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

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3
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2answers
919 views

Showing the identity: $\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$

My knowledge of trigonometry are still insufficient to resolve this problem. Any help would be greatly appreciated. $$\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$$
3
votes
2answers
1k views

How to calculate angles and X,Y coordinates for drawing a hand of playing cards on a canvas

Scenario: I'm programming a module to draw a deck of cards on a canvas as though they were being held by a person. Edit: I've cleaned up the question as best I can to be clearer. What I'm looking ...
2
votes
1answer
201 views

Finding launch angle help.

A smooth spherical object (the first object) is projected horizontally from a vertical height of $26.83$ metres above horizontal ground with a launch speed of $23.44\textrm{ ms}^{-1}$. A second ...
0
votes
1answer
92 views

A problem with a trigonometric equation

I'm trying to solve this problem but I can't figure how. Can you help me? $$A=\frac{\sin \alpha+\cos(3\pi/2-\alpha)+\tan(5\pi+\alpha)}{\csc(2\pi-\alpha)+\sin(5\pi/2+\alpha)}$$ If $\tan \alpha=-2/3$ ...
4
votes
2answers
333 views

Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$

All the angles in a triangle $A,B,$ and $C$ are less than $120^{o}$ Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$
5
votes
1answer
533 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
1
vote
0answers
2k views

Direction ratio of a vector in 3d?

Suppose I have a vector whose starting end is the origin & the other end can be in any of the 8 quadrants (in 3d). I can easily get direction ratios for the vector & also know in which ...
1
vote
1answer
375 views

How did they get this solution?

I'm looking at the solution manual and I have no idea how they convert this. $$ k \cos {3\theta} = k [4\cos^{3} {\theta} - 3 \cos{\theta}] = k[\alpha P_{3}(\cos\theta) + \beta P_{1}\cos{\theta}] $$ ...
6
votes
1answer
533 views

Evaluate $\int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx$

How to evaluate $$ \int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx $$ I don't know how to deal with it.
0
votes
1answer
127 views

A computation with vectors

I'm going through these lecture notes, and I don't understand how one of the example problems was solved. Can anyone show me a step by step solution? Question: Suppose a Canada goose is flying ...
-1
votes
2answers
111 views

What is $\cos^2(x)$ in relation to $\sin(x)$?

How would you find $\cos^2(x)$ in terms of $\sin(x)$? Please explain clearly, I am a high school student.
0
votes
3answers
146 views

Integral of a trigonometric function [duplicate]

Possible Duplicate: Evaluating $\int P(\sin x, \cos x) \text{d}x$ How do I integrate the following function? $$\frac{\sin 2x}{(1 + \cos^2x)^2}?$$ Thanks.
2
votes
4answers
7k views

What is $\sin^2(x)$ equal to?

Let's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something ...
1
vote
1answer
70 views

Computing angle

See the drawing for the situation. Given lenghts a, b and c and also L, but k and angle alpha are unknown. How to compute this angle alpha? I know it is possible to compute if we first compute k in ...
3
votes
1answer
645 views

Distance between two gears surrounded by a known-length belt

This question is very similar (but not identical) to this one: Finding the distance between two gears (actually, we are trying to solve it on Bicycle Exchange: ...
0
votes
2answers
248 views

Find the length (in cm.) of the hypotenuse?

A right angled triangle has sides of length X, Y and Z (all lengths in cm.). It is known that Z is the length of the longest side. The lengths of the other two sides satisfy the inequality ...
1
vote
1answer
3k views

Reciprocal of sin(x)

What is the reciprocal of sin(x), or what is 1/sin(x) equal to in terms of trigonometric ratios? Please answer simply, as I am a high school student, not a mathematician.
0
votes
1answer
2k views

Finding the position of a point after rotation: Why is my result incorrect

I am attempting to calculate the position of a point after it has been rotated I have been using an algorithm but I am getting incorrect values which makes me think I am using the incorrect algorithm ...
-1
votes
1answer
522 views

Where is zero degrees on a graph

I am using the following formula to calculate the position of a point after rotation in my web application. x' = xcos(0) - ysin(0) y' = xsin(0) + ycos(0) But ...
2
votes
2answers
711 views

Calculated rotated point coordinate: is my solution correct

Is my calculation correct for this rotation around a point? A point a(-19.94,392.11) is rotated -49.45 degrees, what is the new coordinates of point a? My solution: ...
9
votes
2answers
527 views

Sum of the reciprocal of sine squared

I encountered an interesting identity when doing physics homework, that is, $$ \sum_{n=1}^{N-1} \frac{1}{\sin^2 \dfrac{\pi n}{N} } = \frac{N^2-1}{3}. $$ How is this identity derived? Are there any ...
1
vote
2answers
433 views

Given an angle, get the trigonometric circle point.

