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
6 views

Peripendicular Line at distance d from point in a given direction

I have a line given by $Ax + By + C= 0$, and a point $x_0,y_0$. From that point $x_0,y_0$ in the direction of the line up to distance $d$, I want to find the equation of the line that is perpendicular ...
3
votes
1answer
32 views

Relationship between $\sin(a+b)$ and derivative product rule?

I noticed this interesting correlation between the sine angle addition formula and the derivative product rule. The sine addition formula is $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$ The ...
-2
votes
2answers
37 views

How do I find the solution(s) to $\cos^2x=2-2\sin x$?

In the interval of $[0,2\pi)$, how do I find the solutions to $$\cos^2x=2-2\sin x$$
0
votes
4answers
26 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
11
votes
3answers
134 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
22
votes
3answers
4k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
0
votes
0answers
18 views

How to calculate the radius of a circle which must have a number of nodes at its ends

Hi I am trying to create a text wheel very similar to this. here's my image The large ring is made up of circular nodes each 80 units in diameter. How can I calculate the radius of the large circle ...
-3
votes
0answers
18 views

Trigonometric problem with two angles [on hold]

$$(a+b) \tan( \theta -\phi) = (a-b) \tan( \theta +\phi)$$ and $$a \cos \phi + b \cos 2\theta = c$$ Prove $a^2 -b^2 +c^2 = 2ac \cos 2\phi$
1
vote
3answers
42 views

Find the number of solutions of the trigonometric equation in $(0,\pi)$

Find the number of solutions of the equation $$\sec x+\csc x=\sqrt {15}$$ in $(0,\pi)$. The question is easy. But when you solve, you get would get $4$ as the answer. I am sure the method gives $4$ as ...
0
votes
4answers
57 views

trigonometry expression simplification with inverse cosine

While working on a problem, I ended up with this expression for y: $$ y=x\sin\left(\arccos\left(\frac{\sqrt{x^2-y^2}}x\right)\right) $$ Is there any way to express $y$ in terms of $x$ only, with no ...
4
votes
2answers
83 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
2
votes
2answers
117 views

$\frac{AB}{A'B'}+\frac{BC}{B'C'}+\frac{CA}{C'A'} \geq 4 \left(\sin{\frac{A}{2}}+\sin{\frac{B}{2}}+\sin{\frac{C}{2}}\right). $

Let be a circle inscribed in the triangle $\triangle ABC$ wiht the center $I$. The intersection of the circle with $AI$ is $A'$, with $BI$ is $B'$ and with $CI$ is $C'$. Prove that: ...
0
votes
3answers
36 views

differential equation with substituion

Solve for y: $y'tan(x+y)=1-tan(x+y)$ so far I have made the substituion $u=x+y$, which yields $\frac{du}{dx}=1+\frac{dy}{dx}$. However, I am not sure what to do from here.
1
vote
1answer
24 views

Find the measurement of line BD

So I was trying to find the measurement of $BD$ I drew green lines to make myself some angles, the measurement $3$ is from the point A to C, If only I can line $AE$ or $CE$ then I will just use the ...
6
votes
3answers
108 views

Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
1
vote
3answers
222 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
2
votes
2answers
42 views

How do I properly read a clinometer?

If the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?
-1
votes
4answers
73 views

If $\cos x = \frac{3}{7}$, then $\sin\frac{x}{2} = ?$ [on hold]

If $\cos x=\frac{3}{7}$, then find $\sin\frac{x}{2}$. I tried everything, but it seems I'm stuck forever in this problem.
2
votes
3answers
129 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
-3
votes
2answers
32 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
0
votes
0answers
17 views

Figure out the component of a value in X and Y coordinates using trigonometry.

