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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1answer
39 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [on hold]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
1
vote
1answer
22 views

Square Wave Intuition

As I understand it, a square wave can be produced as follows: $$y = \cases{ 1 & \text{if } \sin(x) > 0\cr 0 & \text{if }\sin(x) = 0\cr -1 & \text{if } \sin(x) < 0} $$ What I'm ...
0
votes
2answers
48 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
0
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2answers
35 views

Why does the following equality hold? $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$?

Why is $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$ true?
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1answer
26 views

Find all angles that satisfy $6\cos^2(x)+5\cos(x)-6=0$ [on hold]

Find all angles that satisfy: $$6\cos^2(x)+5\cos(x)-6=0.$$
0
votes
1answer
25 views

angle $0$ to $2\pi$ between two 3Dvectors

Ok this is for a computer game I'm learning to program with. How do you find angle between two normalized 3D vectors so that you get the resulting angle in the range $[0,2\pi]$ or $[-\pi,\pi]$? Using ...
6
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3answers
91 views

How prove $\sin \left( \alpha+\frac{\pi }{n} \right) \cdots \sin \left( \alpha+\frac{n\pi }{n} \right) =-\frac{\sin n\alpha}{2^{n-1}}$?

How prove $$\prod_{k=1}^{n}\sin \left( \alpha+\frac{\pi k }{n}\right) =-\frac{\sin n\alpha}{2^{n-1}}$$ for $n \in N$?
2
votes
1answer
57 views

Why is arcsin represented with the ^(-1) notation?

So in trigonometry, we have sin, secant (which is one over sin) and arcisn. Why is arcsin sometimes represented with sin^-1? sin^2 means sin to the second power, but sin^-1 explicitly does not mean ...
7
votes
3answers
222 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
1
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1answer
25 views

How to divide trigonometric ratios using identities?

$$\frac{1-\tan^2x}{1+\tan^2x}$$ We know: $$\frac{1-\frac{\sin^2x}{\cos^2x}}{1+\frac{\sin^2x}{\cos^2x}}$$ Now what? Flip denominator and times numerator? Which equals ??? Please help - Thanks
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2answers
33 views

Find Coefficients from already fourier function

Hello I have this function and I'm asked 1.Find the period for $f(t)$ 2.Find the coefficients $a_n$ and $b_n$ $$f(t)=2(cos(2t+\frac{\pi}{4})-sin(6t-\frac{\pi}{2}))$$ I know that the period for ...
0
votes
1answer
30 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
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0answers
23 views

What is the best trigonometry book available free?

I am not a rich person but I really want to have a look on the trigonometry book
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1answer
14 views

Find the area of trapezium given certain angles and length of diagonal

In the trapezium $MNOP$, $MP$ is the major base and $NO$ is the minor base. Knowing that the angle $P$ is $58° 15'$, the angle $OMP$ is $21° 45''$, and the diagonal $OM$ is of $6.5$ cm, calculate the ...
-1
votes
2answers
29 views

Solve trigonometric equation $\sin(2t) = −\frac{\sqrt{2}}{2}$ on interval $[0, 2 \pi]$

Solve the following equation on interval $[0, 2 \pi]$: $$\sin(2t) = −\dfrac{\sqrt{2}}{2}$$ I got $t=\left\{\dfrac{9\pi}{8},\dfrac{15\pi}{8}\right\}$, but website for math assignment said that it is ...
1
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1answer
29 views

Find sides and height of isosceles trapezium given information about its diagonals

In an isosceles trapezium the diagonals cut at a point $O$ which divides them in two segments of $3$ cm and $7$ cm. If one of the angles formed between them is of $120°$, find the measures of the ...
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0answers
24 views

What is the maximum area of a trapezium with 3 known sides and unknown angles. [on hold]

The Question: A major company in your city has both new equipment capable of making guttering in the shape of an open top trapezium. The sheet metal used is 22 cm wide and bent such that the base s ...
-4
votes
0answers
31 views

Trigonometric math problem [on hold]

A camera is mounted at a point 3000 ft from the base of a rocket launching pad. The Rocket rises vertically when launched, and the camera's elevation angle is continually adjusted to follow the bottom ...
1
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2answers
38 views

Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative

The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$ thanks for helpers
0
votes
1answer
40 views

Radians or degrees?

In problem 2 from this page: http://www.analyzemath.com/calculus/Problems/rate_change.html The last couple steps including the equation: $$\frac{da}{dt} = \left[-\frac{\sin ...
0
votes
2answers
15 views

Finding upper and lower bounds on a trigonometric function

I've been tasked with finding the upper and lower bounds of the element: $A = sin(\frac{\pi.n}{2n+3}) | n\in\mathbb{N}$ I think I have found the upper bound by doing: $\lim_{n\to +\infty} ...
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4answers
83 views

Integrating $\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt$.

I am trying to compute $$ \int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt. $$ This is what I got so far: $t=\sec(x)$ and $dt=\sec(x)\tan(x)x\,dx$ So plugging this in gives me $$ \int ...
0
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2answers
23 views

Solving a Cartesian and parametric equation at a intersection.

A curve C has parametric equations: $x=4cos(2t)$ and $y=3sin(t)$ $-\frac{\pi}{2} < t < \frac{\pi}{2}$ The normal of a point A$(2,1.5)$ on curve C has the equation $6y-16x+23=0$ The curve and ...
-1
votes
3answers
39 views

Finding $\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }$.

How do I get the value of $$\lim_{x \rightarrow \frac{1}{4} \pi } \frac{\tan x-\cot x}{x-\frac{1}{4} \pi }?$$ I need the steps without using L'hospital.
4
votes
2answers
95 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
0
votes
1answer
51 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
6
votes
1answer
53 views

How to prove $\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$

I am trying to prove the following: $$\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$$ I tried some values and it seems convincing. I wonder if this is a ...
9
votes
2answers
214 views

How to Solve Trigonometric Equations?

