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

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31 views

Proving a limit of a trigonometric function

I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions
7
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0answers
93 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
2
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2answers
58 views

Supremum of a sine integral

Let $M_T=\int\limits_{0}^{T}\frac{\sin(t)}{t}dt$ be a sine integral. Why is $2\displaystyle\sup_{T}M_T < \infty$?
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20 views

Trig sub and Integration of Squareroot divided by polynomial squared

Question #2 What am I doing wrong? Do not give me the answer but rather a hint.
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4answers
185 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
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1answer
43 views

How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?

Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$ Now, ...
2
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4answers
78 views

prove that $\sqrt{2} \sin10^\circ+ \sqrt{3} \cos35^\circ= \sin55^\circ+ 2\cos65^\circ$

Question: Prove that: $\sqrt{2} \sin10^\circ + \sqrt{3} \cos35^\circ = \sin55^\circ + 2\cos65^\circ$ My Efforts: $$2[\frac{1}{\sqrt{2}}\sin10] + 2[\frac{\sqrt{3}}{2}\cos35]$$ $$= 2[\cos45 \sin10] ...
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3answers
37 views

trigonometric identity of sin squared in terms of tan squared.

Why is $\sin^2(x)=\frac{\tan^2(x)}{1+\tan^2(x)}$? And why is $\sin^2(x)=\frac{1}{\cot^2(x)}$? I've tried starting from $\tan^2(x)=\frac{\sin^2(x)}{1-\sin^2(x)}$ but that wasn't really working out ...
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1answer
26 views

Integral of Euler's formula

Why is $=\int\limits_{-\infty}^{\infty}\cos(-tx)dF(x)+i\int\limits_{-\infty}^{\infty}\sin(-tx)dF(x)=\int\limits_{-\infty}^{\infty}\cos(tx)dF(x)-i\int\limits_{-\infty}^{\infty}\sin(tx)dF(x)$? I know ...
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39 views

can somebody help me with this? [closed]

Just gonna ask you guys if it's possible to prove that $\sec A\tan A - \sin A\sec A = 1 - \tan A$
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1answer
39 views

Finding the third side of a triangle, given ratio of two sides and difference of two angles [closed]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
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1answer
34 views
2
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2answers
61 views

Limit of an integral

I'm not sure how to approach (no pun intended) the following limit: $$\lim_{x \to 0^{+}} \sqrt{|\sin x - \tan x | } \int_{\cos x}^{1+ \sin x} e^y \, \, \mathrm{d}y$$ I know that the indefinite ...
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votes
3answers
86 views

Simplify a quick sum of sines [duplicate]

Simplify $\sin 2+\sin 4+\sin 6+\cdots+\sin 88$ I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.
3
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5answers
95 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
2
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1answer
22 views

sine and cosine and difference of angles

I'm doing a math review, and I am getting a different answer than the guide, and I need some guidance. Here is the problem: Suppose $\cos(x)=\frac{1}{2}$ and $\sin(y)=\frac{1}{2}$, where $x$ ...
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3answers
67 views

Find $\int\sin^4(x)\cos^2(x)\,dx$

Find $$\int\sin^4(x)\cos^2(x)\,dx$$ My Attempt: $$\int\sin^4(x)\cos^2(x)\,dx = \frac18 \int ((1-\cos(2x)-\cos^2(2x)+\cos^3(2x))\, dx$$ How to proceed from here?
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2answers
29 views

Finding the minimum point looks easy with a graph but hard with a formula

My research has lead me to the following function: $$ \frac{\sin(x) [\sin^2(x)\cdot F+ \cos^2(x)/F ]} { 1 - \cos(x) } $$ $F$ is a parameter, and I would like to find the minimum value of this ...
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2answers
38 views

Question about sines of angles in an acute triangle

Let ABC be a triangle such that each angle is less than 90 degrees. I want to prove that sinA + sinB + sinC > 2. Here is what I have done: Since A+B+C=180 and 0 < A,B,C < 90, at least two of ...
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3answers
61 views

Proof that $\cos(\pi/4)=\frac{\sqrt2}{2}$

Normally I just look up or remember that $\cos(\pi/4)=\frac{\sqrt2}{2}$, or type "$\cos(\pi/4)$" into WolframAlpha to check the answer. But what about the first time someone wanted to know what ...
2
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3answers
41 views

Triangle of maximum perimeter for a given area

What type of triangle has the maximum perimeter for a certain area? Suppose I start with a rectangle of that area (axb=Z). I can stretch one dimension of the rectangle until infinity, reducing the ...
3
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4answers
54 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
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2answers
29 views

In a triangle ABC , $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to…

In a triangle ABC, prove that $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to $\frac{abc}{R^2}$, where $a$, $b$, and $c$ are sides of the triangle and $R$ is the circumradius. My work:- By ...
12
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1answer
208 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
4
votes
3answers
43 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
2
votes
2answers
45 views

Equivalence of trigonometric identity

Is writing $$ \cot{2\theta}=\frac{a-c}{2b} $$ equivalent to $$ \cot{\theta}=\frac{a}{b},\tan{\theta}=\frac{c}{b} $$ becuase of the trigonometric identity $$ ...
6
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2answers
80 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
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2answers
40 views

How to solve $|\tan x| \ge 1$?

