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)

1
vote
2answers
239 views

How do I get $\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$?

How do I get: $$\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$$
0
votes
1answer
84 views

Behaviour of $\tan$

What happens to $\tan(u+iv)$ as $u^2+v^2\to \infty$ via a path where $v\neq 0$ and $u,v\in \mathbb R$? How can I tell?
10
votes
1answer
484 views

Why $\arccos(\frac{1}{3})$ is an irrational number?

I was reading the following question. It is a very nice question with a very nice answer! I would like to know why $\arccos(\frac{1}{3})$ is an irrational number.
2
votes
2answers
270 views

What does $\sin^{2k}\theta+\cos^{2k}\theta=$?

What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to? Besides Mathematical Induction,more solutions are desired.
4
votes
1answer
453 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
1
vote
1answer
680 views

Solve x = sin(t) for t

How can I solve: $(\frac{x}{16})^{\frac{1}{3}} = \sin(t)$ for t?
3
votes
3answers
547 views

unit circle, derive number for any degree, cosinus and sinus

$\sin(90°)= \sin(\frac{1}{2}\pi)= 0$ $\cos(90°)= \cos(\frac{1}{2}\pi)= 1$ $\sin(60°)= \sin(\frac{1}{3}\pi)=\frac{\sqrt{3}}{2}$ $\cos(60°)= \cos(\frac{1}{3}\pi)=\frac{1}{2} $ $\sin(45°)= ...
0
votes
2answers
456 views

Solving $\arctan(a) + \arctan(b) + \arctan(c) = \pi$ for $0 < a < b < c < 10$

This is a trigonometry math contest problem. Which ordered triple of numbers $(a,b,c)$ with $0 < a < b < c < 10$ satisfies the equation $$\arctan(a) + \arctan(b) + \arctan(c) = ...
1
vote
1answer
3k views

How do you find the value of theta in this example?

My problem says to find the measure of each acute angle $\theta$ to the nearest degree. $$\large\cos\theta = 0.2249$$
24
votes
2answers
2k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
2
votes
0answers
139 views

How can this trigonometrics equation be solved exactly, if possible?

I was working on an approximation for the sine function, in which I needed to calculate the maximum error to work on a compensation polynomial. My approximation was this: $$f(x) = \frac {4} {\pi^2} x ...
3
votes
5answers
790 views

Number of Solutions of $3\cos^2(x)+\cos(x)-2=0$

I'm trying to figure out how many solutions there are for $$3\cos^2(x)+\cos(x)-2=0.$$ I can come up with at least two solutions I believe are correct, but I'm not sure if there is a third.
2
votes
3answers
115 views

Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?

For some reason, I cannot see a clever way to solve this (I know the way of doing it like in Wolframalapha) but I am pretty sure there is a double angle identity to crack this puzzle. Could someone ...
1
vote
1answer
2k views

How to calculate relative degree changes in 0 to 360.

I'm working on a project that measures wind direction and I'm stuck on this what appears to be a simple degree problem. Example: Lets say I'm a compass (0' .. 360' ) now I'm pointing due north 0' , ...
0
votes
1answer
98 views

Basic Trig problem

This is a very basic question but that I need to know to solve a harder calculus question. How do I solve for $x$, for the problem $\tan(x) = \sqrt{3}$?
2
votes
1answer
93 views

I need to factor this function so it is entirely dependent on x- semicircle displacement.

In the attached image are three functions. The first is a displacement function which takes angle $t$, and returns a radius. The third one is a semicircle and the second one is a semicircle with the ...
0
votes
2answers
779 views

Notation of inverse trigonometric functions and exponentiation [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ I have worked a bit on trigonometry today, and something strikes me as inconsistent. In the book, the notation for the inverse sine ...
1
vote
2answers
372 views

How to find $\int_{\frac{\pi}{2}}^{\frac{\pi}{4}}\cot^5x\,\csc^3xdx$

I stack about following problem... $\int_{\frac{\pi}{2}}^{\frac{\pi}{4}}\cot^5x\,\csc^3xdx$ I tried to change $\cot^5x=\frac{\cos^5x}{\sin^5x}$ I got ...
1
vote
4answers
160 views

How can I plot the point of $y=\sin(x)$?

