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

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-4
votes
2answers
27 views

How do u prove this?

How to solve? It is asked to prove $$ LHS=RHS $$ Please Which identity should I use and how to know which to use when $$ \cos (2A/1)-\sin (2A) = \cot(π/4-A) $$
1
vote
1answer
27 views

Prove that $\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$

As the title suggests, what is required to prove is that $$\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$$ I was looking back through my ...
0
votes
0answers
5 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
-5
votes
0answers
23 views

How to find the trigonometric functions? [on hold]

Knowing that $\tan a = -\frac 1 2$ and that $a$ is in Quadrant 2, find the other five circular functions ($\sin, \cos, \cot, \sec, \csc$) using the identities in trigonometry. Please help, thank you.
2
votes
1answer
37 views

Strange trigonometric proof.

I was trying to find out how to prove $$ \sin(A-\arcsin(0.3 \ \sin \ A)) \ \cdot \ \sin(A+\arcsin(0.3 \ \sin \ A)) \ = \ 0.91 \ \sin^2 \ A \ \ . $$ When I put this equation into my calculator both ...
2
votes
0answers
39 views

Is this expression for $x\pmod n$ interesting; nontrivial?

For example, we would get several interesting results if we had a formula for $x\pmod n$ that was uniformly convergent, however, according to Wikipedia (Floor and Ceiling Functions) these formulas do ...
0
votes
2answers
61 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
6
votes
2answers
115 views

Find the limit analytically when the sine functions have square roots?

Find the limit analytically of the following: $\lim \limits_{x \to 0} \frac {\sin(\sqrt{2x})}{\sin(\sqrt{5x})} $ The closes thing we learned in class about this was that $\sin(x)$ over $x$ will ...
0
votes
2answers
34 views

Solving an equation with $\sin(x)$ in the exponent: $2^{\sin(x)} \cdot \cos(x) + 1 = 1$

Hi I need help with a trig problem: I have $2^{\sin(x)} \cdot \cos(x) + 1$, and I need this to equal $1$ between $x = -3$ and $3$. I keep going in circles with substitution, etc. Any help would be ...
3
votes
2answers
36 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
0
votes
2answers
15 views

Impossible solutions in trigonometric equations

I'm trying to solve $\sin{4v} + \cos{4v} = 0$ I get 4 equations which I can solve for the solutions, including these 2: $4v_1 = \frac{\pi}{2} + 4v_1 + 2\pi n$ $4v_2 = -\frac{\pi}{2} + 4v_1 + 2\pi ...
0
votes
2answers
33 views

Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$

Problem : Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$ where $x \in [0,2\pi]$ My approach : ...
1
vote
5answers
52 views

cos(4v) + cos(v) = 0

I am given the following equation: $$\cos 4v + \cos v = 0$$ My attempt: $$\cos4v = -\cos v$$ $$\cos4v = \cos(\pi \pm v)$$ $$4v = \pm \pi \pm v + 2\pi n$$ $$4v_1 = \pi + v_1 + 2\pi n$$ ...
5
votes
4answers
90 views

Showing that $\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x$

Show that $$\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x,\forall x\in \left ( 0;\frac\pi2 \right )$$ Firstly, I had use the differentiation of $f(x)=\left (\frac{\sin x}{x} \right )^3- ...
0
votes
0answers
27 views

Epsilon-Delta Limit for Trigonometric Function

I'm studying an Epsilon-Delta proof for a trigonometric function: $$\lim_{x \to 1/9} \sin(x) = \sin(1/9)$$ This is the procedure from my (Italian) book: $$−\epsilon < \sin(x) − \sin(1/9) < ...
-1
votes
0answers
19 views

Prove that $\sin\theta_1.\sin\theta_2.\sin\theta_3=\frac{r^2_1}{16R^2}$

If $2\theta_1,2\theta_2,2\theta_3$ are the angles subtended by the circle escribed to the side $a$(opposite to vertex $A$) of a triangle at the centers of the inscribed triangle and the other two ...
1
vote
1answer
24 views

