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
41 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
40 views

TRIGONOMETRICAL IDENTITIES [on hold]

Prove that 4sinAsin(60+A)sin(60-A)=sin3A
2
votes
0answers
18 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
32 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 ...
3
votes
4answers
55 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) \\ ...
1
vote
2answers
32 views

I need help with this trigonometric integral

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
24 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: $$ \begin{cases} 17+7\sin(2x)=(3\sin(x)+5\cos(x))^2 \\ 3\sin(x)+5\cos(x)\ge 0 \end{cases} $$ ...
-5
votes
0answers
20 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?
3
votes
1answer
116 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 ...
-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 ...
3
votes
3answers
145 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 ...
0
votes
4answers
48 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 ...
3
votes
3answers
54 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 ...
2
votes
4answers
64 views

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

How can we prove the following? $$\cos \left( \arctan \left( \frac{1}{2}\right) \right) =\frac{2}{\sqrt{5}}$$
1
vote
4answers
59 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 ...
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 ...
-5
votes
0answers
16 views

prove the given question [on hold]

Prove that $\sec(2 \alpha)\cos(45^{\circ}-\alpha)\sin(45^{\circ}+\alpha) = \dfrac{1}{2}$.
1
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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 ...
2
votes
2answers
52 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 &= ...
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 ...
-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 $\cos( 18^{\circ})=\frac{\sqrt{5+2\sqrt{5}}}{\sqrt[5]{176+80\sqrt{5}}})$ $\sin( 18^{\circ})=\frac{1}{\sqrt[5]{176+80\sqrt{5}}})$
-1
votes
0answers
44 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
43 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 ...
0
votes
1answer
20 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
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 $$ ...
12
votes
3answers
132 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
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 ...
1
vote
4answers
35 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
666 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?
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 ...
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 ...
1
vote
2answers
64 views

Trig equation: $a \sin \frac{a \pi}{2} = 1$

How do I solve the following? I am having a bit of a slow moment. $$a \sin \frac{a \pi}{2} = 1$$
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
5answers
204 views

can a real number be added to a complex number [on hold]

does it make sense to add a real to a complex given that addition binary operation is only defined for set of complex numbers OR real numbers also a related question: how can exponential $e^x$ which ...
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?
1
vote
1answer
53 views

a simple question: whence the $\pi$ symbols in the solution of a trig equation?

There's a step-by-step discussion of an example irrational trig equation in my textbook. $$\sqrt{3\sin(2x)}=\sqrt{-5\cos(x)\cot(x)}$$ One of the solutions is $$\cos(x)=-\frac23$$ The solution to ...
1
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2answers
53 views

Does $\sin^2(-x)$ simplify?

Does $\sin^2(-x)=-\sin^2(x)$, if not, does it simplify to something else?
0
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7answers
71 views

Why is $1+\cos(\theta)=2\cos^2(\frac{\theta}{2})$

Why is $1+\cos(\theta)=2\cos^2(\frac{\theta}{2})$? Where this comes from? I don't get it. From $\sin^2\theta+\cos^2\theta=1$?? I search everything but I really don't find that.
0
votes
2answers
31 views

how to find the value of the trigonometric function in the question

$$\text{if }\frac{\sin\theta}{\sin\phi}=\frac12 \text{ , }\frac{\cos\theta}{\cos\phi}=\frac32 \text{ ; if both the angles are the acute angle, then find } \tan\theta \text{ and } \tan\phi.$$ this ...
2
votes
8answers
83 views

Prove the trigonometric identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$

While solving an equation i came up with the identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$. Prove whether this is really true or not. I can add that $$\tan\left(\frac{x}{2}\right) = ...
1
vote
1answer
37 views

Dividing a trigonometric expression

Given: $$\sin {x} ⋅ \cos {3x} = \sin {x} ⋅ 2\sin {3x} ⋅ \cos {3x}$$ Can I divide by $\sin {x} ⋅ \cos {3x}$ ? If I check $\sin {x} ⋅ \cos {3x} = 0$ I get 2 more answers that are correct to the ...
0
votes
1answer
42 views

I apply the sum-to-product identity for $\sin$, but my result differs from the textbook's

I don't understand the last transformation here: $$\sin x - \cos 3x = 0\iff \sin x -\sin\left(\frac\pi2 - 3x\right) =0\iff 2\sin\left(\frac\pi4-x\right)\cos\left(2x-\frac\pi4\right)=0$$ When I apply ...
-1
votes
2answers
42 views

Simplify $\tan3x/\tan x$. Answer given is $(2\sin 2x +1)/(2\sin 2x-1)$ [on hold]

The question is to simplify $\displaystyle \frac{\tan{3x}}{\tan x}$. The answer given in my book is $\displaystyle \frac{2\sin 2x+1}{2\sin 2x- 1}$ but I am not getting this answer by solving it. Can ...
1
vote
1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
1
vote
4answers
63 views

Trignometric Identities and Equations

For the following problem(s) I cannot get any answer(s). I would appreciate your help very much. $$\tan { \theta -\sec { \theta } =\sqrt { 3 } } $$ TI get 30 degrees as the reference angle. What ...
0
votes
2answers
42 views

Infinite series of trigonometric ratios

The question is to compute: $$(1+\cos A)+2(1+\cos A)^2 + 3(1+\cos A)^3+\ldots = \sum_{k=1}^{\infty}k(1+\cos A)^k.$$ I tried by setting $1+\cos A=y$, then the serie becomes $$y+2y^2+3y^3+\ldots = ...
2
votes
1answer
42 views

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$. Can I use Taylor's polynomial?
2
votes
1answer
47 views

Gaussian function in the limit of trigonometric functions

I've noticed that $$ (\sin\theta \cos\phi)^{2n} + (\sin\theta \cos\phi)^{2n-1} $$ increasingly resembles a Gaussian function of $(\theta, \phi)$ as $n$ goes to infinity. In particular, when I take ...
0
votes
0answers
15 views

Trigonometric identities for Bessel Functions?

I'm wondering if there exists extensions of trigonometric identities to special functions like Bessel? For example, is there an alternative way to express the following? $J_0((a+b)x) = ?$ Thanks