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

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0
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3answers
85 views

Showing that $A = \{\tan x \mid x \in (-\frac{\pi}{2}, \frac{\pi}{2})\}$ is not bounded, without calculus.

I have to show that $A = \{\tan x \mid x \in (-\frac{\pi}{2}, \frac{\pi}{2})\}$ is not bounded. Since $\tan\colon (-\frac{\pi}{2}, \frac{\pi}{2}) \to \mathbb R$ is surjective $\Rightarrow ...
1
vote
6answers
694 views

Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$

Show that $$\frac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x} = 4\cos(2x).$$
7
votes
1answer
147 views

tough algebric problem?

I wanted to know how can i prove that if $xy+yz+zx=1$, then $$ \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} = \frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$$ I did let $x=\tan A$, ...
1
vote
4answers
255 views

Solve $\cos x>1/2$ for $-180<x<180$

Solve $$\cos x>\frac12\quad\text{for}\quad-180^{\circ}<x<180^{\circ}\;.$$ Hey guys, Ive got the solution to this question although I cant seem to figure out how the textbook did it. Can ...
0
votes
2answers
221 views

Trigonometry : angle between 2 known lines

First of all, I am sorry if my English is super terrible. I'm going to make a program to calculate Zoeprit's equation. I'm stuck at making the algorithm because of my poor math skill. the sketch is ...
8
votes
12answers
1k views

How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method besides L'Hospital's rule.
2
votes
3answers
101 views

If $p =\frac{4\sin\theta \cos\theta}{\sin\theta +\cos\theta}$ Find the value of $\frac{p+2\sin\theta}{p-2\sin\theta}$

Problem : If $\displaystyle p =\frac{4\sin\theta\cos\theta}{\sin\theta +\cos\theta}$, find the value of $\displaystyle \frac{p+2\sin\theta}{p-2\sin\theta} + \frac{p+2\cos\theta}{p-2\cos\theta}$. ...
2
votes
0answers
56 views

Reference request: Another tangent half-angle formula

Wikipedia's "Tangent half-angle formula" article lists these: \begin{align} \tan\frac\theta2 & = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta} = \frac{\tan\theta}{1 + ...
0
votes
2answers
281 views

Solutions to Trigonometric Problem Class

Is there a way to prove that the general solution of: $\sin^2 \pi x + \sin^2 \frac {ab\pi}{x} = 0$ is: $x = \pm 1,\pm a, \pm b, \pm ab$ and more specifically to derive the proof analytically, ...
5
votes
1answer
278 views

Evaluating $\cos(\alpha+\beta+\gamma)$

I am trying to evaluate $\cos(\alpha+\beta+\gamma)$ This is what I have done so far: I know $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha+\beta) = ...
1
vote
0answers
30 views

How to figure out if a point is inside the bounds of a rotated rectangle [duplicate]

I'm trying to figure out an approach (or a formula) to determine if a point (x,y) is inside the bounds of a rotated rectangle. For this test we can assume that the rectangle is centered on the ...
2
votes
2answers
115 views

proving that the differences of squares of hyperbolic sin/cos is an integer.

The hyperbolic sine and cosine are defined as following: $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ $\cosh(x)=\dfrac{e^x+e^{-x}}{2}$ How do I show that their differences of squares are always an integer for ...
3
votes
1answer
188 views

Express $\arccos (\frac 12 \sin x)$ as an algebraic function?

I am trying to express $\arccos (\frac 12 \sin x)$ as an algebraic function on the intercal $[0, \frac {\pi}{2})$ . I tried to find this by setting up a triangle with sides $1$ , $x$, and $\sqrt ...
18
votes
5answers
355 views

If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?

Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$. If $A+B+C+D+E = 540^\circ,$ what is ...
1
vote
2answers
812 views

word problem about right-angle triangle

An observer who is standing $47$ m from a building, measures the angle of elevation of the top of the building as $17°.$ If the observers eye is $167$ cm from the ground, what is the height of the ...
5
votes
4answers
531 views

If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$

If $\alpha +\beta = \dfrac{\pi}{4}$ prove that $(1 + \tan\alpha)(1 + \tan\beta) = 2$ I have had a few ideas about this: If $\alpha +\beta = \dfrac{\pi}{4}$ then $\tan(\alpha +\beta) = ...
1
vote
2answers
214 views

Trigonometry Identity: $3(\cot^{2}\theta+1)-\csc^2\theta-1=\cot^2\theta+\csc^2\theta$

I need some help with solving this Identity. I've tried several methods, but I haven't been able to come up with the correct answer and/or method. I've tried several things. Ranging from converting ...
21
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8answers
1k views

Mental estimate for tangent of an angle (from $0$ to $90$ degrees)

Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable. I can ...
2
votes
0answers
111 views

What's the acceptance of rational trigonometry in current mathematics courses?