I need to solve $$|\tan x| \ge 1$$ So I separated it to cases $$-{\sin x\over \cos x}\ge1$$ And $${\sin x\over \cos x}\ge1$$ But now I've got so many cases to check it seems like I'm doing it in the ...
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vote
1answer
34 views

Calculating XY coordinates on line

I have been working on this problem for a while now and can’t figure out the solution. Hence my post on this forum. I’m trying to figure out the position of a symbol on a line. These lines are located ...
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1answer
28 views

trigonometry issue: can i continue solving with a negative angle?

i'm on the verge of solving a basic sine equation that presented 2 sides (b,c) and 1 angle (B), i found angle C (which had two possibilities) and started solving for angle A using the two ...
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0answers
18 views

Combinatorial proof for the formula of $\tan n\theta$

Is there any combinatorial proof of the formula for $\tan n\theta$ where $n\in \mathbb{N}$? Then proofs that I know are by Induction and using de Moivre's Formula but recently one of my friend asked ...
0
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1answer
29 views

What is the radius of a circle tangent to two lines with a known angle between them

Given angle, $\alpha$, and distance, $d$, what is the radius, $r$, and angle, $\theta$, in the image below in terms of the known quantities?
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2answers
34 views

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly.

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly. I know that we are looking for the smallest positive value ...
0
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1answer
33 views

Drawing lines from tangents from two circles on both sides.

I need to draw two red lines connecting the tangents from two circles on both sides. I need an algorithm that would get them based on any angle these circles are in relation to another. I need the ...
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5answers
116 views

Why sin of supplementary angles have equal values?

Given two supplementary angles (for instance, 30 degrees and 150 degrees), why is $\sin(30^\circ) = \sin(150^\circ)$? Where can I find a proof for this? Or the derivative of such proofs?
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3answers
52 views

Can someone explain this question about trig substitution?

I am not sure what I am doing wrong! Here is the question: Evaluate the definite integral $$\int_0^{7/2}\sqrt{49-x^2} \ dx.$$ What I have gotten to is that the integral (through trig ...
3
votes
2answers
77 views

How to evaluate $\lim\limits_{n\rightarrow \infty} \sin (1/n)$?

I know that the $\lim\limits_{n\rightarrow \infty} \sin \left( \frac{1}{n} \right)= 0$ But I am not sure of the right workings. My attempt: As $n$ tends to infinity, $\frac{1}{n}$ will tend to $0$. ...
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1answer
48 views

Prove the following identity in complex analysis (trigonometry) [closed]

We know $\cos(z)^2 + \sin(z)^2 = 1, \forall z \in \mathbb C$. Prove that, on the other hand, $|\cos^2(z)| + |\sin^2(z)| > 1, \forall z \in \mathbb C$ with $\text{Im}(z) \not = 0$.
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3answers
83 views

I am working on proving or disproving $\cos^5(x)-\sin^5(x)=\cos(5x)$

True or false? $$\cos^5(x)-\sin^5(x)=\cos(5x)$$ for all real x. I have no idea how to prove or disprove this. I tried to expand $\cos(5x)$ using double angle formula but I wasn't sure how to go from ...
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1answer
55 views

In a Right Angled Triangle.

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$. Options are a+b b+c c+a a+b+c My approach. Radius of ...
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0answers
28 views

How can you calculate the module of a gear?

How can you calculate the module of a gear, knowing only the space between the teeth, the number of teeth and the contact angle? On the website "http://woodgears.ca/gear_cutting/template.html" Can ...
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2answers
48 views

Solving the equation: $3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)$

Solving the equation: $$3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)\tag{1}$$ I tried to write $(1)$ becomes $$\sqrt{3}\sin \left(\frac{\pi}{3}-x\right)=\sin \left(\frac{\pi}{3}+2x \right)$$ Now, ...
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2answers
24 views

Even and odd functions

Given $f(x)= \sqrt{1-\cos x}$. Period $0<x<2 \pi$ Is it a even function or a odd function? Whether the $f(x)$ has to be converted to square root of $2$ multiplied by $\sin(x/2)$.
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2answers
49 views

If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$

If $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = $ $\bf{My\; Try::}$ Given $\displaystyle ...
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1answer
25 views

Given an angle and opposite side in a triangle, find constraints on the side adjacent to the angle

In triangle ABC, we are given an angle A = 42 degrees, and its opposite side length, a = 38. i) For what values of adjacent side b such that we have one unique triangle ? ii) For what values b ...
2
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1answer
35 views

Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
0
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1answer
23 views

Law of sines solving for triangles

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers ...
2
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0answers
43 views

How to rewrite this trigonometric formula in terms of scalar and vector products between vectors?

Given two angles $\alpha$ and $\gamma$ such that $$ \cos(\alpha) = v\cdot v' $$ and $$\cos(\gamma) = f\cdot f',$$ what is the simplified form of $\cos(\alpha + \gamma)$ in terms of the vectors ...
0
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1answer
18 views

Two questions regarding the angle of reflection

I have two problems regarding the calculation of angles given certain values. In the first problem I need to calculate the angle X given that both angles Y are identical In the second problem I ...
2
votes
5answers
84 views

Evaluate $\lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $

I have the following limit to evaluate: $$ \displaystyle \lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $$ What's the trick here?