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
34 views

Prove that $\sin^2\frac{A}{2}\csc2A$, $\sin^2\frac{B}{2}\csc2B$, $\sin^2\frac{C}{2}\csc2C$ are in harmonic progression

If sides $a,b,c$ of $\triangle ABC$ are in arithmetic progression (AP), then prove that $$\sin^2\frac{A}{2}\csc2A, \quad\sin^2\frac{B}{2}\csc2B, \quad \sin^2\frac{C}{2}\csc2C$$ are in harmonic ...
1
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1answer
32 views

Prove that there are two values to the third side,one of which is $m$ times the other.

Let $1<m<3$. In $\triangle ABC$, if $2b=(m+1)a$ and $\cos A=\frac{1}{2}\sqrt{\frac{(m-1)(m+3)}{m}}$, prove that there are two values to the third side, one of which is $m$ times the other. ...
2
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4answers
64 views

Solve equation $8\cos x - 6\sin x = 5$ where $0 \le x \le 360$

Solve equation $8\cos x - 6\sin x = 5$ where $0 \le x \le 360$. I am not asked to use any form, so I am going to use $k\cos(x-\alpha)$. $8\cos x - 6\sin x = k\cos(x-\alpha)$ $$=k(\cos x\cos\alpha + ...
1
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0answers
27 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
0
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1answer
67 views

Trigonometric equation: $\sin^3x+\cos^3x+\sin^2x+\sin x+\cos x=2$.

Well, the title says it all. I've tried utilizing the fact that $\sin^3x+\cos^3x=(\sin x+\cos x)(1-\sin x\cos x)$ and then the equation becomes $(\sin x+\cos x)(2-\sin x\cos x)=1+\cos^2x$. Squaring ...
1
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1answer
45 views

Estimate trigonometric functions with complex argument

I would like to prove the following estimates $\vert \sin(z)\vert\leq \sinh(s)$ and $\vert \cos(z)\vert\leq \cosh ( s )$ ,where $z\in D_s(0)\subset\mathbb{C}$ and $D_s(0)$ denotes the disc with ...
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2answers
48 views

Why does $\cos(\pi x)$ approach $0^-$, as $x\to{\frac{1}2}^-$

In problem 220, I understand the rest of the solution except that I don't know why $\cos(\pi x)$ approaches to $0$ from the left, making the solution $-\infty$. I have seen a few problems like this, ...
0
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1answer
23 views

Rearrangement of Complex Sin and Cos

From my complex numbers course notes, there is the following derivation: The definitions of sin and cos I'm very comfortable with, but I cannot see how we get from the definition to the given ...
3
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2answers
97 views

Prove that $6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$ [closed]

Prove that: ​$$6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$$ Expression can be verified for different values of $A$ such as $\frac{\pi}{4},\frac{\pi}{2}$ etc. But ...
2
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0answers
51 views

Integral with irrational functions and polynomials

I thought this integral was simple, but it turns out it's not. $$\int \frac{xdx}{\left(1-x^3\right)\sqrt{1-x^2}}$$ I tried the substitution $1-x^3=\frac{1}{t}$, but that leaves me again with $x = ...
0
votes
3answers
48 views

Prove that $ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + … + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$

Prove that $$ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + ... + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$$ I have been trying to solve it step by like $ \tan^{-1}\frac{1}{3} + ...
0
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1answer
61 views

How to solve $\cos3\theta + 7\cos\theta=4$

If we expand $\cos 3\theta$ in the equation $\cos3\theta + 7\cos\theta=4$ then we end up with $\cos^3\theta+\cos\theta-1=0$ but is it possible to solve the first one directly?
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2answers
41 views

Find $\frac{\cos A}{p_1}+\frac{\cos B}{p_2}+\frac{\cos C}{p_3}=$

Let $p_1,p_2,p_3$ be the altitudes of $\triangle ABC$ from vertices $A,B,C$ respectively, $\Delta$ is the area of the triangle,$R$ is the circumradius of the triangle,then$\frac{\cos ...
0
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1answer
57 views

Does there exists an addition formula for $\arctan(2a)$? [closed]

I want to find an expression for $\arctan(2a)$ in terms of a sum of $\arctan$s, so $\arctan(2a) = c_1\cdot \arctan(b_1)+ c_2\cdot \arctan(b_2)$. Does there exists such a formula? I tried to derive it ...
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3answers
39 views

Solve limit $\lim_{x\to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2}}$

Hi I must solve the next limit $\infty - $\infty$ usibg L'Hopital and Taylor series. $\lim\limits_{x \to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2} }$ I tried to eliminate the root with ...
0
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0answers
30 views

If $\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}$, then how does $\theta$ compare to $\frac{\pi}{3}$?