How are you supposed to go about solving equations such as: $$-\sqrt{3} = \frac{\sin{4\theta}}{\sin{7\theta}}.$$ I know that $\theta = 30^{\circ}$ is one such solution, but how do I find all ...
5
votes
3answers
61 views

Range of trigonometric functions

I would like to know if there is a simple approach to find the range of functions in the form: $$\sin x\sin2x$$ $$\cos x\cos3x$$ $$\sin 2x\cos 4x$$ For example, finding the range of a function in ...
0
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1answer
9 views

Find Adjacent only knowing Angle and Opposite

Can you find the length of the adjacent side of a right triangle only knowing the length of the opposite side and the angle? If so how do you calculate it?
1
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4answers
29 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
3
votes
1answer
44 views

Triple Angle Condition

Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter. Essentially we want sinb, sin3b, sin4b to have rational ratios (manipulate ...
1
vote
1answer
26 views

Data transformation of angles such that $90^\circ$ is equal to $-90^\circ$

Is there a transformation I can perform on a dataset of angles (from $-90^\circ$ to $90^\circ$) such that the transformation of $-90^\circ$ is equal to that of $90^\circ$? I am only interested in what ...
1
vote
2answers
14 views

Centripetal acceleration for a polyline

Given a polyline (x and y coordinates) in Cartesian coordinate system and time component, how can I estimate centripetal acceleration (let's say an average one)? (I have a list of pairs. Each pair ...
0
votes
1answer
21 views

How to find the domain of this trig function?

f(x)=sqrt(tan(2x+π)) Allright, so i know you cannot have a number less than zero under the square root sign and that tangent cannot equal π/2+nπ. So should i try to find the domain of the tan ...
2
votes
2answers
33 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
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votes
1answer
27 views

How to show the following inverse trigonometric equation? [on hold]

Let $a$ and $b$ be real numbers so that $ b\neq 0 $. Show that $ \dfrac{tan^{-1}\left( \dfrac{a}{b}\right)}{\pi}=\dfrac{\ln \left( b/ \sqrt{a^{2}+b^{2}}+ai/ \sqrt{a^{2}+b^{2}}\right) }{\ln (-1)}$.
1
vote
3answers
36 views

Translate a point on a circumference

If I have a point $A$ on the circumference of a circle with origin $O$ and radius $r$, how would I find the coordinates of point $B$, which is also on that circumference, but is $d$ units away from ...
0
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3answers
38 views

Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$

Let $z=\cos\theta+i\sin\theta$. Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$ Can anyone show me how to show the equation? I can't think of how to get $\frac 12 ...
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0answers
12 views

An isosceles triangle of wood is placed in vertical plane, vertex upwards and faces Sun. If 2a be base of triangle…

Problem : An isosceles triangle of wood is placed in vertical plane, vertex upwards and faces Sun. If 2a be base of triangle, h its height and $30^{\circ}$ altitude of Sun, then prove that tangent ...
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0answers
11 views

From a light house L two ships P and Q are observed in direction South West and $5^{\circ}$ East of South respectively. At same time Q…

Question : From a light house L two ships P and Q are observed in direction South West and $5^{\circ}$ East of South respectively. At same time Q is observed from point P in South East direction of ...
16
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4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
0
votes
1answer
18 views

Trig Word Problem Involving System of Equations

Hobbyists often compete with their model rockets to determine which rocket flies the highest. On one test launch, a rocket was fired vertically upward. The angle of elevation to the top of the flight ...
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0answers
27 views

Triangles congruency exercise

This exercise tells that: $$AD = AE, \angle A \cong \angle DEC, \angle ADE \cong \angle BDC$$ Then I have to show that $$\triangle ADB = \triangle EDC$$ The exercise solution highlights that we ...
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2answers
24 views

Are there infinitely many pairs $(x, y)$ satisfying $\cos(x+y)=-1$?

Consider the equation $$\cos(x+y)=-1.$$ I think $\cos(x+y)=-1$ only when $x+y=(2n+1)\pi$ for $n$ any integer. If $S$ is the set of all pairs $(x, y)$ in $\mathbb{R}^2$ satisfying $\cos(x+y)=-1$, is ...
0
votes
1answer
35 views

If the cos of 27 is 0.89, how much is the csc of 27

Hey guys for my trig class we're viewing trigonometric functions and their properties. So far I have understood but I came across this problem and can't seem to solve it: Given the approximation cos ...
0
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3answers
49 views

Trig Substitution Integral Question

My class is going over trig substitution, but I can't figure this one out, mostly because it's not in the correct form. Could someone help explain how to set up this problem? $$ \int \frac ...
9
votes
7answers
137 views

Finding $\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$

How would I go about solving this following limit? $$\lim_{x\to 0} \frac{(1+\tan x)^{\frac{1}{x}}-e}{x}$$ My attempts: Direct substitution yields the limit to be undefined, also ruling out the ...
0
votes
0answers
36 views

Trigonometry graph sketch [on hold]

Sketch the graph of $y=|\tan x|$ for $0 \leq x \leq 360^{0}$ showing clearly the position of the asymptotes. Solve the inequality $|\tan x| < 1$ for $0 \leq x \leq 360^{0}$.
3
votes
4answers
33 views

Use substitution to solve for $x$ in $\frac{1}{2-\sin x}=\sin x$

Use substitution to solve for $x$ in the following equation: $$\frac{1}{2-\sin x}=\sin x$$ This is what I have done so far: $$\sin^2x-2\sin x+1=0$$ $$\arcsin(1)=\frac{\pi}{2}=x$$ The correct ...