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

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2
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3answers
23 views

Equation of a tangent to the graph of a function parallel to a line

Please help me find the answer to this question. Thanks. What is the equation of a tangent to the graph of a function $y=x-\frac{1}{x^2}$ which is parallel to the line $y=3x$?
0
votes
2answers
28 views

Nice little arcsin problem

$\arcsin(\sqrt3\sin x)=1$ I've tried substituting various x's in, but not exactly sure what it means to find x fitting tovthe arcsin.
5
votes
1answer
39 views

Exact value of $\frac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$

Let $\alpha\in\left(0,\dfrac\pi2\right)$. What is the exact value of $$\dfrac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$$ Firstly, I tried to simplify $1-2\tan^2\alpha$ and got ...
0
votes
2answers
62 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
1
vote
3answers
47 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
5
votes
2answers
69 views

Trigonometric equation, find $\sin \theta $

Find $\sin \theta $ if $a$ and $c$ are constants $$ 1-\left(c-a\tan\theta\right)^2=\frac{\sin^2\theta\cos^4\theta }{a^2-\cos^4\theta } $$
4
votes
3answers
112 views

Angle in a triangle within a circle.

A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC $\perp OB$. AC = 12 cm. BC = 5 cm. Calculate the size of $\angle AOB$, marked $\theta$ on the ...
3
votes
2answers
27 views

$x$-intercepts of secant function

I have tried setting $f(x) = 0$ and solving for $x$ by undoing the operations, and what I end up with is $x= -\pi/6$. The book gives the answer as B, however, and I haven't been able to obtain those ...
0
votes
2answers
30 views

The exact value of csc -420 degrees (Find the exact value of each trigonometric funtion)

I'am very confused, I have looked all over google and I can not find out how too do this problem. I have the answer its number 14 since our teacher gives us the answer but we need to show work. I ...
2
votes
2answers
50 views

Determinant of a 4x4 matrix with trigonometric functions

I am stuck with my homework from math. I should calcutate the determinant of a matrix: $$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ...
0
votes
3answers
97 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
0
votes
1answer
15 views

Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...
0
votes
0answers
8 views

How can I make this tangent function only appear once (or be spaced very widely)?

I only want the function to go from $x=5$ to whenever the function is 4.5 (in other words, when $y=4.5$). Is there any way to do this without specifying the domain? It has to have the shape of the ...
2
votes
2answers
76 views

Computing $ \lim_{x \to 0} \left( \frac {1}{x} - \frac {1}{\sin x } \right) $

How to calculate this limit: $$\lim_{x\rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$$ All I know is: $$\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$$ $$\lim_{x\rightarrow 0} \, x = 0$$ ...
3
votes
1answer
64 views

What does “versin” mean?

$$\newcommand{\versin}{\operatorname{versin}}2\versin A+\cos ^2 A= 1+\versin ^2 A$$ I don't understand the word 'ver' in this equation. What does it mean?
0
votes
1answer
14 views

How to find initial direction and angle of collision of a ball with a vertical wall?

I have a problem in my game. I have a wall where a ball hit to a wall from anywhere. I need to give it to the direction according to the collision law. Let suppose if a ball thrown from $(0, 0)$ and ...
1
vote
1answer
51 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
0
votes
1answer
28 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
1
vote
3answers
68 views

Find $ \int \frac {\mathrm{d}x}{(4x^2-1)^{3/2}}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
2
votes
2answers
49 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
5
votes
3answers
98 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
0
votes
4answers
38 views

Equivalence of equations

$ \sin ^2 \alpha = \frac{\tan ^2 \alpha}{1+\tan^2 \alpha} $ $ 1+\tan^2 \alpha = \frac{\tan ^2 \alpha}{\sin ^2 \alpha} $ It is said that these two equations are equivalent. How can that be? I know ...
5
votes
6answers
105 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
0
votes
4answers
51 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
1
vote
0answers
27 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
0
votes
1answer
20 views

The vertical projection of a chord of a circle?