I have $$\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}\qquad\qquad\text{where}\quad 0 < \epsilon < 1$$ What can be said about the value of $\theta$ ...
0
votes
1answer
30 views

Operator confusion when writing trigonometric expression in $k\sin(x-\alpha)$

If I want to rewrite the following 2 expressions in the form $k\sin(x-\alpha)$ a) $4\cos x - 3\sin x$ b) $\cos x + \sin x$ I am confused what happens to the operator in each case. In case a ...
1
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1answer
39 views

Representing a number as $a^2+db^2$ given $d$

Given integers $n$ and $d$, how can I find integers $a$ and $b$ (or show that they do not exist) such that $n=a^2+db^2$? If it helps, in my present application I know the factorizations of $n$ and ...
0
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2answers
38 views

How to find $\sin\left(-\frac{11\pi}{12}\right)$ from the unit circle?

I Want Above To Be In Unit Circle.... unit circle. like convert $\frac{\pi}{3}$ to $\frac{4\pi}{12}$ to make it same denom... and then do stuff.... i tried to do with $\frac{\pi}{2}$ (convert to ...
2
votes
2answers
22 views

Understanding simplifications of complex terms $\exp(-ik\pi/4)$

I read that $1\over{2}$$\pi$$i$($\exp[-3i\pi/4]+\exp[-9i\pi/4])$ = $1\over{2}$$\pi$$i$($-\exp[i\pi/4]+\exp[-i\pi/4])$ = $\pi$$\sin(\pi/4)$ = $\pi\sqrt{2}$ Can you help me to understand how we move ...
0
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1answer
27 views

Equating coefficients explanation for trigonometric identities

Equating coefficients for the trigonometric identities does not make sense for me and I will explain why: If I use the example Write $5\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$, where $0 ...
11
votes
6answers
292 views

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ ...
0
votes
2answers
21 views

Simplifying Inverse Trig Function

I'm trying to figure out how to simplify this expression but I'm not quite sure on how to approach this question. How should I approach this question? Any help is greatly appreciated! ...
3
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1answer
67 views

proof of a definite integral

Is there a proof of the definite integral function given below. I found the result by trial and error method and is stumped by the simplicity of the result. It seems to work for all the values for n > ...
0
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1answer
77 views

Prove that $R_1+R_2+R_3=R+r$,where $R$ is the circumradius and $r$ is the inradius of $\triangle ABC.$

Consider a triangle $DEF$,the pedal triangle of the triangle $ABC$ such that $A-F-B$ and $B-D-C$ are collinear.If $H$ is the incenter of $\triangle DEF$ and $R_1,R_2,R_3$ are the circumradii of the ...
0
votes
1answer
56 views

Prove that $2A+A_0=A_1+A_2+A_3$,where $A$ is the area of the triangle $ABC.$

If $A_0$ denotes the area of the triangle formed by joining the points of contact of the inscribed circle of the triangle $ABC$ and the sides of the triangle;$A_1,A_2,A_3$ are the corresponding areas ...
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3answers
41 views

Write $4\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$ where $0 \le \alpha \le 360$

Write $4\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$ where $0 \le \alpha \le 360$. $4\cos x - 3 \sin x = k\sin(x - \alpha)$ => $k(\sin x \cos \alpha - \cos x \sin \alpha)$ => $k\cos ...
0
votes
0answers
18 views

write $-10\cos x -7\sin x$ in the form $k\cos(x - \alpha)$ where $0 \le \alpha \le 360$.

write $-10\cos x -7\sin x$ in the form $k\cos(x - \alpha)$ where $0 \le \alpha \le 360$. $-10\cos x -7\sin x= k\cos(x - \alpha)$ => $k(\cos x \cos \alpha + \sin x \sin \alpha)$ => $k\cos \alpha ...
3
votes
3answers
87 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
1
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3answers
68 views

How to simplify $y = \sin(\frac{\arcsin(x)}{n}), n≥1$?

$y = \sin(\frac{\arcsin(x)}{n}), n≥1$ I know that: $\lim \limits_{x \to 0} \frac{x}{y} = n$ But I can't figure out what the curve of $x/y$ practically represents. Is there an obvious simple ...
0
votes
0answers
55 views

Sinusoidal function from $4$ knowns

I am trying to find out what sinusoidal function is represented by a minimum of $(0, -10)$ and a maximum of $(2, -4)$ I know that the period is $4$ and that the midpoint would be $(1, -7)$ but that ...
0
votes
3answers
41 views

How to prove that $\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ [duplicate]

Prove that $\displaystyle\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ for $n\in\mathbb{N},n>1$ I'm thinking at a demonstration by induction, as base case $n=2$ ...
3
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2answers
44 views

If $r$ is the inradius of $\triangle ABC$,then prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$

