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
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
vote
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 ...
0
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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
votes
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
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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 ...
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1answer
58 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
40 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
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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
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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 ...
12
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7answers
342 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 $2\pi$ ...
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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
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1answer
57 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
42 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
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0answers
19 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
4answers
90 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
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0answers
56 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
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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
votes
4answers
78 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
84 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 = ...
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2answers
35 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 ...
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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., ...
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2answers
31 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
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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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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 ...
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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 ...
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1answer
42 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, ...
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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
39 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
34 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 ...
1
vote
2answers
59 views

Show that the trigonometric integral $\frac{\pi}{2}-\sin (x) \ll (1+x)^{-1}$.

How can I show that for non-negative $x$ we have $$ \int_x^{\infty} \frac{\sin(t)}{t} dt \ll (1+x)^{-1}. $$ I think this should be an easy task but still I'm unable to solve it. I tried to estimate ...
-1
votes
0answers
14 views

Maxima and minima of a specific function

For $0\leq x\leq360$ ($x$ in degrees), hat are the maxima and minima of $f(x)=\frac{sin(x)+cos(x)}{cos(x)^2+1}$ in surd form?
-1
votes
1answer
28 views

Show that $\sin \theta = \frac{\cos(0.5\theta)-\cos(1.5\theta)}{2\sin (0.5\theta)}$ [closed]

How to show that $$\sin \theta = \frac{\cos(0.5\theta)-\cos(1.5\theta)}{2\sin (0.5\theta)}?$$
2
votes
0answers
24 views

Perpendicular Vectors in 3D space

I was wondering whether given two Vector's v0 and v1 whether I could find the two perpendicular vectors at a given distance, d, from v1, perpendicular to the v0/v1 line. I know that v0 and v1 will ...