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

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49 views

How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $?

How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x? $x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $ for $\sin 2\theta ...
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2answers
40 views

Solving equations with powers without logarithms

Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms. I was asked to find, for example, $27^x = 3$. (without the use ...
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2answers
46 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
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1answer
35 views

If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is…

Problem : If $\alpha, \beta \in [0,\pi]$ then the minimum value of $\sin(\frac{\alpha +\beta}{2})$ is a) $\frac{\sin\alpha +\sin\beta}{2}$ b) $|\sin\alpha -\sin\beta|$ c) $\frac{\cos\alpha ...
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4answers
177 views

Solve The Triangle

I am having a tough time trying to solve this problem. I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its ...
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1answer
43 views

Trigonometry expression 171 [closed]

Simplify step-by-step $$ \frac{\cos(12)\sin(18)}{\sin(12)(2(\cos(6)+\sin(12))-\cos(18))} $$ Result: $1$. (All values are in degrees). I tried, but I cannot solve.
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2answers
46 views

Trigonometric solution of a 6 degree polynomial

How do i prove that $\sin^2 \frac{\pi}{13}$ is a root of the equation $$2^{12}.x^6-13(2^{10}.x^5-5.2^8.x^4+3.2^8.x^3-7.2^5.x^2+7.2^2x-1)$$? Any hints/answers would be appreciated.
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2answers
207 views

How do you find this product?

Is there a way to find the exact value of the product $$P=\displaystyle\prod_{n=1}^{1007} \sin {\left(\dfrac{n\pi}{2015}\right)}$$
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3answers
111 views

Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
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1answer
58 views

Easier way to solve this problem of trigonometry.

Prove that $\sin x \sin y \sin(x-y) + \sin y \sin z \sin(y-z) + \sin z \sin x \sin(z-x) + \sin(x-y)\sin(y-z)\sin(z-x) = 0$ . When I expanded them ,it became horrendous. Is there any easy way or trick ...
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2answers
53 views

Finding the derivative of sinus and cosinus. Trigonometric identities

How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$ How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$ Do ...
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2answers
42 views

Finding a 3rd coordinate of the rectangle points in 3d

I have a 4 3-D-points, each of them has only 2 of 3 known coordinates, as follow (? is unknown here): P5 (P5x, P5y?, P5z) P6 (P6x, P6y?, P6z) P3 (P3x, P3y, P3z?) P4 (P4x, P4y, P4z?) They build ...
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2answers
47 views

Trigonometry sum of solutions question

Problem: For which $a$ will the sum of solutions be equal to $100$, in $\sin(\sqrt{ax-x^2})=0$. The attempt at a solution: For $\sin(x)=0$, $x$ must be equal to $0$, so we get ...
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1answer
15 views

Calculating amount of rotation to straighten an imaginary line created by 2 points.

I am trying to build a small app where my users can straighten up a tilted face with just 2 clicks I ask my users to click on the middle of the nose and the middle of the eyebrows of the face ...
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5answers
378 views

About Trigonometry

Is there anything cool about trigonometry? I was just curious. I'm learning trig right now and I often find myself asking myself, "What's the point?" I feel if I knew what I was working on and why, ...
2
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1answer
34 views

Trigonometric identity proof problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 5: d) $\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha}$ I would appreciate some hints on how to ...
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0answers
21 views

Negative zero in atan2 calcs - what's the application?

