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
325 views

I've seen “hyperbolic rotation” - from this: generalization to multisection rotation: is this possible?

This question is more in recreational mathematics area By accident I came across the concept of "hyperbolic rotation" where we use a matrix containing $\cosh$ and $\sinh$ instead of the ...
1
vote
1answer
75 views

Express distance between two points in given proximity of other points

I have two movable receivers, both receivers get straight line signals from fixed transmitters, where the strength of the signal is given in DBm. Assuming the signal strength as a measure for ...
0
votes
2answers
39 views

Convert output of atan to range -1, 1,-1

I'm trying to convert the output of an arctan function from a range of -PI,0,PI to -1,1,-1 - in fact I've succeeded! But it's not very elegant: ...
1
vote
1answer
102 views

Fitting data with Cosine

So I'm trying to wrap my head about Graphing Cos and Sin but I have several questions about graphing. I know that the Formula is $$Y= A \sin (t \pm h) + K$$ where $A$ is Amplitude , $t$ is ...
1
vote
1answer
38 views

Deriving and applying the cosine law

A brake lever for a crane is 0.9 meters long and hinged at its base. To fully activate brake, the top end of the lever must move 0.4m horizontally. Through what angle must the lever rotate?
0
votes
1answer
45 views

Scalar product in vector/coordinate form

As I know, $a*b = |a|*|b|*cos(a,b)$ in vector form And $a*b = (a_1,a_2)*(b_1,b_2) = a_1*b_1+a_2*b_2$. 1) $$a*b=?$$ $$a=2i-3j+5k$$ $$b=i+2j+8k$$ SOLUTION: $a*b = (2,-3,5)*(1,2,8) = ...
1
vote
2answers
126 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
1
vote
0answers
42 views

Trigonometry: Isosceles Triangle [duplicate]

I saw the following problem on Facebook (figure not drawn to scale): ...
3
votes
1answer
101 views

Solve an inequality $\sin t > \sin \left(t+\frac{\pi}{3}\right)$

I need to solve an inequality. $$\sin t > \sin \left(t+\frac{\pi}{3}\right)$$ I simplified the inequality to: $$\sin t > \sqrt3\ \cos t$$ I'm not sure what to do next. I know I can't square ...
23
votes
4answers
703 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
6
votes
4answers
8k views

Determine third point of triangle when two points and all sides are known?

Determine third point of triangle (on a 2D plane) when two points and all sides are known? A = (0,0) B = (5,0) C = (?, ?) AB = 5 BC = 4 AC = 3 Can someone ...
14
votes
3answers
458 views

How prove this inequality $\sin{\left(\dfrac{\pi}{2}ab\right)}\le\sin{\left(\dfrac{\pi}{2}a\right )}\sin{\left(\dfrac{\pi}{2}b\right)}$

let $$0\le a\le 1,0\le b\le 1$$ prove or disprove $$\sin{\left(\dfrac{\pi}{2}ab\right)}\le\sin{\left(\dfrac{\pi}{2}a\right )}\sin{\left(\dfrac{\pi}{2}b\right)}$$ My try: since ...
7
votes
5answers
500 views

$\cos x+\cos 3x+\cos 5x+\cos 7x=0$, Any quick methods?

How to solve the following equation by a quick method? \begin{eqnarray} \\\cos x+\cos 3x+\cos 5x+\cos 7x=0\\ \end{eqnarray} If I normally solve the equation, it takes so long time for me. I ...
3
votes
4answers
25k views

How to find the equation of a line tangent a circle and a given point outside of the circle

I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outside ...
3
votes
2answers
101 views

Prove that for $\cos (\alpha ) = \frac{1}{3}$, $\alpha < \frac{\pi}{2} - \frac{1}{3}$

I have the following question in a mock exam: $\beta = \frac{\pi}{2} - \alpha$, show that $\beta > \frac{1}{3}$ From the earlier part of the question we know that $\alpha$ is an angle between two ...
2
votes
1answer
39 views

Why do I get 251 square dm the correct answer is 252 square dm? (Error by a fraction)

I am sorry if I am bothering you folks, I've recently started to play with Trigonometry, it's really cool, but trying to understand what mistakes I am making, anyhow, I am guessing that I shall use ...
3
votes
2answers
525 views

$\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)$, Any quick methods?

