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

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1answer
41 views

Exact solution to equation

I am trying to find out how to get some form of exact nonzero solution with isolated x for the equation $x^2 = \sin x$. I am pretty sure my TI 89 is using Taylor series expansions to solve this ...
1
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1answer
30 views

Formula to convert Cartesian coordinates to spherical coordinates?

I have this formula: x, y, z = cos(vertical)*sin(horizontal), sin(vertical), cos(vertical)*cos(horizontal) Which maps a spherical coordinates (horizontal and ...
-2
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1answer
40 views

Can anyone help me solve this [on hold]

Given that $\tan(2x) = \frac34$ and $0 < x < \frac{\pi}4$ find the exact value of (a) $\cos (2x)$ (b) $\cos (x)$.
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2answers
34 views

North Pole, 40 degrees north, and South Pole: question on why first two seem so close relative to third?

Based on the tunnel distance formula from Wikipedia, I calculate that the tunnel distance (shortest distance between two points on Earth's surface, straight through Earth, based on a spherical Earth) ...
4
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2answers
42 views

Different results in integrating both sides of $\sin{2x}=2\cos x\sin x$

I feel like there is something I am missing here. When integrating both sides of the trigonometric identity $\sin{2x}=2\cos x\sin x$ I get different results. The left side of course results in ...
1
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4answers
73 views

How to solve$\frac {1}{\sin {x}} + \frac {\sqrt {3}}{\cos {x}} = 4$ [on hold]

$$\frac {1}{\sin {x}} + \frac {\sqrt {3}}{\cos {x}} = 4$$ Can you help me solve this?
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3answers
61 views

What does $2\cos^2(\theta)−1$ equal to in radians? [on hold]

$$2\cos^2(\theta)−1$$ How would I go about simplifying an expression like this?
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1answer
27 views

Need one example solving trigonometry.

Calculate $\sin \beta, \tan \beta, \cot \beta, \cos(2\beta)$ if $\cos \beta = {5 \over 13}$ and $\beta \in (0^{\circ},90^{\circ})$. I'm a student and I forgot how to solve it correctly...I need just ...
2
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2answers
36 views

Proving that $\tan^n\angle A + \tan^n\angle B + \tan^n\angle C \ge 3 + \frac{3n}{2}$

Given a acute $\triangle ABC$. Prove that $$\tan^n\angle A + \tan^n\angle B + \tan^n\angle C \ge 3 + \dfrac{3n}{2}$$ I have tried by using a inductive proof. In case $n=0$, the equality holds. ...
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1answer
15 views

Prove that the envelope of the family of lines $(\cos\theta+\sin\theta)x+(\cos\theta-\sin\theta)y+2\sin\theta-\cos\theta-4=0$

Prove that the envelope of the family of lines $(\cos\theta+\sin\theta)x+(\cos\theta-\sin\theta)y+2\sin\theta-\cos\theta-4=0$ I did not know much about how to find envelope of a curve.I read on ...
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4answers
58 views

Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$

I have two similar looking questions. $(1)$Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$ $(2)$If $\Delta ABC$ is acute angled,then prove that ...
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1answer
41 views

Jensen inequality conceptual doubt

Prove that in a triangle $ABC$,$\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\geq\frac{3}{4}$. I tried to solve it by Jensen's inequality.I let $f(x)=\sin^2\frac{x}{2}$ ...
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2answers
43 views

How to prove $ \sin^2 {2x} - \sin^2 {x} = \sin {3x}\sin {x} $ [duplicate]

How do I prove: $ \sin^2 {2x} - \sin^2 {x} = \sin {3x}\sin {x} $ ? I'm lost
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0answers
9 views

Representing trigonometric functions in a form of rational functions

Here I introduced a non-Archimedean numerical system in which the real numbers are extended by elements $\omega_-$, $\tau=\omega_-+1/2$, $\omega_+=\omega_-+1$ in such a way that standard parts of ...
3
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0answers
38 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
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2answers
33 views