I was wondering if anyone could help me with the problem below (finding x): So we are given t_i (the initial tangent angle to the circle), t_o (the exiting angle of the tangent of the circle), the ...
3
votes
1answer
19 views

mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
1
vote
3answers
45 views

What is the required radius of the smaller circles around a larger circle so they touch?

I am trying to determine how to calculate the required radius of the smaller circles so they touch each other around the larger circle. (red box) I would like to be able to adjust the number of ...
3
votes
3answers
49 views

Using trig substitution to solve for integration?

So I used a trig sub for this problem: $$\int \frac{1}{x^2\sqrt{9-x^2}}dx.$$ ${x=3\sin\theta}$ ${dx=3\cos\theta\ d\theta}$ ${\sqrt{9-x^2}= 3\cos\theta}$ I ended up with $$\frac19 \int \frac{ ...
1
vote
2answers
70 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
3
votes
1answer
43 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
0
votes
1answer
32 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
2
votes
0answers
24 views

Find the fundamental period

How do I find the fundamental period of this function? $$y = \sin x + \cos(1,01x)$$ I know that the fundamental period of $\sin x$ is $2\pi$ and the fundamental period of $cos(1,01x)$ is ...
9
votes
0answers
63 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
0
votes
2answers
50 views

How to solve $\sin(\arctan((\frac{1}{2}))$ [on hold]

Can you solve $\sin(\arctan((\frac{1}{2}))$? It says I have to use a right triangle
4
votes
4answers
56 views

Evaluate $\int\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm dx$

I have tried to evaluate $$∫\frac{\sin(8x)}{9+\sin^4(4x)}\,\mathrm d x$$ using the following identity: $$\frac{d(\sin^{-1}{u})}{du} = \frac{du}{1+u^2}$$ So I then reformed the integral to this: ...
3
votes
4answers
104 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
3
votes
2answers
99 views

Simple Equation Does my proof work?

Its the inequality equation $|a+b| \leq |a|+|b | $ I managed this by cases. Let $c = a$ and $d=b$ if $a>b $ let $c = b$ and $d = a$ if $b>a $ if $a=b$ let $a=c$ Hence we have $|c+d| \leq ...
7
votes
0answers
87 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-2mx+N = ...
-2
votes
2answers
94 views

Prove that $\lim_{x\to\frac{2}{\pi}}\big\lfloor\sin\frac{1}{x}\big\rfloor=0$ [on hold]

Prove that $$\lim_{\large x\to \frac{2}{\pi}} \left\lfloor\sin\left(\frac{1}{x}\right)\right\rfloor=0$$ using the $\varepsilon$-$\delta$ definition of limits. Note that $\lfloor 0.1\rfloor = 0,\; ...
1
vote
2answers
52 views

How to find $\theta$ at which $d$ is the maximum possible?

I have an equation: $$d=\dfrac{v\cos \theta}{g}\left(v \sin \theta + \sqrt{v^{2} \sin^{2}\theta + 2gh} \right),\ g≈9.81 \dfrac {m}{s^{2}}$$ How to find $\theta$ at which $d$ is the maximum possible? ...
0
votes
1answer
24 views

Isosceles has maximum vertex angle between triangles of equal area

I'm trying to prove the following that in the image below (E1 & E2 are parallel, AB=AC) no matter where I move the vertex point A on line E1 (keeping BC as is), the vertex angle A is going to ...
2
votes
2answers
58 views

Evaluating inverse of trigonometric function

I have this function, $$\sin\left[{\arctan\left({\frac{x}{\sqrt{1-x^2}}}\right)}\right]$$ I drew a right angled triangle putting $x$ on the opposite side and the square root on the adjacent which ...
1
vote
0answers
26 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
0
votes
2answers
39 views

Sine on a Circle

I'm walking a quarter mile circular walking track. The width of the track is 8 feet across. If I walk from one side of the track to the other, walking a sine wave that has a 20 foot period, how much ...
0
votes
0answers
31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
7
votes
0answers
89 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
2
votes
2answers
57 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
0
votes
0answers
20 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
2
votes
4answers
185 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...