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

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0
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1answer
17 views

Trigonometric equation $\sin v = -1/\sqrt{2}$

I'm trying to solve the following: $$\sin(v) = -\frac{1}{\sqrt{2}}$$ My attempt: $$-\sin(v) = \frac{1}{\sqrt{2}}$$ $$\sin(-v) = \frac{1}{\sqrt{2}}$$ $$v_1 = -\frac{\pi}{4} - 2\pi n $$ $$v_2 = ...
8
votes
4answers
761 views

How to raise a complex number to the power of another complex number?

How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
-3
votes
3answers
68 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
-2
votes
0answers
20 views

Triangular Identity. [on hold]

I have an equation $f(x)=5x+2$.I know the slope is 5 and I take the $5^2$ which is 25. I add $25+1=26$ and take the inverse of 26 which is$\frac{1}{26}$ and subtract it from 1, which is the ...
1
vote
2answers
71 views

Trying to solve $\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$

The equation is $$\sqrt{7-4\sqrt2 \sin x}=2\cos(x)-\sqrt2 \tan(x)$$ We get the system $$ \begin{cases} 7-4\sqrt 2 \sin(x)=4\cos^2(x)-2\sqrt2\cos(x)\tan(x)+2\tan^2(x) \\ 2\cos(x)-\sqrt2 \tan(x)\ge 0 ...
-4
votes
0answers
24 views

Verification of an indefinite integral with trigonometric functions

I was making this integral $\int \frac{dx}{\sin(x) + \cos(2x)}$ and i end up with this result: $\frac {2}{\sqrt3}\ln({\frac{\tan(x/2) + 2 -\sqrt3}{\tan(x/2) + 2 +\sqrt3}})\ - \frac ...
6
votes
3answers
173 views

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $\cot\angle MAB + \cot\angle MBC + \cot\angle MCA \geq 3\sqrt{3}$

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $$\cot\measuredangle MAB + \cot\measuredangle MBC + \cot\measuredangle MCA \geq 3\sqrt{3}$$
3
votes
0answers
17 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
3
votes
3answers
76 views

Factorize Trigonometric Equation: $ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $

I have a problem with the following trigonometric equation: $$ 3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0 $$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
13
votes
3answers
157 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
0
votes
1answer
45 views

Resolve $A=\cos{(\pi/7)}+\cos{(3\pi/7)}+\cos{(5\pi/7)}$ using $u=A+iB$

With these two sums: $$A=\cos(\pi/7)+\cos(3\pi/7)+\cos(5\pi/7)$$ $$B=\sin(\pi/7)+\sin(3\pi/7)+\sin(5\pi/7)$$ How to find the explicit value of $A$ using: $u=A+iB$ the sum of $n$ terms in a ...
-6
votes
1answer
43 views

TRIGONOMETRICAL IDENTITIES [on hold]

Prove that 4sinAsin(60+A)sin(60-A)=sin3A
0
votes
2answers
23 views

Right triangle trigonometry help?

I've got a right triangle where I know the slope of side $c$ based on the two points $(-150,200)$ and $(0,0)$. Also I know the length of side $a$. I was wondering based on these two known factors how ...
8
votes
1answer
236 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
3
votes
0answers
22 views

Get the largest rectangle in a quadrilateral

So I have coordinates for a few shapes with 4 sides of varying angles. I need to find the largest rectangle in them, even if the rectangle is rotated. Is there an algorithm for this? In the example ...
0
votes
3answers
33 views

If limit of $ \lim_{x\to0}(\frac{sin2x}{x^3} + \frac{a}{x^2} + b) $ is zero, then find a+b? [on hold]

If limit is zero: $$ \lim_{x\to0}\left(\frac{\sin 2x}{x^3} + \frac{a}{x^2} + b\right) = 0 $$ then find $ a+b=? $ please help me to solve this question, thanks.
1
vote
3answers
25 views

the double angle identities-sin2A

I have a question that asks: Express each of the following in the form $a\sin bA$. The first part of the question asks me to do this for $a) 6\sin A\cos A$ The answer they give is $3\sin 2A$, but I ...
1
vote
4answers
62 views