I've been reading about Wildberger's rational trigonometry and I'm willing to learn it. I'm wondering if it's usage is accepted in undergraduate mathematics courses. It seems there's a redefinition on ...
2
votes
2answers
135 views

Evaluating $\int\cos\theta~e^{−ia\cos\theta}~\mathrm{d}\theta$

Is anybody able to solve this indefinite integral : $$ \int\cos\theta~e^{\large −ia\cos\theta}~\mathrm{d}\theta $$ The letter $i$ denotes the Imaginary unit; $a$ is a constant; Mathematica doesn't ...
1
vote
2answers
90 views

Solving the inequality: $1\leq \cot^2(x)\leq 3$

I want to solve the following inequality: $$1\leq \cot^2(x)\leq 3.$$ but I'm unsure of how to handle the positive and negative square roots. If I take the square of the inequality, just focusing on ...
1
vote
2answers
375 views

Solving the trigonometric equation $A\cos x + B\sin x = C$ [duplicate]

I have a simple equation which i cannot solve for $x$: $$A\cos x + B\sin x = C$$ Could anyone show me how to solve this. Is this a quadratic equation?
4
votes
1answer
80 views

Is there an intuitive way to understand why a frequency cannot be writen as a sum of other frequencies?

Let's say we have a sequence of functions $(f_n)$ and for $n$, $f_n$ is a cosine at frequency $k$, i.e. $f_n(t)=\cos(2\pi kt)$ for some $k$ which depends of $n$. Let $n_1,\dots n_k$ be natural ...
16
votes
4answers
1k views

About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Consider the functional equation $$f(x+y) = f(x)g(y)+f(y)g(x)$$ valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$. ...
3
votes
2answers
314 views

Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$

Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$ I have got as far as showing that: $\cos(x+y)\cos(x-y) = \cos^2x\cos^2y -\sin^2x\sin^2y$ and $\sin(x+y)\sin(x-y) = ...
2
votes
1answer
84 views

find parameter for maximize area

suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are $A(cos(3-t),sin(3-t))$ $B(cos(t),sin(t))$ ...
3
votes
3answers
709 views

Find the value of $\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^{8}\theta + \cos^6\theta + 2\cos^4\theta + 2\cos^2\theta - 2$

We are given that $\sin\theta + \sin^3\theta + \sin^2\theta = 1$ Find the value of $\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^{8}\theta + \cos^6\theta + 2\cos^4\theta + 2\cos^2\theta - 2$ Now, I ...
6
votes
4answers
297 views

An inequality for $\cos$ and $\sin$

Let $\alpha, x,y\in\mathbb{R}$. Do the inequalities $$|\sin(\alpha x)-\sin(\alpha y)|\le \alpha|x-y|$$ and $$|\cos(\alpha x)-\cos(\alpha y)|\le \alpha|x-y|$$ hold?
5
votes
1answer
289 views

Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$

I would like to show that for $A(x) = \int_{0}^{x}\frac{1}{1+t^2}dt$, we have $A\left(\frac{2x}{1-x^2}\right)=2A(x)$, for all $|x|<1$. My idea is to start with either $2\int_0^x\frac{1}{1+t^2}dt$ ...
3
votes
2answers
319 views

Prove that $x^2<\sin x \tan x$ as $x \to 0$ [duplicate]

$$x^2<\sin x \tan x \quad as \; x \to 0$$ I made the substitution $x \to \arctan x$ . $\arctan^2 x<x\sin (\arctan x)$ $\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$ There are two ...
2
votes
5answers
303 views

What is the derivative of $x\sin x$?

Ok so I know the answer of $\frac{d}{dx}x\sin(x) = \sin(x)+ x\cos(x)$...but how exactly do you get there? I know $\frac{d}{dx} \sin{x} = \cos{x}$. But where does the additional $\sin(x)$ (in the ...
2
votes
3answers
873 views

Visual proof of the addition formula for $\sin^2(a+b)$?

Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here : Also it is easy to generalize (in any way: algebra , picture ...
2
votes
2answers
218 views

Can it happen that an object will not cast any shadow at all?

I am puzzled by a question in Trigonometry by Gelfand and Saul on p. 57. Can it happen that an object will not cast any shadow at all? When and where? You may need to know something about ...
1
vote
5answers
223 views

Expressing a combination of sine and cosine as a single cosine

Prove that: $$\dfrac{\sqrt2}2 \cos \omega t - \dfrac{\sqrt2}2 \sin \omega t = \cos \left(\omega t + \dfrac\pi4\right)$$ Obviously, if we are evaluating the right side of the equation, it would a ...
2
votes
1answer
297 views

A strange trigonometric identity in a proof of Niven's theorem

I can't understand the inductive step on Lemma A in this proof of Niven's theorem. It asserts, where $n$ is an integer: $$2\cos ((n-1)t)\cos (t) = \cos (nt) + \cos ((n-2)t)$$ I tried applying the ...
0
votes
3answers
85 views