Finding cubed roots of complex number

Is this correct? $a^3 =r^3e^{i3\theta}= 5\sqrt{5}e^{i\arctan(11/2)}$ $$\implies r=\sqrt{5}, 3\theta = \arctan(11/2)+2\pi n,n\in\Bbb Z$$ $$\theta = \frac{\arctan(11/2)+2\pi n}{3}$$ $$\theta = ...
0
votes
2answers
16 views

$4\sin^2\frac{\theta}{2}.S=(n+1)\sin n\theta-n \sin (n+1)\theta$, and $4\sin^2\frac{\theta}{2}.C=-1+(n+1)\cos n\theta-n \cos (n+1)\theta$

If $S\equiv \sin\theta+2\sin2\theta+3\sin3\theta+......+n\sin n\theta$ and $C\equiv \cos\theta+2\cos2\theta+3\cos3\theta+......+n\cos n\theta$,prove that $4\sin^2\frac{\theta}{2}.S=(n+1)\sin ...
-5
votes
0answers
17 views

can you illustrate this problem please? [on hold]

2 forest rangers observed a camp fire in the directions S60W and S66E from their stations. If the 2nd ranger was 2.76 miles due west of the 1st, which is the closer to the fire and how much closer is ...
1
vote
1answer
36 views

ASTC: Finding exact values of trigonometric functions

Our teacher showed us this really dodgy way of finding exact values by drawing up the 4 ASTC (all stations to central diagram) quadrants and making a right angle to the x axis. So how would I do a ...
-1
votes
4answers
76 views

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$. I only know how to solve using factoring and the basic trig identities, I do not know reduction or anything of the sort, please ...
3
votes
1answer
38 views

The Rhombohedron

I am trying to model a rhombohedron (using Blender) as a first pass to building Dürer's solid so I am trying to calculate the (x,y,z) values for a given side length 'a' and angle 'theta' (starting ...
4
votes
0answers
58 views

Prove $\cos(\sin x)>\sin(\cos x)$ [duplicate]

Prove that $\cos( \sin x)>\sin(\cos x), \forall x\in\mathbb{R}$. I have thought that we should consider their difference and show it is positive for all x, so: Let $$A=\cos\sin x-\sin\cos ...
-3
votes
1answer
18 views

Find the parameter a of function $y = 2\sin(\frac{\pi}{4}x+a)$ [on hold]

Find the parameter a of the function $y = 2\sin(\frac{\pi}{4}x+a)$ so that the corresponding trigonometric function would be even, and the value at point $x = 0$ positive. What is the fundamental ...
0
votes
0answers
23 views

Get coordinates to rotate a path around a circle JS (d3.js)

I'm trying to use the formula from this question Calculating the coordinates of a point on a circles circumference from the radius, an origin and the arc between the points to rotate a line around 180 ...
1
vote
2answers
62 views

trying to solve $\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$

The equation is $$\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$$ The system is $$ \begin{cases} \cos(x)-2\cos(2x)=2\cos^2(2x) \\ -\sqrt{2}\cos(2x)\ge 0 \iff \cos(2x)\le 0 \end{cases} $$ The ...
0
votes
3answers
58 views

Find period of $y=\sin\frac1x$

Find period of $$y=\sin\frac1x$$ We knew that function $y=\sin x$ has period $2\pi$, $y=\sin2x$ has period $\pi$. And $y=\sin \frac1x$ has period $2\pi$, but when I see its graph, I think I was ...
0
votes
2answers
60 views

Periodic function without trigonometry and complex numbers [on hold]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
-2
votes
0answers
29 views

Exercise about factorization

I've just started a new year at school, and I learned these formulas: $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ We used them in class to do some factorization ...
1
vote
3answers
36 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 = ...
-1
votes
0answers
29 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 ...
-3
votes
0answers
27 views

Verification of an indefinite integral with trigonometric functions [on hold]

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 ...
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 ...
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
48 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
3
votes
0answers
23 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
36 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
62 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
32 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? [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
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 ...
-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
158 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
60 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
66 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
63 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 ...