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
57 views

Evaluate $\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}$

Evaluate $$\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}.$$ The first thing i noticed was that $$\cos \frac{\pi}{7}=\frac{\zeta_{14}+\zeta_{14}^{-1}}{2},$$ where $\zeta_{14}=e^{2\pi i/14}$...
1
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2answers
35 views

Simplify $\frac{\cos(2x)}{\cot(x)-1}-\frac{\sin(2x)}{2}$

I am given this expression to simplify: $\frac{\cos(2x)}{\cot(x)-1}-\frac{\sin(2x)}{2}$ and I know the correct answer is $\sin^2(x)$ I was able to reduce the second fraction to a bit nicer $\frac{\...
0
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1answer
29 views

Sine and cosine solutions of a differential equation

I have to solve a differential equation with constant coefficient such as$$ay'''+by''+cy'+dy=f(x)$$ which has for a characteristic equation$$P_c(\lambda)=a\lambda^3+b\lambda^2+c\lambda+d=0$$First I ...
2
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2answers
226 views

How to integrate $\cos^2x$? [duplicate]

It seems like I am stuck on such a simple problem: How to I find the antiderivative of $\cos^2x$? I have tried partial integration, it doesn't seem to work (for me). Some help on how to integrate it ...
0
votes
1answer
70 views

A function in terms of trigonometric ratios [closed]

Given the function $$f(x) = \frac { 1-\sin2x+\cos2x }{ 2\cos2x }$$ find the value of $8\cdot f(11)f(43)$. I found the answer to be $4$. May I know if the answer is right?
0
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4answers
46 views

Solve Trigonometric Equation $\csc^2x + 2\cot x - 5 = 0$

I'm stuck on this question. I've tried looking at online trig calculators and I still don't understand what to do. Solve the following equation algebraically for $0 ≤ x ≤ 2\pi)$. $\csc^2x + 2\cot x -...
8
votes
2answers
243 views

Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern

Consider the sequence: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \...
4
votes
1answer
64 views

In $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot \frac{B}{2}$?

In a $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot(B/2)$? $\bf{My\; Try::}$ Here $A+B+C=\pi\;,$ Then $\tan A+\tan B+\tan C=\...
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3answers
447 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
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2answers
34 views

In $\triangle ABC$ with $A = \frac{\pi}{4}$, what is the range of $\tan B\tan C$?

In a $\triangle ABC\;,$ If $\displaystyle A=\frac{\pi}{4}\;,$ and $\tan B\cdot \tan C = p\;,$ Then range of $p$ $\bf{My\; Try::}$ For a $\triangle ABC\;, A+B+C=\pi.$ So we get $\displaystyle A+B=\...
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1answer
90 views

Why is De Moivre's theorem not generalised for $(\sin x+i\cos x)$?

A representation of the form $(\sin x+i\cos x)^n$ can be reduced as follows $$( \sin x + i \cos x )^n$$ $$( \cos (90-x) + i \sin(90-x) )^n$$ $$( \cos (90n - nx) + i \sin(90n - nx) )$$ Now for all ...
0
votes
1answer
36 views

Illumination of light on wall

A search light rotating from point $P$ is positioned $50$m from two walls that are opposite eachother. The walls is long enough to make the light almost invisible at each end. The illumination of the ...
1
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3answers
121 views

Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution

What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \...
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0answers
77 views

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $ 2 $ ,...
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0answers
34 views

Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
0
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0answers
13 views

Where did I make a mistake in this transformation of random variable?

The arctangent of a standard Cauchy random variable $Z\sim\text{Cauchy}(0,1)$ is uniformly distributed in $[-\frac{\pi}{2},\frac{\pi}{2}]$. The proof is straightforward: $$P(\arctan(Z)\leq t)=P(Z\...
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2answers
25 views

Trigonometry-circumcircle and sides of triangle.

How to prove that $$4R\sin A\sin B\sin C=a \cos A+b \cos B+c\cos C$$ where R is the radius of the circumcircle and $a$,$b$ and $c$ the respective sides of the triangle. I wrote $R=a/2\sin A$ and ...
3
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0answers
36 views

Using trig substitution, how do you solve an integral when the leading coefficient under the radical isn't 1?

I'm currently studying for my calculus exam, and i've run into a problem that has given me tons of issues. It's not one i've worked on before(or even seen before) so i'm worried if I see one on my ...
2
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5answers
76 views

How to find the coordinates where the altitude of a triangle intersects the base in 3 dimensions?

Assuming I know three completely random coordinates in 3d space that correspond with vertices of a triangle, how can I then find the point at which the altitude intersects the base? I know how to ...
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1answer
31 views

Finding out various sine values from its graph.