If in a triangle $ABC$,$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A$,then prove that triangle will be equilateral triangle

If in a triangle $ABC$ we have $$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A\ ,$$ then the triangle will be equilateral triangle. I tried but except few steps,could not prove it. ...
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1answer
51 views

Series of $\csc(x)$ or $(\sin(x))^{-1}$

In some cases I found that $$\csc(x)= \lim\limits_{k\rightarrow \infty}\sum_{n=-k}^{k}(-1)^{n}\frac{1}{x-n\pi}$$ Is anything to prove or disprove that?
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1answer
58 views

Prove that $\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}$ [on hold]

Let $D,E,F$ be the feet of the perpendiculars from the incenter $I$ to the sides $BC,CA$ and $AB$ respectively. If $r,r_1,r_2$ and $r_3$ are the inradius of the triangle $ABC$ and radii of the circles ...
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0answers
32 views

Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
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0answers
23 views

Trig substitution triangle restrictions

I apologize if this is a dumb question, or if I am a little slow, but I've been thinking about this for all of yesterday and today and I just can't figure it out, despite googling it. I am confused ...
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1answer
23 views

Area under a parabolic trajectory

I have this problem: "prove that the area under the trajectory described by a parabolic shot that has: $f(x)=\tan(\theta)x - (\frac{g}{2v^2\cos^2(\theta)})x^2$ and $x=v\cos(\theta)t$ is defined ...
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0answers
19 views

With 2 as smallest period of the function $f(x)$= $\tan^2[(\frac{\pi x}{n^2-5n+8})]$ + $\cot(n+m)\pi x$ ;the period m can't belong to is?

Here n $ \in N$ , m $\in Q$. Options are: A) $(-\infty, -2) \cup (-1, \infty)$ B) $(-\infty, -3) \cup (-2, \infty)$ C) $(-2,-1) \cup (-3,-2)$ D) $(-3, -5/2) \cup (-5/2, -2)$ I have an answer to ...
2
votes
3answers
66 views

Minimum value of $\cos x+\cos y+\cos(x-y)$

What is the minimum value of $$ \cos x+\cos y+\cos(x-y). $$ Here $x,y$ are arbitrary real numbers. Mathematica gives (with NMinimize) $-3/2$. But I don't know if this is correct and if so, how to ...
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2answers
20 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle ...
-1
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1answer
52 views

Proving a trigonometry question. [on hold]

If $$ \tan A \cdot \tan B= \sqrt{\frac{a-b}{a+b}}, $$ then prove that $$ (a - b \cdot \cos2A)(a - b\cdot \cos2B) = (a^2 - b^2).$$
3
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2answers
68 views

Since $\lim\limits_{x\to0}\frac{\sin kx}{kx}=1$ for constants $k$, is it also true for general arguments?

To be more specific, is it true that $$\lim_{x\to0}\frac{\sin f(x)}{f(x)}=1~~?$$ I'm tempted to say yes at first glance, so long as $f(x)\to0$ as $x\to0$. The reason I ask is to verify this limit ...
1
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1answer
25 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
0
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2answers
33 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
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2answers
42 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
6
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3answers
48 views

Prove that $3x-x^3<\frac2{\sin2x}$

Prove that $$3x-x^3<\frac2{\sin2x},\forall x\in\left(0,\frac\pi2\right)$$ I have tried by proving that $$3x-x^3<\frac9{5\pi}x+\frac32<\frac2{\sin2x},\forall ...
2
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2answers
36 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
0
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2answers
34 views

Which is the justification for this indefinite integral relation? [on hold]

Why is the following indefinite integral equation correct: $$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$ What are the necessary steps?
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2answers
34 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
2
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4answers
142 views

How to solve $2 \tan x / (1 - (\tan x)^2) = (\sin 2x)^2$? [on hold]

$$\frac {2\tan {x}}{1-(\tan {x})^2} = (\sin {2x})^2$$ I tried a lot but I get nowhere
2
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0answers
31 views

Geometry/Trigonometry Determine angle in a Triangle [duplicate]

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
0
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0answers
28 views

Trignometry, bearings yacht race question [on hold]

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
2
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0answers
33 views

How to find the period of $\cos(|\sin x|-|\cos x|)$?

