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

learn more… | top users | synonyms (1)

1
vote
2answers
83 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 ...
0
votes
1answer
50 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
44 views

TRIGONOMETRICAL IDENTITIES [on hold]

Prove that 4sinAsin(60+A)sin(60-A)=sin3A
3
votes
0answers
24 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
37 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
67 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
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
33 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
22 views

Solving Trigonometric Equations? [closed]

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
124 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 [closed]

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
161 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
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 ...
3
votes
3answers
61 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
67 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
65 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 [closed]

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

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

What are the dihedral angles in a disphenoid with four identical triangles, each having one edge of length $2$ and two edges of length $\sqrt{3}$? Tried to look it up, but couldn't find it...
2
votes
2answers
59 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
1answer
65 views

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

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
46 views

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

Is the following expression valid? $$ \tan^2 \theta \sin^2 \theta = \sin^2 \theta \tan^2 \theta$$
2
votes
0answers
59 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
24 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 $$ ...
14
votes
3answers
187 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
42 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
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
677 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
97 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
65 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
48 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
206 views

can a real number be added to a complex number [closed]

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
42 views

Calculating the resultant of two forces and angle? [closed]

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
vote
2answers
54 views

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

Does $\sin^2(-x)=-\sin^2(x)$, if not, does it simplify to something else?
0
votes
7answers
73 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 $(2\sin 2x +1)/(2\sin 2x-1)$ [closed]

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
34 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
44 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 ...