Question (and Answer): The answer is written in thin black, inc = increasing, dec = decreasing. Am I wrong anywhere? Thanks!
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0answers
32 views

Get content of transformation matrix from transformed vectors

In the following example: $$ \begin{pmatrix} X\\ Y\\ \end{pmatrix} = \begin{pmatrix} \cos\alpha & 1\\ 0 & \sin\beta\\ \end{pmatrix} \begin{pmatrix} A\\ B\\ \end{pmatrix} $$ $X$, $Y$, $A$ and $...
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3answers
89 views

Elementary trigonometry question [closed]

$\tan \theta$ = $n\tan \phi$ then the maximum value of $\tan ^ 2 (\theta - \phi )$ is? The answer is $\frac{(n-1)^2}{4n}$. How do I solve to get the required answer?
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1answer
60 views

Trigonometry question proof [closed]

If $0 < a$, $b < \pi$ , $\cos a + \cos b - \cos ( a + b) = 3/2 $, then show that $a = b= \pi/3$ I tried expanding $cos(a+b)$ but what to do next?
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1answer
39 views

Parameterise linear combination of cosines

How do I parameterise the following implicit surface? $$ \cos x + \cos y + \cos z = 0 $$ Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. ...
3
votes
1answer
45 views

$\sin(nx)$ espansion into $n$-th grade $\sin(x)$ polynomial

Maybe this is a well-know question, anyway I haven't found an exact duplicate. It is possible to express $\cos (nx)$ as a polynomial of degree $n$ in $\cos(x)$. As stated in this answer, it is ...
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1answer
29 views

Triangular sides

In a triangle the least angle is $45º$ and the tangents of the angle are in arithmetic progression. If its area is $27\text{cm}^2$, find the length of the sides. I tried to solve the problem in this ...
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0answers
49 views

How do I show the relationship between $I_n:=\int_{0}^{\pi}sin(x)^ndx$ and $I_n:=\frac{n-1}{n}I_{n-2}$

How do I show the relationship between $$I_n:=\int_{0}^{\pi}sin(x)^ndx$$ and $$I_n:=\frac{n-1}{n}I_{n-2}$$ for when $n \in \mathbb{N}$ and $n≥2$
0
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0answers
43 views

If $\sin x + \cos x +\tan x + \cot x +\sec x +\csc x=7$, then $\sin 2x$ is a root of $x^2 -44x + 36$ . [duplicate]

If $$\sin x + \cos x +\tan x + \cot x +\sec x +\csc x=7$$ then show that $\sin(2x)$ is a root of the equation $x^2 -44x + 36$. How do I solve this question?
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1answer
33 views

Sum of sinx+sin3x+sin5x+…sin(2n-1)x [duplicate]

Options are n/2 cosx- 1sin(nx)/2sinx . cos(n+2)x n/2.sinx-1/2sin(nx) n/2.cosx - cos(n+2)x sinx +sin(nx)
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0answers
17 views

how do project a point to a line?

I have a situation like this enter image description here I know the point n and the point s, and the distance between those two, and the tangent of the point n. I need to somehow project the ...
5
votes
1answer
132 views

If $\sin x + \csc x =2 \tan x$. Find value of $\cos^9x +\cot^9x +\sin^7x$

Problem: If $\sin x+\csc x=2\tan x$, Find value of $\cos^9x+\cot^9x+\sin^7x$ Solution: \begin{align*}&\sin x+\csc x=2\tan x \\ &\sin x+\frac{1}{\sin x}=2\frac{\sin x}{\cos x} \\ &\...
3
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1answer
49 views

Trigonometry + Geometry

In the given triangle we have this point $O$ such that $\angle OAB=\angle OBC=\angle OCA=\omega$ Hence prove that $\cot\omega=\cot A+\cot B+\cot C$. I figured out the RHS by using sine and cosine ...
2
votes
1answer
38 views

Need help in simplifying $(\arcsin x)^3 + (\arccos x)^3$

If $a<\frac{1}{32}$, then what's the number of solutions of $$(\arcsin x)^3 + (\arccos x)^3 = a \pi^3\quad ?$$ I don't know what this condition restricts it to. Finally I get a quadratic equation ...
1
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1answer
35 views

Trigonometric function- Equation

Let $n$ be a positive integer such that $$sin\frac{\pi}{2n}+cos\frac{\pi}{2n}=\frac{\sqrt{n}}{2}$$ then n lies is what interval? Its easy to see that $$|\frac{\sqrt{n}}{2}|\le \sqrt{2}$$ and hence ...
3
votes
4answers
64 views

Help to simplify $\arctan\left(\frac{\sqrt{1 + x^2} -1}{x}\right)$

Can someone help me simplify the argument of $\arctan$ in this problem ? $$\arctan\left(\frac{\sqrt{1 + x^2} -1}{x}\right)$$
2
votes
1answer
35 views

Trigonometric equation solutions.