My book did provide a rule as: If $f_1(x),f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then we have $h(x)= f_1(x) + f_2(x)$ has period, as $\bullet$ LCM of $\{T_1, ...
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1answer
39 views

Integrating $\frac{x^3}{(81-x^2)^2}$

I've been trying to figure out this integral for an hour or so now, but keep failing. I can't figure out where I go wrong: $$I = \int \frac{x^3}{(81-x^2)^2} dx$$ Let $x = 9sin\theta \implies dx = 9 ...
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0answers
14 views

Trig substitution using reference triangles

Suppose we are doing a trig substitution and make some substition $x = a \sin \theta \equiv \sin \theta = \frac{x}{a}$ where the domain of x is $|x| \le a$ Then from the reference triangle we can ...
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3answers
42 views

Integrating trig substitution triangle equivalence

When we integrate certain integrals, such as $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We can make a substitution like $x = 4 \sin \theta$ Then we can simplify the above integral to the following: $$8 ...
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2answers
74 views

How to find the period of $\cos(\cos\theta)$?

How to find the period of $f(\theta)=\cos(\cos\theta)$? For this, I've taken the easiest approach: Let $T$ be the least positive value for which the function is positive. Then $$f(\theta)= ...
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0answers
21 views

Polynomial function for arctan(tanx) [on hold]

What is the Equivalent polynomial function for arctan(tanx), arccos(cosx), arcsin(sinx)?
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1answer
23 views

Multiplicity in Solutions to Trig Function Equations

This is a very simple problem, but I can't figure out where I am going wrong! Say you have the following: $a \sin\theta + b \cos\theta = c. \tag{1}$ Now, this for example can be rewritten using: $R ...
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1answer
45 views

Determining North-South Line Via Watch Method: Theory & Reason

I recently read that if you're in the northern hemisphere and have an analog watch, then you can point the hour hand at the sun and know that a south line lies between (bisection) the hour hand and ...
2
votes
3answers
43 views

Prove that $\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$

Prove that $$\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$$ I have known the solution that uses convex function. But I'd like another solution don't use it. :D
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1answer
21 views

Solve the following Trigonometric Equation

I am not sure what to do with this; $-\csc^2x + (\sqrt 2)\csc x \cot x = 0 \text{ between} (0, \pi)$ Do I convert to sine and cosine and then add the identities together?
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0answers
16 views

simplifying distance equation

The optimal angle for throwing a ball from a cliff is $$\sin \theta = \frac{1}{(2+ \beta)^{1/2}}$$ the original distance equation is $$ x = \cos \theta (\sin \theta + (\sin^{2} \theta + ...
4
votes
2answers
60 views

Why $\tan x>\sin x$ in this question?

The question asks me to prove the identity $\tan ^2x-\sin ^2x=\tan^2 x \sin^2 x$ and use this result to explain why $\tan x>\sin x$ for $0<x<90$ I've proved the identity and I can't explain ...
1
vote
1answer
20 views

Rewriting a trig function into a sum of exponential functions

Rewrite the function $2 + 4\sin(\pi t + \frac{\pi}{6})$ into a sum of exponential functions. By that I mean using Euler's formula $\sin(x) = \dfrac{e^{i\pi x} - e^{-i\pi x}}{2i}$. If it wasn't for ...
5
votes
1answer
34 views

Determining if a sum of trig function is periodic

$$2 + \sin(2\pi\cdot t) + 3\cos(3\pi\cdot t) - 5\sin(7t-\frac{\pi}{4})$$ Is there any manual, easy, way of knowing such a function is not periodic? I'd love to know if there's any method which ...