Given an angle, in degrees, how can I get the trigonometric circle point coordinates for it? For instance, given the angle 0, I would get (1,0). 90 would be (0,-1). Clockwise.
2
votes
1answer
8k views

Getting the angles of a non-right triangle when all lengths are known

I have a triangle that I know the lengths of all the sides. But I need to know the angles. Needs to work with non-right triangles as well. I know it is possible, and I could have easily done this ...
3
votes
1answer
209 views

Logic proportions problem

What the problem says: When a screen is placed 3 m from a projector, the picture occupies 3 m^2. How large will the picture be when the projector is 5 m from the screen? My direct answer ...
2
votes
2answers
131 views

Did I write the right “expressions”?

$9$. Consider the parametric curve $K\subset R^3$ given by $$x = (2 + \cos(2s)) \cos(3s)$$ $$y = (2 + \cos(2s)) \sin(3s)$$ $$z = \sin(2s)$$ a) Express the equations of K as polynomial ...
1
vote
2answers
167 views

Express some equations as polynomial equations

Given $$\begin{align*} x&=(2+\cos(2s))\cos(3s)\\ y&=(2+\cos(2s))\sin(3s)\\ z&=\sin(2s),\end{align*}$$ I was wondering how to express these equations as polynomial equations in $x$, ...
1
vote
3answers
386 views

Constructing a line with a known line, intersection point and angle.

I am creating a Java game with collisions. I found myself stuck on the following problem. I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, ...
5
votes
2answers
3k views

Ease-in-out function

I am trying to create a nice ease-in-out function that given values from 0 - 1 produces an output of 0 - 1 which accelerates slowly up to full speed then slows down again as it nears 1. I currently ...
0
votes
2answers
214 views

Convergence of $\sum_{n=1}^{\infty} \sin^{-1}{\frac{1}{n}}$

I am trying to find the convergence of $$\sum_{n=1}^{\infty} \sin^{-1}{\frac{1}{n}}$$ I tried divergence test but the $\lim = 0$, which is inconclusive? I believe the ratio/root test won't work ...
1
vote
3answers
6k views

How do I Find all Angles of 4-sided polygon given side lengths?

I have a program that lets users draw custom 4-sided shapes using java 2d. I want to calculate the angles inside the shapes so I can rotate text to the proper angle and label each side. I am trying ...
4
votes
1answer
595 views

What's wrong with this proof that $e^{i\theta} = e^{-i\theta}$?

I recently learned that $\cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2}$ Based on this, I managed to "prove" that: $$e^{i\theta} = ...
4
votes
2answers
1k views

If z is one of the fifth roots of unity, not 1…

If z is one of the fifth roots of unity, not 1, show that: $1+z+z^2+z^3+z^4=0$ Which wasn't too bad, but the next part is killing me: show that: $z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$ Can ...
1
vote
1answer
98 views

Help with trigonometry?

My question says to find the range of the following functions: $$y = \csc x$$ $$y = \sec x$$ $$y = \cot x$$ Only I don't know what they mean by range and how to find the answer.
2
votes
2answers
415 views

Trigonometry Min/Max Problem

$f(x) = 2\sin x \hspace{10pt}(0 \leq x \leq \pi)$ $g(x) = -\sin x \hspace{10pt}(0 \leq x \leq \pi)$ Rectangle ABCD is enclosed between the above functions' graphs (its edges are parallel to the ...
4
votes
5answers
712 views

How to prove a trigonometric identity

Show that $$ \tan(A)=\frac{\sin2A}{1+\cos 2A} $$ I've tried a few methods, and it stumped my teacher.
0
votes
1answer
74 views

Find a rotation where the shape has the least width possible on the x-axis

I am toying around with a shape problem and I am looking for a more clever solution than what I have been able to come up with. Here is the problem: I have a set of points that form an enclosed ...
3
votes
2answers
4k views

Equivalent sine / cosine functions?