Trying to solve $\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$

The equation is $$\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$$ I solve it thus: $$ \begin{cases} 2\cos^2(x)-\sqrt3=2\sin^2(x) \\ -\sqrt2 \sin(x)\ge 0 \iff \sin(x)\le 0 \end{cases} $$ The first ...
3
votes
4answers
59 views

Trying to solve the trig equation $\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$

The equation is $$\sqrt{3+4\cos^2(x)}=\frac{\sin(x)}{\sqrt 3}+3\cos(x)$$ My solution goes like this $$ \begin{cases} 3+4\cos^2(x)=\frac{\sin^2(x)}{3}+\frac{6}{\sqrt 3}\sin(x)\cos(x)+9\cos^2(x) \\ ...
3
votes
3answers
58 views

Reduction formulae in definite integration

$$I_n = \int_0^{\pi}\frac{\sin^2(nx)}{\sin^2(x)}dx $$ Find relation between $I_n$, $I_{n+1}$ and $I_{n+2}$ I tried integration by parts by taking $\sin^2(nx)$ as the first function, but reached ...
1
vote
2answers
49 views

I need help with this trigonometric integral [on hold]

I dont know how to do this integral $\int \dfrac{dx}{\sin(x) + \cos(2x)}$ i have tried the fundamental trigonometryc identity $(\sin x)^2 + (\cos x)^2 = 1$ but that does not work out the way i ...
0
votes
2answers
29 views

Getting two different sets of results for $\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$

The equation is $$\sqrt{17+7\sin(2x)}=3\sin(x)+5\cos(x)$$ My solution is, first, to define a system: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
3
votes
3answers
151 views

How can the trigonometric equation be proven?

This question : Whats the size of the X angle? has the answer $10°$. This follows from the equation $$2\sin(80°)=\frac{\sin(60°)}{\sin(100°)}\times \frac{\sin(50°)}{\sin(20°)}$$ which is indeed ...
1
vote
3answers
65 views

Find $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.

$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find $LK_1^2 + LK_2^2 + ...
3
votes
1answer
122 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
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votes
0answers
22 views

Solving Trigonometric Equations? [on hold]

I was just wondering if there was any way to solve #4(d,e,f) by hand without using a graphing calculator?
-2
votes
0answers
15 views

get rectangle size out of 2 corners and rotation [on hold]

okay so i have two corners top Left corner, bottom Right corner and rotation of the rectangle which is rotated from its center.I need to find out the size of the rectangle. I guess that I should get ...
1
vote
2answers
75 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
0
votes
4answers
50 views

How to calculate the tangent of x?

I've looked it up of course and got $\tan(x) = \cos(x)/\sin(x)$. For example $\tan(60) = \cos(60)/\sin(60)$ I get $0.32004$ but when I use a calculator I get $1.7320508075688772935274463415059$? Is ...
2
votes
4answers
65 views

Prove that $\cos \arctan 1/2 = 2/\sqrt{5}$

How can we prove the following? $$\cos \left( \arctan \left( \frac{1}{2}\right) \right) =\frac{2}{\sqrt{5}}$$
3
votes
2answers
42 views

Envelope of two sine waves interfering

I have two waves interfering with different amplitudes, frequencies and phases. $$ x(t) = X_1 \sin(\omega_1 t+\varphi_1) + X_2 \sin(\omega_2 t+\varphi_2) $$ I am able to convert this wave into the ...
1
vote
1answer
293 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
2
votes
2answers
54 views

Find min of $M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$

Find min of $$M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$$, where $A, B, C$ are three angle of triangle $ABC$ Using Cauchy-Schwarz, we obtain: \begin{align*} M &= ...
1
vote
0answers
8 views

trying to grasp disphenoid tetrahedral honeycomb, what are the dihedral angles?

what are the dihedral angles in a disphenoid with 4 identical triangles of 1 edge length '2' and 2 edges of length 'sqrt(3)'? Tried to look it up, but couldn't find it... perhaps there is no such ...
0
votes
1answer
47 views

Length of all sides of a triangle, knowing one angle one length and the perimeter of the triangle.

