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)

6
votes
4answers
635 views

Simple Trig Equations - Why is it Wrong to Cancel Trig Terms?

In the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...
1
vote
1answer
22 views

Finding the distance from a mirror

My friend sent me this problem, and my efforts to solve it have thus far been frustrated. I need some insight! Joe is 6 feet tall, and standing in front of a mirror that is at eye level, and is 3 ...
0
votes
2answers
13 views

how to find the other find circular functions using trigonometric identities. [on hold]

How can I find the other five circular functions of this problem using the trigonometric identities? $\tan x = -\frac{1}{2}$, $x$ is in Quadrant 2
0
votes
1answer
18 views

Basic Trigonometry problem regarding angles, sines and cosines

PRoblem: A long cable is thrown over a high tree branch, pulled tight, and anchored on the ground, with each end some distance from the branch. On the right side, the cable makes a 60 degree angle ...
2
votes
1answer
57 views

How do I simplify $\tan(\theta)\sin^2(\theta)$?

I am trying to simplify this trigonometric expression, so I can solve for theta. $$\tan(\theta)\sin^2(\theta) = k(q^2)/[4(L^2)mg]$$ I can't seem to simplify it down to one instance of theta despite ...
0
votes
0answers
24 views

Expected value of product of sinusoids

In the book Adaptive Signal Processing by Widrow, an equation (2.20) on page 23 is presented without proof as: $$E \left[ x_k x_{k-n} \right] = \frac{1}{N} \sum_{k=1} ^{N} \sin\left(\frac{2 ...
6
votes
3answers
211 views

Finding the definite integral of a trigonometric expression

Find the integral of $$ \int_0^{\frac{\pi}{2}}{{\sqrt{\sin(2\theta)}} \cdot \sin(\theta)d\theta}$$ I got $$I=\int_0^\frac{\pi}{4}{\sqrt{\sin(2\theta)} \cdot (\sin(\theta)+\cos(\theta))d\theta}$$ But, ...
34
votes
4answers
4k views

Why do both sine and cosine exist?

Cosine is just a change in the argument of sine, and vice versa. $$\sin(x+\pi/2)=\cos(x)$$ $$\cos(x-\pi/2)=\sin(x)$$ So why do we have both of them? Do they both exist simply for convenience in ...
1
vote
0answers
17 views

How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
-3
votes
1answer
31 views

Trigonometric Problems [on hold]

$$X=A\sin(wt+L)\\ Y=B\sin(vt+Q)$$ Prove that, $$\frac{X^2}{A^2}+\frac{Y^2}{B^2}-2\left(\frac{XY}{AB}\right)\cos(L-Q)=\sin(L-Q)$$
-4
votes
2answers
36 views

How do u prove this? [on hold]

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
35 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
12 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
26 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
2answers
57 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
40 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
64 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
120 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
38 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
42 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
18 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
54 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
97 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
28 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
22 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
17 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
38 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
42 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
24 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
63 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
63 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
38 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
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) \\ ...