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

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18 views

How to find the root of this non-linear equation?

I am trying to solve this non-near equation using Matlab but it doesn't give me the correct answer (as shown in the document that I am doing it from). The Matlab code gives me imaginary root. Could ...
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0answers
23 views

Trig word problem [on hold]

A game warden at tower A sights an injured zebra at a bearing of 295 degrees. A warden in tower B, which is located 45 miles at a bearing of 45 degrees from tower A, sees the same zebra at a bearing ...
2
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3answers
34 views

Finding $\lim_{x\rightarrow 0}\frac{x}{2}\sqrt{\frac{1+\cos(x)}{1-\cos(x)}}$

We know that $$\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\tan({x}/{2})$$ so we can change the above function to another form as follow ...
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1answer
45 views

For which positive integers $n$ does $P(n)$ fail to hold?

Let $n$ be a natural number and let $z$ be a complex number. Consider the following proposition: $P(n)$: If $\cos (nz)$ is bounded above by one in absolute value, then $\cos z$ ...
3
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0answers
21 views

Probability density function of $x$ in the unit circle?

I'm trying to work out how to find the probability density function (PDF) for $x$ values on the unit circle - not within the unit circle but on the edge. The reason for doing so is that I'm trying to ...
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0answers
16 views

Split this integral

I need to split this integral if possible: \begin{equation} \int_{\mathbb{R}^d} e^{\sum_{i=1}^dx_iz_i}cos(\sum_{i=1}^dy_iz_i)d\mathbf{z} \end{equation} I wanted split into two part : one with $x_i$ ...
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3answers
52 views

Solving $\cos^6(2x)+\sin^6(2x)=\frac58$ [on hold]

How do I solve the following equation for $x$? $$\cos^6(2x)+\sin^6(2x)=\frac58$$ Thanks
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2answers
21 views

Calculate sum of angles if you know their tan value

$$\tan(u) = 2, \ \ \ 0 < u < \frac{\pi}{2}\\ \tan(v) = 3, \ \ \ 0 < v < \frac{\pi}{2}$$ What is $u + v$? I know that both angles are in the first quadrant in the unit circle. How do I ...
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0answers
8 views

Averaging of a product of fast oscillating function and slow oscillating one

Let's have a product of fast oscillating function and some slowly oscillating function: $$ F(\mathbf r) =f(kz)g(\mathbf r), \quad f(kz) = cos^{2}(kz),sin^{2}(kz). $$ I want to average this quantity ...
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0answers
14 views

Trouble matching two shapes at an angle [on hold]

I want to connect two pieces of wood of the same width at an angle, but the ends are never the same width.
3
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2answers
67 views

Using an Integral to Solve for a Variable a

I am struggling to use the following equation: $$ \int_0^a \sqrt{a^2-x^2}\,\,\text{sgn}(|x|-1)\, dx = 0 $$ where $a > 1$, to deduce that $a = \text{cosec}(\frac{\pi}{4} - \frac{\alpha}{2})$, ...
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5answers
136 views

Why is $(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000\cdots$?

Apparently, $$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$ Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$. Why is that?
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1answer
45 views

How to find this limit without l'Hospital's rule: $\lim_{x\to 0} \frac{\cos(2x)-1}{1-\cos(7x)}$ [on hold]

How to find this limit without l'Hospital's rule? $$\lim_{x\to 0} \frac{\cos(2x)-1}{1-\cos(7x)}$$ Note: Taylor expansion is not available.
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0answers
6 views

Proving a specific case of reduction

Here is the problem: Let I(m) = integral(0-pi/2)(sinx)^m (dx) Prove that I(m) = ((m-1)/m)(I(m-2)) I used the reduction formula for (sinx)^m, which is: Doing this, you find that the integral ...
3
votes
1answer
49 views

Why is trigonometry important in calculus?

