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
votes
3answers
143 views

How do you integrate the following trigonometric function involving sin and cos?

How do you integrate the following functions: $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta $$ ...
4
votes
2answers
123 views

Help me solve a trigonometric equation

I am doing some work in RF circuit design. I need to solve an equation for my design: $$\frac 1{\cos(t_1)}+\frac 1{\sin(t_1)} =\frac 1{\cos(t_2)}+\frac 1{\sin(t_2)}$$ (I created a nicely typed image ...
1
vote
1answer
41 views

Trigonometry Addition Thereom With Only one exact value?

Use the expression of $\sin(A+B)$ to evaluate $\sin 195$. Do I use one exact value like $45+150$ or $60$ or is there another way?
0
votes
2answers
34 views

Trigonometry Addition Thereom

Using the expansion of a. $\sin(𝐴+𝐵)$, prove that $\sin75°=\sqrt 6+\sqrt{24}$ b. $\sin(𝐴+𝐵)$, prove that $\tan75°=2+\sqrt 3$ Where to start? draw up triangle of sin 75? find other values? help ...
1
vote
1answer
113 views

Trigonometric equality $x = 99 \sin (\pi x)$

Find the number of real solutions of $\displaystyle x = 99 \sin (\pi x)$. I am getting stuck in some trigonometric relations.
28
votes
3answers
808 views

$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

Please help me to solve this integral: $$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$ I managed to calculate an indefinite integral of the left part: $$\int\frac{\cos x}{\sin ...
2
votes
3answers
902 views

sin(x) inequality

This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$. I recognize this shouldn't be too difficult ...
3
votes
1answer
116 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
3
votes
2answers
74 views

Why do we need to find the intersection between these lines?

We have the functions $$ x = -1 + 2 \cos(t)$$ $$ y = 3 + 2 \sin(t)$$ They give P's orbit with $t$ on $\left[0, \dfrac{3}{2} \pi\right]$ Find (to 2 decimal places accurate) for which values of t ...
1
vote
3answers
792 views

Manually Finding Values of Inverse Trigonometric Functions

I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc. What is the best way to go about this? So far, I have been trying to solve the problems ...
2
votes
4answers
220 views

if $\sin24^\circ = p$ what is $\cos24^\circ$?

Let $p=\sin 24^\circ$ Then what would $\cos (24^\circ)$ be in terms of $p$? What would $\sin (168^\circ) \cdot \sin(-78^\circ)$ be in terms of $p$? I'm not sure how to approach these as we have ...
11
votes
2answers
999 views

$\cos(x)+\cos(x\sqrt{2})$ is not periodic

Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
1
vote
1answer
3k views

Drawing an arc between two points

I was writing a java program to draw an arc. Arc2D.Double(int x,int y,int width,int height,int startAngle,int arcAngle,int type); Since, I'm not familiar with the mathematics behind drawing arc, I'm ...
9
votes
2answers
631 views

A hard 'if and only if' trigonometric identity proof

Prove $$ \frac{-2+2\tan A+2\cos B\cdot\sin B+\cot^2 A\cdot({\sec^4A-\operatorname{cosec}^2A-2)}}{2+\tan^2A-2\sin^2A} =(\sin A+\cos A)^2 $$ if and only if B is the double angle of A, or ...
0
votes
1answer
139 views

Why is it necessary to use sin or cos to determine heading? (dead reckoning)

Here's the problem: (see pic for problem): https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-ash3/21281_10152793202590262_1804321932_n.jpg You have a robot that is moving forward at a variable rate ...
11
votes
2answers
227 views

Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
2
votes
2answers
94 views

How can I prove this cosine equation?

How to prove that $\cos(90)\cos(\theta)+\sin(90)\sin(\theta)=\sin(\theta)$ ?
0
votes
1answer
410 views

PreCalc problem about high tide?

$$y=3+4\cos\left(\dfrac{\pi}{5.7}(x-2)\right)$$ $x$ is the time in hours after midnight What time did the first high tide occur today? How deep was the water at the time? When will the second high ...
3
votes
1answer
75 views

Show that the following product equals 1 (involves trig)

How can I show that: $$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$ Could you please explain to me how to approach this problem? Thank you.
2
votes
3answers
379 views

Are the names and symbols for common mathematical operators the same in every language?

Do all human languages that have arithmetic use +, -, ×, ÷, and ^? How about sin, cos, tan, asin, acos, and atan?
1
vote
1answer
56 views

prove this trigonometric expression

If $$\tan \theta +\sin \theta =m $$ and $$\tan \theta -\sin \theta =n$$ then prove that $$m^2-n^2=4\sqrt{mn}$$ I've tried to $(m^2-n^2)$ as $(m-n)(m+n)$ but can't get to RHS.
3
votes
2answers
57 views

what is the value of this trigonometric expression

I want to find out value of this expression $$\cos^2 48°-\sin^2 12°$$ Just hint the starting step.Is there any any formula regarding $\cos^2 A-\sin^2 B$?
2
votes
1answer
19 views

value of this inverse trigonometric expression.

How to evaluate this expression. $$\sec^2(\tan^{-1} 2)+\csc^2(cot^{-1}(3))$$ I'm stuck on how to process squares, which is on sec and cosec function?.
10
votes
2answers
364 views

A matrix w/integer eigenvalues and trigonometric identity

Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated: Let $n$ be a natural number. (a) Consider the following Toeplitz/circulant symmetric matrix: ...
1
vote
2answers
303 views

Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?

