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)

0
votes
1answer
15 views

Find coordinates of bounding box corners of rotated rectangle

I have a rotated rectangle inside a bounding box. It can be rotated to any angle. I know the coordinates of the "top left" corner of the inside rectangle (and I am able to work out the other 3 ...
1
vote
1answer
62 views

Prove that $\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$ has no rational roots.

Show that the following equation has no rational roots. $$\sin{\frac{2\pi x}{x^2+x+1}}=\frac{1}{2}$$ This is what I've tried: $$\left ( \frac{2\pi x}{x^2+x+1}=\frac{\pi}{6}+2k\pi ...
-1
votes
2answers
38 views

Is there a way of finding the general solution to this equation

I only know how to solve an equation $\sqrt{x}=\sin(3x)$ by newton raphson method of estimating the zeros of the equation. But I am looking for some other method of generalized solution to such an ...
1
vote
1answer
19 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; ...
2
votes
2answers
41 views

Help with indefinite integration

I am learning indefinite integration, yet am having problems understanding and recognizing where to substitute what. a good trick is to attempt convert algebraic expressions into trigonometric and ...
0
votes
1answer
20 views

How to determine if a node has a positive y axis?

I am trying to determine if a node on this diagram has a positive y axis: Things I know: The blue node in the centre will always be at 0,0. second blue node will always have a positive x axis. ...
3
votes
4answers
48 views

Substituting a value of sine function in a trigonometric equation

I am trying to really understand trigonometric equations and I've stumbled upon a rather confusing example. Solve the following equation: $\sin x= 2|\sin x|+ {\sqrt{3}}\cos x$ First step is to ...
23
votes
3answers
426 views

Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

Let $\varphi=\frac{1+\sqrt5}2$ (the golden ratio). How can I simplify the following expression? $$7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$$
-4
votes
1answer
27 views

Sin theta equation [closed]

Good afternoon, I just started this new class, and am having a slight issue. I understand the $s= r \times θ$ formula to find the length of a circular arc, but I don't understand how to plug in the ...
2
votes
2answers
84 views

Prove or disprove $\frac{\sqrt{1+\tan x}}{\cot x} = \frac{1+\sin x}{\cos x}$

I have tried to prove the identity \begin{equation} \frac{\sqrt{1+\tan x}}{\cot x} = \frac{1+\sin x}{\cos x} \end{equation} by $t$-substitution but seem not to work. Please don't solve(don`t post the ...
6
votes
3answers
651 views

Two different trigonometric identities giving two different solutions

Using two different sum-difference trigonometric identities gives two different results in a task where the choice of identity seemed unimportant. The task goes as following: Given $\cos 2x ...
0
votes
3answers
50 views

What is this question asking? Geometry

Rhombus $ABCD$, with side length 6, is rolled to form a cylinder of volume 6 by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin\left(\angle ABC\right)$? Is it asking for the sin of the angle ...
0
votes
8answers
83 views

Trigonometric Property

How can I show that the following property holds? $2\cos(4a)+2\cos(2a)+1=\displaystyle\frac{\sin(5a)}{\sin(a)}$ I've been trying to derive it to no avail. What would be a way to approach similar ...
1
vote
3answers
24 views

Trigonometry on circle as function of distances to (-r,0 ) and (r,0)

I have two point A and B on a circle centered at the origin $ O = (0,0)$ with radius r And I am only told: A and B are both on the upper half plane ($ y \ge 0 $ ) the distance $a_1$ from A to the ...
1
vote
5answers
66 views

Proving $1+\cot^2(-\theta)=\csc^2(\theta)$

I'm stuck on this one proof that I just can't get for some reason. It seems really simple too, and I've tried just about everything I can think of, but I just keep going in circles. ...
3
votes
2answers
99 views

Trigonometric Limit without L'Hopital [closed]

I am having problems solving this limit without L'Hopital or series. $$ \lim_{ x\to 0 } \frac{x\cos(x) - \sin(x)}{2 x^3} $$ I tried some trigonometric manipulations without success. I tried ...
1
vote
6answers
123 views

Best way find $\lim_{x\to 0}( \frac {\sin x}{x})^{\frac 1x}$

$\lim_{x\to 0}( \frac {\sin(x)}{x})^{\frac 1x}$ $$$$ I can use Tailor to get to $\lim_{x\to 0}(1+\epsilon(x))^\frac 1x$ $$$$ $(\epsilon(x)\underset{x\to\infty}\to 0) $ $$$$ but does that mean that ...
0
votes
0answers
39 views