How to prove the following equation by a quick method? \begin{eqnarray} \\\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)\\ \end{eqnarray} If I use so much time to expand it and take extra care ...
0
votes
2answers
166 views

Prove tangent of angle in scalene triangle inscribed in right triangle

I'm given the following: EDIT: The actual problem was "Show that $\tan(\alpha+45^\circ) = 1 + \frac{120\sqrt{2}}{x})$", which I had already proved: From this I can see that $\tan{(\alpha + ...
3
votes
1answer
231 views

Finding modulus and argument of z³ - 4√3 + 4i = 0

I think I am messing up somewhere as the principle argument should be a nice number from the standard triangles such as $\fracπ4$, $\fracπ3$ or $\fracπ6$ or something close. (That's what we have ...
3
votes
2answers
58 views

trigonometric equations calculations and their implementation in practise.

$$\sin (11x)\sin (x) = \cos (10x) $$ How can one solve this ? The formulas for summation or doubling dont seem to be of much help here.
0
votes
1answer
99 views

How to find angle $v$ in a rectangular diagonal

I am unable to find a way to find the angle $v$ (in degrees) in a rectangular diagonal. Here's what I have: Opposite: $15.1$ m Hypotenuse: $23.5$ m Adjacent: $x$ $v$ = measure (in degrees) of the ...
1
vote
1answer
68 views

Simplifying trigonometric and cyclometric functions

could anyone show me the steps, to reducing this expression: $ \cos(\cot^{-1}{x}) $ so it doesn't contain any trigonometric or cyclometric (inverse trigonometric) functions? Thanks
5
votes
2answers
74 views

How prove $A=B=C$?

in $\Delta ABC$, such $$\sin{A}+\cos{B}+\tan{C}=\dfrac{3\sqrt{3}+1}{2}$$ prove that $$A=B=C=\dfrac{\pi}{3}$$ My try: use $$\sin{x}+\sin{y}=2\sin{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$$ ...
2
votes
3answers
159 views

Simple circle geometry/ similarity question

How would you prove that a=b? Would i be possible to solve this using similarity or trigonometry? Thankyou in advance for any help. Any theorems or links would be appreciated
1
vote
2answers
44 views

Derivative of trigonometric function

How i can find the derivative of this trigonometric function $csc^4(8x^4-5)$ i tried to do it my self and i got to this $ 4[csc(8x^4-5)]^3 * [-csc(8x^4-5)cotan(8x^4-5)] $ The answer in the book ...
1
vote
1answer
90 views

What is the standard notation for $\arcsin$

I found a lecture notes that claims the following. Is this standard? The notation $\overline{\text{arc}}\text{ sin }x$ is the inverse function of $\sin x$ restricted to $\left [ ...
1
vote
2answers
270 views

If a line makes angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes, then $\sin^2{\alpha} + \sin^2{\beta} + \sin^2{\gamma} = 2 $

The following is the question in my textbook:- If a straight line makes angle $\alpha$, $\beta$, $\gamma$ with the $x, y, z$ axes respectively, then show that $\sin^2{\alpha} + \sin^2{\beta} + ...
4
votes
5answers
206 views

Trigonometry confusion

I was doing a bit of trigonometry, as I have been for a couple of years and it suddenly dawned on me that I don't really understand the trigonometric functions, at all. You first learn the basic trig ...
3
votes
6answers
344 views

How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$

Prove the following equation. \begin{eqnarray} \\\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1\\ \end{eqnarray} I can't prove it by many methods I use. Please give me some hints. Thank you ...
5
votes
1answer
89 views

$\frac{\cos x}{1}+\frac{\cos(2x)}{2}+\cdots+\frac{\cos (nx)}{n}\gt -1$ is true for $n\in\mathbb N, 0\lt x\lt \pi$?

Let $n$ be a natural number and let $0\lt x\lt{\pi}$. Then, here are my questions. Question 1 : Is the following true? $$\sum_{k=1}^{n}\frac{\cos(kx)}{k}\gt -1$$ Question 2 : Is the ...
1
vote
2answers
140 views

What is $\int \frac1{1+(a\tan x)^2}dx$?

What is $\int \frac1{1+(a\tan x)^2} \mathrm dx$? This is a difficult integral. If you can, please give a step-by-step solution - I would be delighted.
1
vote
1answer
61 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
0
votes
1answer
22 views

Solving for x - Trig

Someone mind helping on this? I think have done the question correct but the system isnt accepting my answer.
1
vote
1answer
99 views

Problem involving trigonometry and cubics

One of my teachers proposed me the following problem: $$\text{If } (3\sec x+\csc x)\sin x=5\cos^2 x\text{, calculate } z=\tan x+\sec x$$ I started by manipulating $$3\tan x +1=5\cos^2 x$$ $$\sec^2 ...
1
vote
1answer
307 views

Drawing a fitted wave sine between any two points in 2d

I'm trying to draw a sine wave with specified start point, end point and length. To do so I have already done finding Amplitude and Period for sine function. The problem is that sine wave is not ...
3
votes
6answers
320 views

Does $\cos(x+y)=\cos x + \cos y$?

