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

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11
votes
6answers
226 views

Ways to prove $\displaystyle \int_0^\pi dx \dfrac{\sin^2(n x)}{\sin^2 x} = n\pi$

In how many ways can we prove the following theorem? $$I(n):= \int_0^\pi dx \frac{\sin^2(n x)}{\sin^2 x} = n\pi$$ Here $n$ is a nonnegative integer. The proof I found is by considering ...
1
vote
1answer
40 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.
3
votes
1answer
42 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 ...
0
votes
1answer
8k views

Finding functions for an angle whose terminal side passes through x,y

How would I "Find the six trigonometric functions for the angle theta whose terminal side passes through the point (-8,-5)"?. I learned this material over 2 years ago and since then have forgotten. I ...
0
votes
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 ...
1
vote
0answers
38 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 ...
0
votes
7answers
46 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 + ...
0
votes
1answer
17 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 ...
2
votes
1answer
71 views

How to prepare myself for an advanced trignonometry exam

I'm gonna have a trigonometry/general algebra exam soon. My teacher has told us about some trignometric proofs, and we defined the $\sin$ and $\cos$ int he right way, doing all formal proofs for the ...
0
votes
1answer
9 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
0
votes
2answers
38 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 ...
0
votes
1answer
20 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.
0
votes
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 ...
4
votes
3answers
111 views
+100

the first $2k$ terms of the power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
2
votes
2answers
30 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 : ...
1
vote
1answer
827 views

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side.

A Ferris wheel has a radius of 20 m. Passengers get on halfway up on the right side. The direction of rotation is counter clockwise. The bottom of the Ferris wheel is 2 m off the ground. It rotates ...
7
votes
1answer
211 views

How to find this integral $\int_{0}^{1}\frac{x}{1-x^4}\arctan{\frac{x-x^5}{1+x^6}}\,dx$

Find the integral value $$ I=\int_{0}^{1} {x \over 1 - x^{4}}\,\arctan\left(x - x^{5} \over 1 + x^{6}\right)\,{\rm d}x $$ My good friends gave me this problem, and I can't solve it. Using computer I ...
0
votes
4answers
63 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 $x=2\cos^2\theta+5\sin^2\theta$)
1
vote
4answers
34 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?
5
votes
4answers
70 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 ...
1
vote
4answers
54 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 ...
1
vote
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.
4
votes
3answers
54 views

Finding the limit of a function with ArcTan

I've found difficulties finding this limit ( without using Taylor series approximation, as it's intended for the secondary-school ): $$ \lim_{x\ \to\ \infty}\left[\, {x^{3} \over \left(\,x^{2} + ...
1
vote
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 ...
0
votes
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 ...
3
votes
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
votes
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?
1
vote
1answer
22 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), ...
2
votes
2answers
85 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 ...
2
votes
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 ...
0
votes
2answers
42 views

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

How do you prove that $8 \cos{(x)}\cos{(2x)}\cos{(3x)} - 1 = \dfrac{\sin{(7x)}}{\sin{(x)}}$?
2
votes
4answers
63 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) ...
1
vote
1answer
596 views

Sum of Sinusoids with Same Frequency = Sinusoid (proof)

I am studying Fourier analysis on my own, I realised that probably the first thing you want to proof in Fourier transform is that the sum of 2 sinuoids (namely a sine and cosine) with the same ...
0
votes
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) $.
1
vote
3answers
36 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
vote
5answers
82 views

Trigonometric Limits - solution needed

how to solve this problem? (without using l'Hopital rule) $$\lim_{x\to π/2} \frac{1-\sin x+\cos x}{\sin 2x -\cos x}$$ thanks for helping.
5
votes
4answers
349 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 ...
0
votes
2answers
46 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
0answers
43 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 ...
1
vote
4answers
59 views

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

How to find the value of $$\lim_{x\to \infty} \frac{x - \sin(x)}{x+\sin(x)}$$
9
votes
4answers
263 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
1
vote
4answers
70 views

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

What is the maximum value of $f(\theta) = \sin\theta \cos\theta$ ?
0
votes
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 ...
2
votes
2answers
41 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
1
vote
1answer
1k views

How to find the critical numbers of a trig function

So here's my function: $g(θ) = 20θ − 5 tan θ$ The instructions are: Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use $n$ to denote any arbitrary integer ...
7
votes
3answers
10k views

Is there a way to get trig functions without a calculator?

In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a ...
7
votes
2answers
253 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 $\arctan{A}-\arctan{B}$ ...
1
vote
4answers
53 views

Solving this trigonometric task

Find the values of $R$ and $\alpha$ in the identities below, given that $R>0$ and $\alpha$ is an acute angle. $$\sqrt{3}\cos{\theta}-\sin{\theta}=R\cos(\theta+\alpha)$$ I'm a bit confused by this ...
0
votes
0answers
19 views

Solving spherical triangle

How do you use Napier's analogies to find the angles $\alpha$ and $\beta$ in here ?
4
votes
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 ...