I am taking an online trig course so I don't have the luxury of asking for help when I don't understand something. How can I plot $y=\sin(x)$?
0
votes
2answers
319 views

How to show this equation is true?

How can I show that this equation is true for all $x \in \mathbb{R}$? $$\sin^6x + \cos^6x = 1 - 3\sin^2x \ \cos^2x$$
1
vote
1answer
211 views

Question on trigonometric linear equations

Doing an exercise on complex analysis where it began by asking me to solve some equations for $u_x$ and $u_y$ I got stuck and looked up the answer. $$u_x\cos\theta+u_y\sin\theta=u_r,$$ ...
1
vote
1answer
178 views

Expressing the four roots of a particular quartic in terms of trigonometric functions

I know one root of the equation (eq.1), $x^4+ax^3+2x^2-ax+1 = 0$ is, $x_1 = \tan\big(\tfrac{1}{4}\arcsin(\tfrac{4}{a})\big)$ How to find the other three roots of eq.1 expressed similarly in terms ...
4
votes
2answers
1k views

What are functions used for?

When I say functions, I don't mean the trigonometric functions like $sin$, $cos$, and $tan$, I mean defined functions like $\large f(x) = 2x + 4$. Why is $\large f(x)$ used and why isn't a single ...
1
vote
3answers
114 views

Simplify $\sin^3{\left(\cot^{-1}{\left(x\right)}\right)}$

How can the following function such that no trigonometric functions are present: $\sin^3{\left(\cot^{-1}{\left(x\right)}\right)}$ Wolfram|Alpha shows the result as $\frac{1}{{\sqrt{x^2+1}}^3}$. ...
5
votes
2answers
333 views

Prove this trigonometric identity in quadrilateral

If $\alpha,\beta,\gamma,\delta$ are angles in quadrilateral different from $90^\circ$, prove the following: $$ ...
2
votes
3answers
296 views

Product of tangents

I was able to reduce an equation I have to: $$f(t) = \tan(\mu) \tan(\nu) - C = 0$$ where $\mu, \nu$ are linear functions of t and $C$ is a constant. Are there any identities for the product of ...
1
vote
1answer
235 views

polar graphs and investigation

I am new to polar graphs and I am trying to investigate some certain cases: What happens when you change the $b$ value to different positive integers in polar equations of the forms: ...
19
votes
8answers
3k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
0
votes
2answers
834 views

What are these specific trigonometric functions used for?

$\cos\theta$, $\sin\theta$, $\tan\theta$, $\csc\theta$, $\cot\theta$, $\sec\theta$ What are these all used for? Are they used to find the measure of a give angle provided the measurements of other ...
1
vote
1answer
193 views

If $\cos x = \frac{1}{\sqrt{5}}$, what is $\cos^{-1} x$?

Given that $\cos x = \frac{1}{\sqrt{5}}$ and $\tan x < 0$, what is the exact value of $\cos^{-1} x$? Since $\sin x = - \frac{2}{\sqrt{5}}$, we can see that $\tan x$ is in fact $-2$. But how do we ...
6
votes
1answer
450 views

Rigorous proof of an infinite product.