I need to write short note why trigonometry is important is calculus and engineering mostly for presentation. I am not focusing on on what topic it specifically it appears (because I am guessing the ...
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votes
1answer
10 views

Graphing and adding trig [on hold]

Cos a= 24/25 Sin is less than 0 Cos(0+pi/6) I graphed 24/25 in the 4th quadrant and then did Pythagorean theorem. After that I don't know what to do
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1answer
18 views

find image and inverse image of function

I have function $f:R\to R^2 , \ \ f(x)=<\cos 3x, \sin 3x>$ and I have to find image on the interval $(0, \pi]$ and inverse image $[0, +\infty) \times[0, +\infty)$ I think the image will be ...
0
votes
2answers
39 views

Trigonometry specific problem

This was all the information given $$\sin^2{2 x} - \sin x-1 = 0, \ x \in [0,2\pi)$$ I did the quadratic formula and ended up with two answers which was a positive and negative. I canceled the ...
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8answers
56 views

Proving $\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$

I got this question from a paper but can't solve it and the question paper has no solutions section.How do you prove this? $$\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + ...
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1answer
21 views

How to prove this, a sin(B-C) + b sin (C-A) + c sin (A-B) = 0

I used Sin rule and I couldn't solve rest of the part.
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1answer
36 views

Proving identity relating to properties of triangles [on hold]

Prove that $\sum a^3 \sin(B - C) = 0.$ (Edit from comment: $a$ is the length of side $BC$. $B$ and $C$ denote angles at vertices $B$ and $C$ respectively.) How can I solve this problem? Any tips ...
2
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2answers
32 views

Trigonometry : Find the length of side

Can someone tell me how to calculate the length 'd' from the below figure? It is from Lecture 06 - Optical flow : ...
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0answers
48 views

Prove that $a^2(p-q)(p-r)+ b^2(q-r)(q-p)+ c^2(r-p)(r-q) =4(\delta)^2$

If $p$,$q$,$r$ are the perpendiculars drawn from the vertices of a triangle ABC upon any straight line meeting the sides externally in D,E,F. where a,b,c are the sides opposite to angles A,B,C in ...
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4answers
69 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
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4answers
35 views

What is the approximation of trigonometric function by simple function

for $f(x)=\sin x$, $g(x)=\cos x$, $h(x)=\tan x$, What is the approximation of each function by using simple function?
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votes
4answers
72 views

How to simpify $\cos x - \sin x$

How does one simplify $$\cos x - \sin x$$ I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse. Yet ...
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3answers
35 views

How to solve $\cos(5\alpha + \pi/2) = \cos(2\alpha + \pi/8)$ for $a$?

I missed the lecture. I don't want you to solve my homework, I just want to learn how to solve equations like this one. Since I have no idea, I'll post the task I got for homework, rather than ...
1
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1answer
38 views

Sum of trigonometric functions

Is the following inequality true? $$\left|\sum_{i=1}^{n}\left(\cos(x_i) \prod_{j\neq i}\sin(x_j)\right)\right|\le 1$$ I tried to count the extremes but it didn't work.
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0answers
20 views

When substituting in integration, do you have to change the limits of integration so long as you keep it consistent?

I have this integral: In order to solve for it, I have to substitute: t=tan(theta) dt=(sec(theta))^2 d(theta) When substituting that, I know I have to change the limits of integration within ...
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1answer
23 views

Find the radius given only a few variables

I'm writing a program that allows someone to generate a vertical road segment in 3D given a HEIGHT and an ANGLE. The road starts off flat, curves (to the ANGLE), has a brief straight segment (SEGLEN), ...
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4answers
56 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
2
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2answers
22 views

Inequalities with arctan

I don't understand how to solve inequalities with arctan, such as: $$\arctan\left(\frac{1}{x^2-1}\right)\ge \frac{\pi}{4} $$ If someone could solve this and give me a very brief explanation of what ...
2
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2answers
88 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
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2answers
43 views

Prove $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$

How do you prove that $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?
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1answer
50 views

Find exact value of $\sin\left(\dfrac x2\right) $

I have tried this problem over and over but can not get it. Can anyone provide a solution? Given $\sin(x) = -\dfrac67$ and $\tan(x)\gt0$ , find the exact value of $\sin\left(\dfrac x2\right) $.
3
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3answers
39 views