The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions). So, ...
0
votes
1answer
123 views

Trigonometry - Addition and subtraction theorem

If $\theta$ and $\phi$ are angles between $0°$ and $90°$, and $\sin \theta=3/5$ and $\tan \phi=7/24$, find without the use of a calculator, the value of each of the following: a. $\sin(\theta−\phi)$ ...
1
vote
2answers
98 views

Derivative of Trig Functions (Intuition Help?)

Looking for some intuition help here. I have the following exercise and these are the steps I take: $$ y = \sin\left(\frac{1}{x}\right) $$ $$ u=\frac{1}{x} $$ $$ y = \sin u,\;\;\frac{dy}{du} = \cos ...
29
votes
1answer
599 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
0
votes
0answers
77 views

Get the entrance point from a straight line in a rectangle

The rectangle is like a street. The right half is to go upwards, the left half to go down. The red lines are paths of vehicles. And my goal is to give every vehicle the right lane. So when you look at ...
0
votes
1answer
138 views

How to find a point on the tangent line whos length is 1?

im trying to figure out a formula to find the point(x,y) on a tangent line whos length is between 0 and 1 while it rotates around the unit circle uniformly, so the point would either be right on the ...
1
vote
3answers
12k views

Trigonometric bearing problem

I have two trigonometric problems that I solved, however it does not match the answer in the book: 1) A yacht crosses the start line of a race on a bearing of $31$ degrees. After $4.3$ km, it ...
1
vote
2answers
1k views

How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $

How to simplify the following expression : $$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
12
votes
1answer
182 views

Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$

How am I supposed to prove this inequality? $$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$ Jordan inequality might be an option but led me ...
1
vote
2answers
2k views

Determining pendulum rise using trigonometry

Everyone in my math class (including the teacher) is having problems with this trigonometry question: I am assuming that you halve the pendulum and the bottom of the triangle would be $\frac{1.8}{2} ...
0
votes
4answers
94 views

How do i prove this trigonometric expression?

How do you prove this? $${(1-2\sin^2A)^2 \over \cos^4A-\sin^4A} = {2\cos^2A - 1}$$
3
votes
2answers
101 views

How to derive $\cos\frac{n\pi}{3}=\frac{1+3(-1)^{[\frac{n+1}{3}]}}{4}$

Consider the following formula to calculate a trigonometric function: $$\cos\frac{n\pi}{3}=\frac{1+3(-1)^{[\frac{n+1}{3}]}}{4}$$ $[x]$ denotes the integer part of $x$. The formula is valid for ...
5
votes
1answer
219 views

Find the following integral: [duplicate]

Find $$\int \sqrt{\tan x}dx$$ My attempt: $$\text{Let}\ I=\int \sqrt{\tan(x)}dx$$ $$\text{Let}\ u=\tan(x), du=(1+\tan^{2}(x))dx$$ $$I=\int \frac{\sqrt{u}}{u^{2}+1}$$ $$\text{Let}\ v=\sqrt{u}, ...
2
votes
1answer
52 views

Trigonometric Integration + Series

I am doing an integration question: $$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$ So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$ How can I do that?
3
votes
2answers
188 views

Define two rational numbers $\alpha$ and $x$ such that $\sin( { \alpha }) =x$

Of course for $x\neq 0 $ and $\alpha$ in radians. Can you define them?
0
votes
3answers
1k views

Distance between two antennas

I am trying to find out the formula to calculate how high antennas need to be for Line of Sight (LoS) propagation. I found: d = 3.57sqrt(h) also ...
7
votes
6answers
446 views

Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $

My attempt: $\int {{{1 + \sin x} \over {\cos x}}dx} $, given : $u = \sin x$ I use the general rule: $\eqalign{ & \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr ...
6
votes
2answers
80 views

Another trigonometric proof…?

...sigh..another problem how shall I prove the following? $$ {\cot A\over1- \tan A} + {\tan A \over 1- \cot A} = 1 + \tan A + \cot A$$ so what now? the following's what I've done: $$\cot A - \cot^2 A ...
1
vote
6answers
76 views

How to prove this trigonometric expression?

How would you go about proving the following? $${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$ This is what I've done so far: $$LHS = {1+\cos^2 A -2\cos A + 1 - ...
-1
votes
2answers
56 views

For a complex number $z$ applies to $\operatorname{Re} (z) = 5$.

For a complex number $z$ applies to $\operatorname{Re} (z) = 5$. What values can $\operatorname{Re} (1/z)$ assume to be?
2
votes
2answers
68 views

Prove that a sum converges to a trigonometric expression

$$2^n \cos \left (\frac{n \pi}{2} \right )=\sum_{k=0}^{n} (-1)^k \binom{2n}{2k}$$ I expanded the LHS and got $$\binom{2n}{0}-\binom{2n}{2}+\binom{2n}{4}-\binom{2n}{6}+\cdots+(-1)^{n}\binom{2n}{2n}$$ ...
1
vote
1answer
2k views

Phase Relationships of Sinsuoidal Waveforms?!

I'm reading through a chapter on sinusoidal alternating waveforms and I'm having some difficulty in the section on Phase Relations. The generic expression is mentioned below for a waveform that has ...
14
votes
3answers
273 views

Proving a trig infinite sum using integration

How can I prove the following using integration and elementary functions? Prove that: $$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$ $0 < \theta < 2\pi$
5
votes
1answer
146 views

Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
0
votes
1answer
5k views

Calculate new positon of rectangle corners based on angle.

I am trying to make a re-sizable touch view with rotation in android. I re-size rectangle successfully. You can find code here It has 4 corners. You can re-size that rectangle by dragging one of ...
8
votes
4answers
271 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...