I'll give a proof of the following expansion: $$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$ $${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$ $${\sin x} = 2^2 \cos \frac{x}{2}\cos ...
0
votes
1answer
1k views

Finding a result vector from 2 vectors without cross product

If I have 2 lines with its symmetric equations I can get the vectors U and V of each line, and with a cross product I can get the vector R; but how can I get the vector R without a cross product?
2
votes
1answer
185 views

Sine rule and equal angles

Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with ...
0
votes
0answers
149 views

Solve this trigonometric system $ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $

$$ \tan x+\tan y=2\sqrt3 \land \tan\frac{x}2+\tan\frac{y}2=\frac{2\sqrt3}3 $$ I need full solution please. I've tried different transformations, but couldn't get much near, I keep getting huge ...
0
votes
1answer
86 views

Elementary trigonometry: $\tan$

How can I assign $a,b,c,d$ values $\pm \tan\theta,\pm{1\over\tan\theta}$ so that ${(a-b)(c-d)\over (a-d)(b-c)}=\tan^2(2\theta)$? Thank you for helping.
-1
votes
1answer
293 views

Absolute value of $\sin z$ on square

Show that $|\sin z|\geq 1$ at all points on the square with vertices $\pm (N+1/2)\pi\pm(N+1/2)\pi i$, for any positive integer $N$.
17
votes
1answer
72k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
0
votes
1answer
467 views

Using Pythagorean Identities to Solve for Values

I'm doing homework for my trig class, and it's asking for us to use Pythagorean identities to solve for other trig values. I got through the first 10 fine, but I'm stuck on the last three. My teacher ...
18
votes
2answers
1k views

De Moivre's Theorem. Motivation and origins.

I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous $$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$ but ...
4
votes
0answers
406 views

Find roots of sum of sinusoids

Given this function and an initial point, find the next root: $$ \begin{align} f(t) & = -L\\ & {} + A \sin(\Theta_1 + \omega_1 t) \\ & {} +B \cos(\Theta_1 + \omega_1 t)\\ & {} - ...
1
vote
1answer
716 views

how to integrate $ \int_{-\infty}^{+\infty} \frac{\sin(x)}{x} \,dx $? [duplicate]

Possible Duplicate: Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? How can I do this integration using only calculus? (not laplace transforms or complex ...
16
votes
3answers
1k views

Sine Approximation of Bhaskara

An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.) $$\sin x \approx ...
0
votes
4answers
4k views

Solving a triangle given two side lengths and the measure of a non-included angle

Let's say given an angle A = 46 °, side a = 2.29 and b = 2.71 I figured that the angle B = 58.4 by saying: $$B = \sin^{-1} \left(\frac{ 2.71 \sin{46^{\circ}}}{2.29}\right)=58.4^{\circ}$$ But I ...
0
votes
1answer
278 views

Finding the direction vectors of two lines that make an angle of 60 degrees

I have this problem, I have to find the values of $a$ so the direction vectors of the lines make an angle of $60$ degrees. $$\frac{x-3}{2} = \frac{y+5}{2} = z+2$$ $$ x-1 = y-1 = ...
0
votes
2answers
3k views

Solving a triangle, given two sides and the measure of the included angle

Let say you have a triangle Angle A = 41 degrees , side b = 3.41 and c = 5.83 can you use pythagoras theorem to find the side a? and how can you find Angle B and C
2
votes
1answer
444 views

Find angles using the Law of Cosines

if you must find the Angle C based on the sides of a = 2, 3 b = 4,6 og c = 5, 9  I have used the formula: $$\cos (C) =\frac{a^2 + b^2-c^2}{2ab}$$ use, but I think i'm doing something wrong: ...
0
votes
1answer
1k views

Solving triangles and quadratic equations

When calculating the pieces in a triangle with only two sides and an intermediate angle is known, one must solve a quadratic equation. By solving the equation are 2, 1 or 0 solutions, as ...
1
vote
2answers
146 views

Trigonometry & circle math

I tried to solve this Trigonometry question, but I do not know how to solve. I read that the circle has radius 1 and center at (0.0) as the unit circle is plotted in the coordinate system. I ...
7
votes
3answers
6k 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 ...
1
vote
1answer
910 views

Trigonometry - Bearings

I have a Trigonometry problem I just can't seem to grasp: Given a coordinate, find a bearing to reach that coordinate. What I have tried thus far: I have another problem in the same section that ...