Verifying trig identities specific problem

$$\frac1{1-\cos y} + \frac1{1+\cos y} = 2\csc^2y $$ My attempt was me trying to find a common denominator on the left side but I don't know what to do after that.
11
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2answers
1k views

Proof that $\sin 10^\circ$ is irrational

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin 10^\circ$ is irrational. Any ideas?
0
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2answers
49 views

Fixing the closed form of $\sum_{k=1}^nk\sin^2(kx).$

I've been working on finding the closed form of this:$$\sum_{k=1}^nk\sin^2(kx).$$ Using the fact that:$$\sum_{k=1}^nku^k={u\over (1-u)^2}\bigg[nu^{n+1}-(n+1)u^n+1\bigg]\forall u\ge 1\quad (1)$$ I ...
5
votes
4answers
359 views

Trigonometric identities tan(x/2)

I have this task. I know that i) is $\displaystyle\frac{2t}{1-t^2}$ How do I get to ii) and iii) If $\displaystyle\tan(x) = \frac{2t}{1-t^2}$ I would multiply by $\cos(x)$ to get ...
2
votes
4answers
64 views

Extracting $x$ from $\cos(\arcsin(x))$

The following I know to be valid: $x = \sin(\arcsin(x))$ But is it possible to extract $u$ from $\cos(\arcsin(u))$ ? Should it be: $\cos(\arcsin(t)) = \sin\left(\dfrac{\pi}{2} + \arcsin(t)\right) ...
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3answers
37 views

Maximizing sin(a-b) given a trig relation

Suppose $a$, $b$ are acute angle measures such that $\tan a = 5\tan b$. Find the maximum value of $\sin(a-b)$. $\sin(a-b)=4\sin b \cos a$, but I don't know what to do from here.
1
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4answers
59 views

Evaluating $\lim_{x\to \infty} \frac{x - \sin(x)}{x+\sin(x)}$ [on hold]

How to find the value of $$\lim_{x\to \infty} \frac{x - \sin(x)}{x+\sin(x)}$$
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0answers
32 views

Calculating the length of a circular arc

In the post, How do the power-series definitions of sin and cos relate to their geometrical interpretations?, I am having trouble following the logic the blogger uses in the "Calculating the length of ...
-5
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0answers
44 views

What are the integration of these inverse trigonometric function? [on hold]

Integrate the following: Please Help me, I don't where to start. I used several methods to solve this like completing the squares.. $\int\frac{u^4+4}{u^4+9}du$ $\int\frac{\sin(x)(\cos ...
7
votes
2answers
254 views

Finding an inverse trigonometric sum

How do I prove that the following equality holds- $$\sum_{p=1}^{10} \sum_{q=1}^{10} \arctan \left(\dfrac{p}{q}\right)=25\pi$$ I tried to create telescoping terms by using the $\arctan{A}-\arctan{B}$ ...
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4answers
70 views

What is the maximum value of $f(\theta) = \sin\theta \cos\theta$

What is the maximum value of $f(\theta) = \sin\theta \cos\theta$ ?
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0answers
19 views

Solving spherical triangle

How do you use Napier's analogies to find the angles $\alpha$ and $\beta$ in here ?
4
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1answer
49 views

Why do we have trigonometric functions besides $\sin(x)$?

Probably a terrible question, but I've been curious and can't come up with a reason besides convenience for myself with my limited knowledge. Why do we have $\cos(x)$, $\tan(x)$, etc. when all of ...
-3
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2answers
32 views

Integrate using trigonometric substitutions: [on hold]

Integrate $\frac {\sqrt{4x^2+4}}{x} $ using trigonometric substitutions
-6
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3answers
38 views

Integrate $\frac{x^3}{(1-x^2)^{1/2}}$ using trigonometric substitition [on hold]

Use trigonometric substitition to integrate $$\int\dfrac{x^3}{(1-x^2)^{1/2}}\,dx$$