i am sure this question is answered in a round about way, but my math is not strong enough to put it all together so i need a direct answer for my direct question if you don't mind (: Now i did draw ...
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votes
0answers
16 views

prove the given question [on hold]

Prove that $\sec(2 \alpha)\cos(45^{\circ}-\alpha)\sin(45^{\circ}+\alpha) = \dfrac{1}{2}$.
3
votes
1answer
82 views

Calculating angle on ellipse

This is a really basic question, yet I can't remember my old geometry classes nor could I find an answer via google. Given a circle "tilted" at angle a to the horizontal plane, and given angle b ...
0
votes
1answer
19 views

Intersection of angular ray with circle

I have a geometric/trigonometric problem. I will include a diagram but I know images are not ideal so I will do my best to describe the figure as well. Sorry for the Paint diagram. The angle corner ...
0
votes
1answer
21 views

Rotated parabola 2d vertex

I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get ...
-2
votes
2answers
63 views

Resolving $x^5=i$ using algebra and trigonometry, prove that [on hold]

Resolving $x^5=i$ using algebra and trigonometry, prove that $\cos( 18^{\circ})=\frac{\sqrt{5+2\sqrt{5}}}{\sqrt[5]{176+80\sqrt{5}}})$ $\sin( 18^{\circ})=\frac{1}{\sqrt[5]{176+80\sqrt{5}}})$
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votes
0answers
45 views

Can $ \tan^2 \theta \sin^2 \theta$ be written as $ \sin^2 \theta \tan^2 \theta$? [on hold]

Is the following expression valid? $$ \tan^2 \theta \sin^2 \theta = \sin^2 \theta \tan^2 \theta$$
2
votes
0answers
49 views

Sum of arctans of trignometric expressions

Let $s_k=\sin\frac{2\pi(4k+1)}{4n}$ and $c_k=\cos\frac{2\pi(4k+1)}{4n}$ for some positive integer $n$. If $n=2007$ and $x=3$ , find $\tan \sum_{k=0}^{n-1} \arctan(\frac{s_k}{x-c_k})$ I tried using ...
2
votes
0answers
52 views

Will $x=0$ satisfy the equation $\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$?

The equation is $$\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$$ And the one condition set for the solution is that $x$ should fall within this range: $0\le x < \pi$ The solution process boils down to $$ ...
6
votes
3answers
2k views

How do I determine if the following function is periodic?

A question in my textbook asks me to determine if the function $f(x)=\cos(3x)+\sin(x)$ is periodic. I do not believe this to be the case as the arguments of sine and cosine are different and as such ...
1
vote
4answers
36 views

Epsilon-Delta Limit Proof: Arccos(x) Inequalitiy

I'm studying a Calculus proof using notes (proving that $\lim_{x \to 1} \cos(x) = \cos(1)$ from the definition of limit). The text says that we get from: $\cos(1) −\epsilon < \cos(x) < ...
0
votes
4answers
47 views

Calculate area of a triangle with just one length and a tangent-relation(?)

I am looking through some old mathematics that I did 5 years ago and don't remember 100%. Right now I am learning about trigonometry and have some problem with a question. "The triangle ABC is ...
0
votes
1answer
41 views

Calculating the resultant of two forces and angle? [on hold]

A force of $256 N$ and a vertical load of $537 N$. Trying to work out the resultant of the two forces and the angle at which it acts to the horizontal?
0
votes
2answers
41 views

Maximum of $\cos \alpha_{1}\cdot \cos \alpha_{2}\cdot \cos \alpha_{3}…\cos \alpha_{n}.$

Maximum value of $\cos \alpha_{1}\cdot \cos \alpha_{2}\cdot \cos \alpha_{3}\cdot \cos \alpha_{4}....\cos \alpha_{n}.$ If it is given that $\cot \alpha_{1}\cdot \cot \alpha_{2}\cdot \cot ...
3
votes
3answers
94 views

Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus

In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation ...
0
votes
4answers
667 views

Limits of cosine and sine [duplicate]

When $\theta$ is very small why $\sin \theta$ is similar to $\theta$ and $\cos\theta$ similar to $1$? Is it related to limits or we can prove it simply by using diagrams?