Find the value using a calculator: $\cos 75°$ At first I thought all I need is to separate the simpler known values like this: $\cos 75^\circ = \cos 30°+\cos45° = {\sqrt3}/{2} + {\sqrt2}/{2} $ $= ...
0
votes
1answer
55 views

For what $k$ does $k \sin A + \cos 2A = 2k - 7$ have a solution?

The equation $k \sin A + \cos 2A = 2k - 7$ has a solution, if: $k >6$ $k>2$ $k<7$ $2\leq k\leq 6$ Although I did figure out the answer to be the last option using a ...
1
vote
1answer
90 views

Calculating Value Of Trigonometric Formula

when $~\tan\theta+\sin\theta=\dfrac{1}{2}$, evaluate $~(\sin^{2}\theta-\sin 2\theta)$ is it possible to get the exact value? I got $~\sin^{2}\theta-\sin ...
1
vote
3answers
767 views

Find all complex solutions of $\sin(z)=1$ [closed]

Find all complex solutions of $\sin(z)=1$. How would I go about this?
0
votes
1answer
524 views

Find equation of tangent line

Find the equation of the tangent line at parameter values $\theta=\pi/6$ and $\theta =5\pi/4$ to the cycloid given by $$x(t)=r\theta-r\sin \theta$$ and $$y(t)= r-r\cos \theta$$ with $\theta\in ...
2
votes
1answer
50 views

What is inverse tangent?

I recently started thinking about what inverse tangent is. It is obvious that the definition of tangent is $\frac{\sin x}{\cos x}$, however, what is inverse tangent? I first thought $\tan^{-1} x = ...
1
vote
2answers
282 views

Convergent subsequences of $x_n = \sin n$ and $y_n = \cos n$…

As in title, $x_n = \sin n$ and $y_n = \cos n$. Can we find some index sequence $\{n_k\}$ such that both $\{x_{nk}\}$ and $\{y_{nk}\}$ converge? (Not necessarily to the same limit) I'm fairly ...
1
vote
2answers
85 views

Relationship among $A,B,C,D$ for $\cos A\cos B=\cos C\cos D$

While solving this Question, I could derive the following: As $\displaystyle 2\cos A\cos B=\cos(A-B)+\cos(A+B)$ substituting $A+B=90^\circ\iff B=90^\circ-A$ we get $\displaystyle 2\cos ...
2
votes
1answer
43 views

Trigonometric inequality solving

How to solve this inequality $\left|\dfrac{\cos 2x + 3}{\cos x}\right|\geq 4$ ? I tried to consider 2 cases: 1) When $\cos 2x \geq 0$ and $0<\cos x<1$ 2) $\cos 2x\leq 0$ and $-1 < \cos x ...
0
votes
1answer
2k views

Using angle formula to solve $3\tan\theta = 2\cos\theta$

This may seem fairly straightforward, but I have been stuck on this for the past half-hour. I need to use Double Angle Formulae such as the following: $\sin2A ≡ 2\sin A \cos A$ $\cos2A ≡ \cos^2A - ...
-1
votes
3answers
123 views

What does $\tan^{-1}\left(\frac{\sqrt6}{\sqrt2}\right)$ equal exactly?

$$\tan^{-1}\left(\frac{\sqrt6}{\sqrt2}\right)$$ I can't work out how to answer this in exact form?
4
votes
1answer
92 views

How solve this equation

Solve the equation $$\cos \left(x+30^{\circ}\right)+\cos \left(x+10^{\circ}\right)=\cos \left(2x+10^{\circ}\right)+\cos 10^{\circ}$$ , where $x\in (0,\pi )$ My try: ...
0
votes
2answers
92 views

$\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = 1$. How?

$$\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = 1$$. How? I can do $$\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = \frac{1}{\pi} \int^\infty_{-\infty} \frac{d}{dx} \tan^{-1}{(x)} \; dx$$ But ...
1
vote
1answer
195 views

Rotate a Regular Convex Polygon so vertices are maximum distance from both X and Y axis

I need some assistance with the formula/algorithm to find the vertices of a regular convex polygon (centered at (0,0) and with a circumradius of 1) rotated so that the vertices are as far away from ...
1
vote
3answers
212 views

Proof of Aristarchus' Inequality

Does anyone know how to prove that if $0<\alpha<\beta<\frac{\pi}{2}$ then $\frac{\sin\alpha}{\alpha}>\frac{\sin\beta}{\beta}$. Any methods/techniques may be used.