I was given the following problem on a quiz: I put A, C, and D. The answer was A and D. We were taught four relevant equations: $\sin(x)=-\sin(-x)$ $\cos(x)=\cos(-x)$ ...
2
votes
1answer
68 views

Need help understanding $\frac {dx}{\cos^2(\frac{x}{2})} = 2d(\operatorname{tg}(\frac{x}{2})) $

I have found this statement somewhere, however, I dont really understand it. Could someone explain me where does $2$ before $\operatorname{tg}(x/2)$ come from? $$\frac {dx}{\cos^2(\frac{x}{2})} = ...
6
votes
3answers
252 views

How can I find the integral of this function using trig substitution?

$$\int_0^1x^3\sqrt{1 - x^2}dx$$ I need to find the integral of this function using trigonometric substitution. Using triangles, I found that $x = \sin\theta$, and $dx = \cos\theta d\theta$; so I ...
1
vote
3answers
96 views

Help Needed To Prove A Trigonometric Identity

I'am trying to prove the below identity, but this is what i end up getting. $$\begin{align*} \frac{2\tan(x)}{1 + \tan^2(x)} &= \sin(2x)\\ &= \frac{2\frac{\sin(x)}{\cos(x)}}{1 + ...
2
votes
2answers
707 views

How do I find the discontinuity of the function $f(x) =\cos (x/(x - \pi))$?

I need help in finding the discontinuity of the function: $$f(x) = \cos \left(\frac{x}{x - \pi}\right)$$ Any comments or advice will be much appreciated. Thanks.
2
votes
1answer
162 views

Equivalent to $\begin{align}\int \cos{\left(2x\right)} \ dx\end{align}$?

For some reason, I am lost on one part of this integration problem: $\begin{align}\int \cos{\left(2x\right)} \ dx\end{align}$ $u = 2x$ $du = 2 \ dx$ $\frac{1}{2} du = dx$ ...
3
votes
2answers
222 views

Are there exact expressions for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$?

I was just wondering if there is any way to get an exact expression (with radicals) for $\sin \frac{3\pi}{8}$ and $\cos \frac{3\pi}{8}$. In case it's relevant, I want to express $z = \sqrt[4]{8} ...
2
votes
2answers
168 views

Trigonometric identity, possible error

I need to prove the following trigonometric identity: $$ \frac{\sin^2(\frac{5\pi}{6} - \alpha )}{\cos^2(\alpha - 4\pi)} - \cot^2(\alpha - 11\pi)\sin^2(-\alpha - \frac{13\pi}{2}) =\sin^2(\alpha)$$ I ...
1
vote
3answers
186 views

Integration: How to Begin? [duplicate]

Possible Duplicate: Help evaluating $\int \frac{dx}{(x^2 + a^2)^2}$ How to I begin this integration problem? $\begin{align}\int_{0}^{1} \frac{dx}{{\left(x^2 + 1\right)}^{2}}\end{align}$ ...
6
votes
3answers
335 views

Why Doesn't This Integral $\int \frac{\sqrt{x^2 - 9}}{x^2} \ dx$ Work?

I am trying to solve this integral, which is incorrect compared to Wolfram|Alpha. Why doesn't my method work? Find $\int \frac{\sqrt{x^2 - 9}}{x^2} \ dx$ Side work: ...
1
vote
2answers
119 views

Proving that $\sin^yt-\cos^yt=1$ doesn't have solutions such that $y$ is odd and $\tan(t/2)$ is a positive integer.

How can I prove (if it's correct) the following: $$\sin^yt-\cos^yt=1$$ doesn't have solutions such that $y$ is odd and $\tan(t/2)$ is a positive integer. This is just a small part of a much bigger ...
3
votes
1answer
203 views

An unusual symmetric inequality of trigonometric functions

Given $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2 $. I have to prove that $ \left| \begin{matrix} \cos\alpha & \cos\beta & \sin\gamma\\\sin\alpha & \cos\beta & \cos\gamma\\\cos\alpha ...
0
votes
0answers
88 views

Approximating a function with a sine function: transform into constant amplitude?

I have a smooth function, it is stationary. So I tried approximating my function with regression by fitting a sine function that changes period, phase & frequency every observation to get the ...
1
vote
2answers
170 views

Simple Trig inverse problem

Suppose we have $x=\sin a \cos b, y=\sin a\sin b, z=\cos a$. I want to invert them to get $a,b$ in terms of $x,y,z$. At first sight it appears very simple, for example my first reaction would be to ...