Need pointers on how to do this trigonometric proof

$$ \cos x = \cos y + \cos^3 y$$ $$\sin x = \sin y - \sin^3 y$$ Prove that $\sin {(x - y)} = \pm \frac{1}{3}$. I need a little hint, not a complete answer.
3
votes
2answers
132 views

For what $x\in[0,2\pi]$ is $\sin x < \cos 2x$

What's the set of all solutions to the inequality $\sin x < \cos 2x$ for $x \in [0, 2\pi]$? I know the answer is $[0, \frac{\pi}{6}) \cup (\frac{5\pi}{6}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, ...
4
votes
2answers
234 views

Why does $\lim\limits_{x\to0}\sin\left(\left|\frac{1}{x}\right|\right)$ not exist?

Can someone explain, in simple terms, why the following limit doesn't exist? $$\lim \limits_{x\to0}\sin\left(\left|\frac{1}{x}\right|\right)$$ The function is even, so the left hand limit must equal ...
18
votes
5answers
1k views

Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$

In old popular science magazine for school students I've seen problem Prove that $\quad $ $\dfrac{1}{\cos^2 20^\circ} + \dfrac{1}{\cos^2 40^\circ} + \dfrac{1}{\cos^2 60^\circ} + ...
1
vote
1answer
61 views

Hyperbolic integration solving

$$ \therefore x-x_0 = \pm \int_{\phi(x_0)}^{\phi(x)} \frac{d \phi}{\sqrt\frac{\lambda}{2}\left( \phi^2-(\frac{m}{\sqrt \lambda})^2\right)} $$ How can we write the above equation to as, $$ \phi(x) = ...
1
vote
4answers
176 views

Where does $\sin 3° =3\sin 1° -4 \sin^3 1°$ come from?

Wikipedia makes the claim: "Though a complex task, the analytical expression of $\sin 1°$ can be obtained by analytically solving the cubic equation $\sin 3° =3\sin 1° -4 \sin^3 1°$ from whose ...
13
votes
5answers
2k views

Are there infinitely many rational outputs for sin(x) and cos(x)?

I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? ...
1
vote
0answers
215 views

Gps Coordinate 1 mile away - Haversine

I've come across the need for doing GPS calculations. I've implemented the haversineformula for gps distances but I need the opposite, a point 1 mile away in any direction. The GPS coordinates are ...
4
votes
1answer
122 views

I don't understand the “rational” arguments for trig functions

For example look at $\cos(x)$. I imagine that $x$ can sometimes be rational (or integer) or irrational, and sometimes $\cos(x)$ can be rational (or integer), or irrational. Under what circumstances ...
2
votes
5answers
2k views

$y = -2\sin(x - \pi/3):\;$minimum & maximum values?

I have the following function $$y = -2\sin⁡(x-\pi/3),\quad 0\leq x\leq 2\pi$$ I know that $\sin x$ has max at $\pi/2$ and min at $3\pi/2$ but how would I use this information to find the solution to ...
4
votes
2answers
235 views

$x \sin x=2$ why is my proof that there no solutions wrong?

$\frac 12 x \sin x=1$ . Let's look at a right triangle with base $x$ and altitude $\sin x$ . Then our equation is for the area of this triangle. Let the sides of the triangle be $a=x$ , $b=\sqrt ...
1
vote
2answers
154 views

How to graph the trigonometric function when period is more than the range?

I have to graph this trigonometric equation for a given range, $y = -\sin⁡ \left(\dfrac{x}{3}\right) - 2, ~-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$ But the period is coming out to be $6 \pi$. ...
2
votes
2answers
574 views

Sums $\sum_{k=1}^n \sin(2k-1)\theta$, $\sum_{k=1}^n \sin^2(2k-1)\theta $

To prove: $1.$ $$\sum_{k=1}^n \sin(2k-1)\theta = \frac{\sin^2 n\theta}{\sin \theta}.$$ $2.$ $$\sum_{k=1}^n \sin^2(2k-1)\theta = \frac{n}{2} - \frac{\sin 4n\theta}{4\sin 2\theta}.$$
2
votes
3answers
927 views

Prove that $\sin 10 \sin 20 \sin 30=\sin 10 \sin 10 \sin 100$?

$\sin 10 \sin 20 \sin 30=\sin 10 \sin 10 \sin 100$ (these are all degrees) This is a competition problem which I got from the book "Art of Problem Solving Volume 2". I'm not sure how to solve it ...
2
votes
3answers
121 views

Demonstrate this triangle! Help

Give a triangle ABC with $$\sin{\left(\frac{3A}{2}\right)}+\sin{\left(\frac{3B}{2}\right)}=2\cos{\left(\frac{(A-B)}{2}\right)}$$ Demonstrate that triangle ABC is equilateral triangle!! Thank all! ...