In acute angled triangle $ABC$,a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides.$r_b$ and $r_c$ are defined similarly.If $r$ is the inradius of ...
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2answers
15 views

Show that 3 sinusoidal phasors sum to zero

It is common in electrical power engineering to use three-phase circuits with sinusoidal currents out of phase with each other by 120 degrees. The benefit of this is that the currents sum to zero at ...
3
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4answers
75 views

Convergence and limit of $\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac jn\right)-\cos\left(\frac{j-1}n\right)\right\}$

The title says it all - I'm trying to find a way of proving the convergence and evaluating the limit of $a_n=\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac ...
3
votes
5answers
83 views

Solve the equation $1 / \cos x = \cos x + \sin x$

I'm having trouble solving the equation $$ \frac{1}{\cos x} = \cos x + \sin x $$ For what I understand I have to make the equation $= 0$ So I get $$ \frac{1}{\cos x} - \cos x -\sin x = 0 $$ Any ...
0
votes
1answer
16 views

Angle of rotation based on apparent change in size

I have a camera set up which views an object in 2D in front of it square on that's 309mm away, the object changes in size by 0.073mm. What I am trying to calculate is by what angle has the object to ...
0
votes
1answer
26 views

Solve trigonometric equation with maple

Let $b(q)$ be given by expression b(q) Find $\alpha$ such that b(q) = 0. By hand, we can find the solution $$ \alpha = ...
0
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2answers
34 views

Big O for a $\cos$ series

I have to show that $ \sum_1^N \cos(nx) = O(\frac 1{|x|}), [-\pi, \pi] $, x different from 0. I really don't know how to show that. I obviously know that $\cos(nx)$ is bounded by $1$, I know what ...
0
votes
3answers
69 views

$\tan(\arcsin x)$ if $x\gt0 $

I drew a triangle with $1$ as the hypotenuse and $x$ as the side opposite of sine. I'm getting a blank on how to get the bottom part of the triangle. Would the bottom side just be $\cos\theta$? i.e., ...
1
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2answers
30 views

Simplify 5|secθ| if θ=arcctan(x/5)

I got the answer wrong on a test so I was hoping if someone could help me correct it. I put the answer was 5secθ
0
votes
1answer
35 views

When combining wave functions what do the different forms signify [closed]

An expression like $4 \cos x + 3 \sin x$ can be written in the form: $k \cos (x + \alpha)$ $k \sin (x - \alpha)$ $k \sin (x + \alpha)$ I do not understand what these 3 forms represent or how they ...
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votes
1answer
20 views

Trigonometric Identity Similar to Tangent Addition Identity

I found the following trigonometric relationship in a paper and I can't seem to derive it. $1/2 arctan(\frac{2\sqrt{l}}{l-1}) = arctan(\sqrt{l})$ Where $l$ is a natural number. It seems similar ...
1
vote
2answers
43 views

Limit of tan function

This is a question from an old tutorial for a basic mathematical analysis module. Show that $$\lim\limits_{n\to \infty}\tan^n(\frac{\pi}{4}+\frac{1}{n}) = e^2$$ My tutor has already gone through ...
0
votes
1answer
39 views

Defining sine and cosine via ODE's

So I read in Simmons book on Differential Equations that via the equation y''+y=0 One can define s(x), c(x) as their solutions with some given initial conditions, that is s(0)=0 s'(0)=1 ; c(0)=1, ...
5
votes
2answers
1k views

Isolating y in sin(xy)=cos(xy)

Given $\sin(xy)=\cos(xy)$, what is the best way to isolate $y$? Since $\sin(\frac{\pi}{2}) = \cos(\frac{\pi}{2})$ it would seem intuitive to say that $xy=\frac{\pi}{2}$ and thus that ...
0
votes
1answer
30 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is ...
1
vote
2answers
38 views

Calculus I Integral

$\int(\sqrt x + \sec x \tan x)\, dx$ using $u$-substitution. So far as this class has been taught... the class knows that in order to use $u$-substitution, a $u$ and its derivative must be present ...
4
votes
1answer
45 views

If $\frac{\sin\alpha}{\sin\beta} \le 1+\epsilon$, then $\frac{\alpha}{\beta} \le 1+\sqrt\epsilon\;\;$ (for acute $\alpha$ and $\beta$)

Prove the following: For $0 \le \alpha,\beta \le \frac{\pi}{2}$, if $$\frac{\sin \alpha}{\sin \beta} \le 1+\epsilon$$ then $$\frac{\alpha}{\beta} \le 1+\sqrt\epsilon$$ This is from this ...
2
votes
2answers
33 views

Induction proof of the identity $\cos x+\cos(2x)+\cdots+\cos (nx) = \frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}$ [duplicate]

Prove that:$$\cos x+\cos(2x)+\cdots+\cos (nx)=\frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}.\ (1)$$ My ...