Simplify trigonometric equation

$$ \alpha sin\theta + \beta sin\phi + \gamma sin(\theta+\phi) = 0 $$ where $\alpha, \beta, \gamma$ are constants. I want to simplify this into a linear relation between $\theta, \phi$ I wonder if ...
46
votes
9answers
774 views

Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
-1
votes
1answer
29 views

How to find the length and height of a ladder leaning against a wall [closed]

A ladder is placed on horizontal ground with its foot $2$ metres from a vertical wall. If the ladder makes an angle of $50^\circ$ with the ground, find a. the length of the ladder b. how ...
1
vote
3answers
70 views

$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$

For $0<\theta<360$ $$2\sin(\theta + 17) = \dfrac {\cos (\theta +8)}{\cos (\theta + 17)}$$ $$\Longrightarrow \sin(2\theta + 34)= \sin (82-\theta)$$ since sine is an odd function $$2\theta + ...
-2
votes
2answers
57 views

What is the minimum value $P$ can have in $\cos(P\sin x) = \sin(P\cos x)$ [closed]

What is the minimum value $P$ can have in $\cos(P \sin x) = \sin(P \cos x)$, if there is a solution to the above equation in $x \in [0, 2\pi] $?
0
votes
4answers
53 views

Solving Trig Equation $\cos(2x)=-\sin(2x)$

Proceeding as follows: $$\cos(2x)=-\sin(2x)\Rightarrow \cos \left(2x\right)=-\cos \left(\frac{\pi }{2}-2x\right)$$ How to proceed further? Can I remove the $cos$ from both sides and proceed or no?
2
votes
2answers
45 views

Equal sign or approximation sign?

I want to solve the equation $2\enspace \sin^2\enspace \theta+2\enspace \sin\enspace \theta-1=0$, $0\leq\theta<2\pi$. Using the quadratic formula, I found one of the solution to be ...
1
vote
1answer
22 views

Chain fixed at two points, how far does it drop down?

Not too sure whether this should be in maths or physics, but oh well. If you have a metal chain of length h metres and you have 2 points, the distance between them being x metres, If h is less than ...
1
vote
1answer
32 views

Maximum value of trigonometric expression?

If $ f(x) = \cos x ( \sin x + \sqrt [2] {\sin^2 x + \sin^2 \theta} )$, where $\theta $ is a given constant, then maximum value of $f (x) $ is?The answer is in terms of $\sin \theta$ or $\cos \theta ...
1
vote
2answers
45 views

When is $a \space \sin^2(x) + b \space \cos^2(x) \le 1$?

When is the above expression less than or equal to $1$, meaning for what values of $a$ and $b$ will the above expression be less than or equal to $1$?
0
votes
2answers
25 views

Location of a point given 3 points (latitude, longitude) and their distance to the point

Let's say I got $3$ points: $(lat_1, lon_1), (lat_2, lon_2), (lat_3, lon_3)$ and their distances to the point $p$ I want to know are $d_1, d_2, d_3$. I can also know the distance between the points ...
1
vote
1answer
57 views

How do I parametrize a circle that's not centered at the origin?

If the circle were centered at the origin, of radius r, then r(cos$\theta$, sin$\theta$) traverses the circle once counterclockwise, for 0 $\le$$\theta$$\le$2$\pi$. What if the circle were centered ...
2
votes
4answers
37 views

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

$$\frac{1}{\tan (x)(1+\cos(2x))} = \csc(2x)$$ I really don't know what to do with denominator. Sure, I can use the double angle formula for cosine, and get: $$\frac{1}{\tan(x)(2 - 2\sin^2(x))} = ...
1
vote
0answers
51 views

Integration of certain real functions using Euler's Formula.

I've heard about using Euler's formula $$e^{ix}=\cos(x)+i\sin(x)$$ to transform rational functions of sine and cosine into computable indefinite integrals. However, upon attempting to apply this ...
9
votes
1answer
95 views

What are the formal terms for the intersection points of the geometric representation of the extended trigonometric functions?

Mike Pierce's answer to this question, regarding trigonometric functions beyond the common (co)sine, (co)secant, and (co)tangent, points to a figure on the Wikipedia page on trigonometric functions ...
0
votes
1answer
21 views

Show that a sinusoid having a frequency larger than one corresponds to a sinusoid having a frequency less than one.