For $0<\theta<\pi/6$ all the values of the expression $\tan^23\theta \cos^2\theta-4\tan3\theta \sin2\theta+16\sin^2\theta$ lies in what interval. I actually took $\sin^2\theta$ common out of ...
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1answer
34 views

How do I prove that $\cosh(x)$ is bijective for the interval $[0,\infty[$?

I've found out that $\cosh´(x) > 0$ when $x>0$ by graphing. But how do I show that mathematically? Same thing for when $x$ approaches $\infty$, $\cosh(x)$ approaches $\infty$, but how do I show ...
2
votes
3answers
136 views

How do I find all solutions to $\cos(x)^4-\sin(x)^4 = 1$

The interval, when graphing this function, is that the equation is true every $x \in \{0,\pi,2\pi,3\pi\dots\}$ but how do I prove that this is the only solution? My assumption is that the solution $\...
-1
votes
4answers
49 views

Show that $\sum_{k=1}^{n-1} \sin(\frac{2 k \pi}{n})$ is equal to $0$ [closed]

Proof of $ \sum_{k=1}^{n-1} \sin(\frac{2 k \pi}{n})= 0 $. How can I prove this statement without dividing into cases of odd $n$ and even $n$?
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7answers
80 views

Why is $\sin(\arccos(x))$ a semicircle with radius 1?

It was unexpected to see that from looking at the equation. Is there an intuitive explanation for why it's a perfect semicircle?
1
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2answers
52 views

Cannot understand solutions manual on integral $\int\frac{1}{x\sqrt{x^2-4}}\,dx$

I work on $$ \int\frac{1}{x\sqrt{x^2-4}}\,dx. $$ I set $u = x/2$ just like is is in the solution but I don't understand how it becomes $2\,du = 1/2\,dx$. Wouldn't it just be $2\,du = dx$ ?
0
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1answer
59 views

How to find the value of $\sin24^\circ$ [closed]

Is there any method to do it by hand quickly? i want to show the angle $72$ can be trisected by compass and ruler. so i need to find the way to calculate it... help please!
0
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1answer
46 views

How to find $\tan{\theta}$ when $\theta=\arctan⁡{(8/3)}$

Basically I'm trying to find the exact value of $\tan{\theta}$ when $\theta = \arctan{(8/3)}$. I'm not exactly sure where to start. I know that $\arctan$ is the inverse of $\tan$, but I can't really ...
0
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2answers
68 views

$\sin 2x - \tan 2x = -\sin 2x\tan 2x$ trigonometric identity proof

I need to prove $$\sin 2x - \tan 2x = -\sin 2x\tan 2x$$ I tried simplifying $$ \sin 2x = 2\sin x\cos x;\quad \tan 2x = \frac{2\tan x}{1-\tan^2x}. $$ But it's so long and complicated that I ...
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2answers
54 views

Proving $\cos A \cdot \cos 2 A \cdot \cos 4 A \cdots \cos 2^{n-1} A = \frac{\sin 2^n A}{2^n \sin A}$

Just a bit of background on the question: When proving: $$\cos\frac{\pi}{15}\cdot \cos\frac{2\pi}{15} \cdot \cos\frac{3\pi}{15}\cdot \cos\frac{4\pi}{15} \cdot \cos\frac{5\pi}{15} \cdot \cos\frac{6\pi}...
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2answers
55 views

Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$

The question is: Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$, where $x$ is real. Give the solutions in exact form. What I have done $$\left[\ln(\sin^{-...
2
votes
4answers
76 views

$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
2
votes
0answers
70 views

Can you solve a trig equation with a variable both inside a trig function and outside one?

I have the equation: $$d=\frac{t}{2}-\frac{sin(t)}{4}$$ I'm completely failing at how to get this in terms of $t$ I only care about it for values of $0<t<2\pi $ I've seen the graph so I know ...
0
votes
1answer
9 views

How is the graph of $cot(x)$ valid for negative values of $x$?

In the graph, consider a point between $\frac{-\pi}{2}$ and $-\pi$. We know that $cot(x) = \frac{cos(x)}{sin(x)}$. For negative values of $x$, i.e., $cos(-x)$ is always positive and $sin(-x)$ is ...
0
votes
3answers
74 views

Solve the equation on the interval $0 \le \theta\le2\pi $

I have this Final Math Exam Review, for Math Analysis/Trig = Pre-calculus. So I stumbled upon my review and this section arose where it told me to Solve the equation on the interval $0 \le \theta ...