I hope this is not considered a duplicate -- I asked a question about negative zero on StackOverflow and was told the question was better suited to a math-related SE site. To that end I'll try to ...
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0answers
61 views

Is $\sin (\mathbb N)$ dense in $[-1,1]$? [duplicate]

Let $\mathbb N$ be the set of positive integers, then is it true that $\sin (\mathbb N)$ is dense in $[-1,1]$ i.e. is it true that for every $x,y \in [-1,1]$ with $x<y$ , $\exists m \in \mathbb N$ ...
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2answers
58 views

Trigonometry - simplifying a given equation [duplicate]

Question: $$\tan 9 - \tan 27 - \tan 63 + \tan 81$$ Answer I'm getting : 0 What I did: Well I clubbed together $\tan 9$ and $\tan 81$ and $\tan 27$ and $\tan 63$ (took out negative as common). Then ...
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2answers
21 views

Clarification on the domain of $\arcsin(\sqrt{1-x^2})$

As the title says, I don't understand how to find the domain of $\arcsin(\sqrt{1-x^2})$. I kinda understand how it would equate to it would be -1 < x < 1 (inclusive of 1 and -1) by definition of ...
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1answer
50 views

For which integer n, sin(π/n) can be a rational?

When I was studying about the trigonometric functions, I sow that most of the values of sin(π/n) and cos(π/n) n∈N are irrational. How can we determine all the n∈N such that sin(π/n) or cos(π/n) is a ...
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2answers
12 views

A cycloid that goes through the beginning and through a general point

Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How can we determine $d$ such that the cycloid goes through ...
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2answers
64 views

Solve $\cos(x) = \cos(x+a)$, if $\cos(x)\ge0$

Solve $\cos(x) = \cos(x+a)$, if $\cos(x)\ge0$ This is how I did it: For $\cos(x)\ge0, x\in[(4n-1)\frac\pi2,(4n+1)\frac\pi2]$ Also, $\cos(x+a)\ge0\implies ...
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4answers
45 views

Proving the trigonometrical identities

please prove this answer, step by step.. $$\cos A - \cos 3A = 4 \sin^2A \cos A$$ I had just finished the left side $= -2 \sin 2A \sin A$ but then I have no idea to prove it..
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2answers
29 views

Trigonometric manipulation

From $$\frac{R\sin(\omega t)-\omega L\cos(\omega t)}{\omega^{2}L^{2}+R^{2}}$$ I have to get $$\frac{\sin(\omega t-\alpha)}{\sqrt{R^{2}+\omega^{2}L^{2}}}$$ where $\alpha$ is a constant. How do I do ...
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1answer
27 views

Simple Trig Question / Introduction to Vectors Question

Sorry this is such a simple question; I'm just struggling a little with my trigonometry homework. An example question: "A ship sails due north (relative to the current) with a speed of 20 knots. The ...
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1answer
42 views

How does BC = sin(y)?

This is part of an exercise to work with the proofs of the sum of angles, either sin(x + y) or in this case cos(x + y). The solution steps declare BC = sin(y) without any explanation. I guessed at it ...
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2answers
35 views

Trigonometry question, find the value if $\cos(x) = \frac{5}{13}$ and $\cos(y) = -\frac{5}{13}$

My cousin is working on this and showed it to me. I'm unsure how to solve it. $x$ and $y$ represent two angles in standard position. $x$ has its terminal arm in the first quadrant and $y$ has its ...
0
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5answers
58 views

Taking the sin of arccos

When solving for the value of x in the equation $$\sin^{-1}{(\sqrt{2x})}=\cos^{-1}(\sqrt{x})$$ one would take the sin of both sides of the equation cancelling out the arcsin leaving ...
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1answer
32 views

Proving that pressure at a point does not depend on orientation

In a) the solution states that $dS_1=$cos$\theta dS_2$, in other words it considers the surface area to be equivalent to the length of a line, in order to use basic trigonometry. I understand we are ...
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2answers
85 views

Fun Tan Question [duplicate]

Using only trig identities, how would you approach the following question? Determine the value of $$ \prod_{i=1}^{89} \tan i^° = \tan 1^° \cdot \tan 2^° \cdots \tan 89^° $$
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1answer
36 views

Ratio of sides of Triangle $ABC$

if in a Triangle $\Delta ABC$ with $a$, $b$ and $c$ as sides $$\begin{align}\left(Cot\frac{A}{2}\right)^2 ...
1
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3answers
35 views

Double Angle Trigonometry Question

So there is this question which consists of 2 parts. $$ a) \text{ Simplify } \frac{\sin2x}{1+\cos2x} \\ b) \text{ Hence, find the exact value of tan 15.} $$ So far I've discovered that $ \text{a)} ...
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0answers
29 views

Computing the coordinates of a point, offset from a rotated point.