I am studying electrical engineering for fun online. There is this one solution to a question on an online textbook that does not make any sense to me. The question is: Show that $\cos(2\pi ...
14
votes
3answers
1k views

Find the value of a function whose derivative is zero

The initial function is $$h(x)=\arcsin x + \arccos x$$ The derivative of this function is $0$ since $$h'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}\equiv0$$ This means that $h(x)$ is a ...
0
votes
0answers
36 views

Finding coordinates of position given three coordinates and distances: 3D

I'm hoping to determine the x, y, z coordinates of a 4th position (D) given the coordinates of three other positions and their distances: $A(0.25, 0.25, 0.25), B(0.4663, 0, 0.25)$, and $C (0.3912, ...
-2
votes
3answers
45 views

What is the angle between these unit vectors? [closed]

Let $a$ and $b$ be two unit vectors and $p$ is the angle between them. If $a+b$ is also a unit vector, then: $p = \pi/3$ $p = \pi/4$ $p = \pi/2$ $p = 2\pi/3$
0
votes
1answer
14 views

Prove of right regular pyramid that a perpendicular line in the middle of its height intersects its edge exactly in the middle

Prove of right regular pyramid that a perpendicular line in the middle of its height intersects its edge exactly in the middle. It seems for me to be counter intuitive that this is the case. How ...
-4
votes
0answers
25 views

zeroes of tan function

Currently in preparation for an exam on Separation of variables for PDEs. And I might have conveniently forgotten the zeroes of tan. Can I confirm that the zeroes are located at $n\pi$ where $n$ is ...
0
votes
2answers
36 views

How would you find the length of a side of a triangle where 2 sides are known and the length of a line in the middle is also known?

How would you find the length of a side of a triangle where the other 2 side lengths are known and the length of a another line that meets at the same point is known? I know there has to be an answer ...
24
votes
3answers
4k views

Do “other” trigonometric functions other than Tan Sin Cos and their derivatives exist?

I remember my physics teacher mentioning that other trigonometric functions exist apart from the Sin Cos and Tan, he mentioned a few and they did not sound familiar, nothing like Sec Csc and Cot. I ...
1
vote
1answer
40 views

Optimizing trigonometric equation

I've come across a problem from an old calculus textbook which goes like A tool shed, $250\space cm$ high and $100\space cm$ deep is build against a wall. Calculate the shortest ladder length that ...
1
vote
1answer
29 views

EM waves - orthogonality - amplitude/phase angle

A plane electromagnetic wave has the shape: $\vec{E}(\vec{r},t)=E_0\cdot cos(\vec{k}\vec{r}-\omega t)\cdot \vec{e}_y$ $\vec{B}(\vec{r},t)=(B_1\cdot cos(\vec{k}\vec{r}-\omega t)+B_2\cdot ...
0
votes
1answer
14 views

Value of sin-product only depended of argument difference

I have the product of $$Y_1 = A \sin (\omega t_1 + \phi_1)$$ and $$Y_1 = A \sin (\omega t_2 + \phi_2).$$ I know, since I plotted it with python, that the product $$X = Y_1 \cdot Y_2$$ is indepened ...
0
votes
2answers
93 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between ...
1
vote
0answers
56 views

How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
0
votes
3answers
41 views

Math trigonometry transformation

Hi, I haven't done math in a while, and stumbled upon this thing. The angle ($\arccos 7/25) is given, and i have to calculate the cosine of it's half. I've used the basic formula for cosine of an ...
2
votes
5answers
72 views

Value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$

How can we find the value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$? Note that $\sin^{-1}$ is the inverse sine function. i'm asking for the solution x for this equation Pls workout the ...
2
votes
0answers
42 views

Length between two circles intersection area?

How do I know the (smallest) length of the intersection area between two circles of different sizes? We know both circles radii and the overlapping area. ...
0
votes
1answer
23 views

Taking it a step further with a sum

So I was watching an "old" video from numberphile about the three square problem. https://youtu.be/m5evLoL0xwg Here is also an image: ...
1
vote
0answers
46 views

how to prove this nice trigonometric identity [duplicate]

I was working on a complex analysis problem in "Berkeley Problems in Mathematics", which was asking me to prove that some product is equal to $n$. I had reduced the problem to proving a trigonometric ...