble getting started with this ...
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2answers
60 views

Is there an algebraic solution to this problem?

The base of my pool cartridge filter tank is composed of an arc and a chord and has a total perimeter of 49.5 inches. The length of the chord portion of the shape is 7 inches. The arc portion has a ...
0
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3answers
56 views

What is the intuition behind $\cos^2(x)$ being the same as $(\cos x)^2$?

Shouldn't $(\cos x)^2$ be $\cos^2 \cdot x^2$? as $(xy)^2$ is $x^2y^2$? What is the intuition behind it? Please use simple language as I am not a mathmatician.
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4answers
273 views

How would you solve for all solutions of $\sin(2x)=\cos(3x)$ algebraically?

So my buddy and I (both HS Math teachers) have been messing around with a question about finding all solutions of a "co-function" equation like the one above. The typical HS questions asks students to ...
0
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1answer
39 views

If cosecant is $\frac{1}{\sin\theta}$ why do $\sin^{-1}(1)$ and $\frac{1}{\sin(1)}$ give different answers?

When I go $\sin^{-1}(1)$ I get $90$ degrees but when I put $\frac{1}{\sin(1)}$ in the calculator I get $57.2$ degrees. Why is this?
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4answers
31 views

Trigonometry System of Equations

If $\textstyle \tan x$ +$\textstyle \tan y$=24 and $\textstyle \cot x$ + $\textstyle \cot y$=28 compute $\textstyle \tan (x+y)$ I've tried various approaches to doing this including, using the ...