Good day. I have a question which should be easy but I have not been able to figure it out. The coordinates of a point on a unit circle, given an angle, is $$\begin{align} x &= \cos(\alpha) \\ y ...
2
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1answer
38 views

$\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $?

How to prove, if $\tan( x + i y ) = a + ib$ then $\tan (x - iy) = a - ib $ ? I am not familiar with trignometric identities. So any help will be appreciated. Thanks in Advance.
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2answers
52 views

Find the exact value of the trigonometric function $\sin 7\pi/ 6$

I am finding it a little difficult to solve this problem. The reference angle for $\sin 7\pi/6$ is sin 30 degrees (I think) which is sine 1/2. But that is not the answer. How do I sove this problem?
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0answers
23 views

Finding true bearings?

What is the true north bearing of NNE on 16 point cardinac compass? I just wanna know that is there any exact bearing or do we have to only give an approximate bearing?
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2answers
70 views

if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $?

I would be interest to show : if $ f(x)=x+\cos x $ then find $ \int_0^\pi (f^{-1}(x))\text{dx} $ ? my second question that's make me a problem is that : what is :$ f^{-1}(\pi) $ ? I would be ...
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5answers
148 views

Help me prove: sin(A+B) = sinA cosB + cosA sinB [duplicate]

Can you help me prove that: sin(A+B) = sinA cosB + cosA sinB? Thanks!
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3answers
61 views

$\arctan(-3/2)$ doesn't give expected result.

Let's say I want to find the angle measure (in degrees) such that $\tan(x) = -3/2$. It turns out that $x \approx 123.7$, and when I compute $\tan(123.7)$, I get $\approx -3/2$; so far so good. ...
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5answers
90 views

Solving $\sin x = 4\sin10°\sin40°\sin(70°-x)$

So, I have this equation: $$\sin x = 4\sin10°\sin40°\sin(70°-x)$$ And I'm trying to solve for $x$. Apparently $x=20°$ is the (smallest positive) solution but I can't arrive at it. I'm not very ...
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0answers
27 views

Find the relation the 'maps' 2D points to the corresponding 3D images.

I have this [on hold] question (#857264) re-phrased. Hope that the content is more meaningful now. The following is the picture modified from the original. The question is a rectangular piece of ...
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0answers
107 views

What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
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0answers
32 views

Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
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1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
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1answer
24 views

Finding angle of depression?

The question is that a boy standing on the top of a staircase 33m high while looking at a oatch of grass on the ground 50 m away from him.What is the anglre from where he was looking at. Please draw ...
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1answer
23 views

Finding Angle of Depression

Finding angle of depression but i dont have idea that what i should mark as hyportenus or opposite? Please help!!!! So, here is the question that Chris is standing on the top of the cliff of 70 m and ...
3
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2answers
52 views

Trigonometry calculation problem

Problem: Calculate $\cos(\alpha+60)$, if $\sin(\alpha)=\frac{2}{3}$ and $\alpha\in[90^\circ,180^\circ]$. I have tried following: ...
2
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2answers
116 views

Mathematical Identity

I'm stuck in a path on a paper about thermal conductivity. There is a identity involving an integral that a I can't realize how they've perfomed it. Here is it: $$\lim_{N\to \infty} ...
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1answer
20 views

find period of discrete cosine

let us consider following we should find period of this discrete signal,for periodicity we should have $x[n+kN]=x[n]$ or $10\cos(0.088\pi(n+kN) +\phi)=10\cos(0.088\pi n+\phi)$ or $0.088\pi ...