Alright. It's been long that I studied trigonometry and did Laws of Motion and Free Body Diagrams, and I was decent good at them, but somehow I am having trouble in understanding the following. Note ...
0
votes
0answers
43 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
1
vote
1answer
45 views

Compute angle from vertical at which a sphere strikes the lip of a cup

I'm working on a problem, wherein a sphere of known radius is dropped vertically and strikes the edge of a cup. I need to figure out the angle of deflection, which will be a function of where along ...
1
vote
2answers
33 views

Prove that $\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$

I'm trying to prove by induction that $$\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$$ provided that $$\sum_{k=1}^{n} \theta_k < \frac{\pi}{2}$$ So in ...
6
votes
3answers
177 views

Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$

Use the mean value theorem to prove that if $0 \leq x \leq 1$, then $$x\sqrt{1-x^2} \leq \sin x \leq x$$ The theorem guarantees the existence of a point, but not an inequality, so I don't know how to ...
8
votes
2answers
120 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
6
votes
0answers
37 views

Why do people prefer cosine to sine when speaking of harmonic oscillation?

In almost all of the physics textbooks I have ever read, the author will write the oscillating function as $$x(t)=\cos\left(\omega t+\phi\right)$$ My question is that, is there any practical or ...
2
votes
1answer
313 views

dividing an offset circle into triangles

First of all - I am sorry if it is the wrong forum or if this is a very trivial question. I am not a mathematician nor a trigonometry genius - and therefor I would ask a simple answer that someone ...
2
votes
1answer
37 views

Help with Definite integral question

Anyone please help with this question: (a) Show that: \begin{align} \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \end{align} (b) Hence show that: \begin{align} \int_{0}^{\frac{\pi}{4}} ...
0
votes
1answer
68 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{\sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2\sin(t)\cos(t)} \implies \lim_{t \to 0} \frac{2t}{\sin(2t)}$$ ...
-3
votes
2answers
52 views

general solution to trigonometric equation, help!!! [on hold]

if $$\sin\left(\frac {π}{4} \cot\theta\right)=\cos\left(\fracπ4\tan\theta\right)$$ then find general solution of $\theta$
0
votes
0answers
15 views

Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
2
votes
1answer
358 views

Relative side lengths of dual dodecahedron and icosahedron

If the side length of a dodecahedron is $1$, then what is the side length of its dual icosahedron whose vertices occupy the same space as the mid-points of the faces of the dodecahedron. I've read ...
-1
votes
2answers
63 views

Six variables. System of equations.

$$ \begin{align} x & =\frac{R+\frac{G+B}{-2}}{R+G+B} \\[10pt] y & =\frac{\frac{(G-B) \sqrt{3}}{2}}{R+G+B} \\[10pt] z & =R+G+B \end{align} $$ How do I get the formula for ...
1
vote
1answer
29 views

How high above sea level do your eyes have to be to see a point that is 4.1 miles away “as the crow flies”?

There's a fireworks show going on tonight at a little town that's 4.1 miles away from my house, and I want to watch it from a hill near my house. So I thought I'd set up a simple geometry problem to ...
-3
votes
2answers
70 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$ [on hold]

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$\cos^2 x + \sin^2 x=1$$ Express the number $2$, ...
0
votes
2answers
18 views

Problems identifying harmonic motion

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that ...
1
vote
1answer
29 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
2
votes
3answers
65 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
-2
votes
2answers
79 views

How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?

Is there someone show me how I evaluate this integral:$$\int\frac{\mathrm{d}x}{1+\sin x−\cos x} $$ I used $t=\tan\frac{x}{2}$ but i didn't succeed . Thank you for any help .
1
vote
6answers
57 views

Does the equation $2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$ have real solution?

Do the equation $$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$$ have any real solutions? Please help. This is an IITJEE question. Here $x$ is an acute angle. I cannot even start to attempt ...
1
vote
2answers
865 views

Calculate Triangle Ground using Height and Top Angle

Is it possible to calculate the ground of a triangle only using the height and top angle. Click here to see a poorly draw sketch of what I'm trying to calculate. So is it possible and how, to ...
0
votes
0answers
21 views

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...
3
votes
0answers
37 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
8
votes
3answers
7k views

Find the slope of a line given a point and an angle

I'm trying to figure out this problem and feel like it's something that must be so simple that I could've done in high school no problem, but for some reason my brain is frozen this morning. I would ...
0
votes
4answers
138 views
+100

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
-3
votes
0answers
33 views

Questions of multiple angles [on hold]

$2\sin A/\cos3 A+2\sin3A/\cos9A+2\sin9A/\cos27A= \tan27A-tanA $
10
votes
4answers
462 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
3
votes
4answers
147 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
12
votes
1answer
437 views

What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ ...