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

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2
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2answers
39 views

Finding the function of a sine graph that has both translation and transformation

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ...
1
vote
7answers
116 views

How to integrate $\int{\frac{1}{\cos(x)}}dx$ using the substitution $u=\tan\left(\frac{x}2\right)$?

So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$ to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$ which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$ But I'm not sure if this is ...
1
vote
2answers
40 views

Find solution for a trigonometric equation set

How can I solve $$\begin{cases} \sin^2(y_1) = \frac{1}{2}\sin^2(\frac{y_1+y_2}{2})\\ \sin^2(y_2) = \frac{1}{2}(\sin^2(\frac{y_1+y_2}{2})+1) \end{cases}$$ where $$\space y_1,y_2 \in [0,\frac{\pi}{2}] $$...
0
votes
1answer
30 views

Do we have $a\sin x+b\cos x=c\sin (x+d)$ with $a,b,c\in\mathbb{C}$ and $d\in\mathbb{R}$?

In showing that the set $$ W=\{f:\mathbb{R}\to\mathbb{C}\mid f(x)=a\sin(x+b),a\in\mathbb{C},b\in\mathbb{R}\} $$ forms a complex vector space, I need to answer the following question: Given $a,b\...
1
vote
2answers
75 views

Why is calculus normally taught after trigonometry (instead of more immediately after algebra)?

This question is a little meta. I hope I'm in the right place. In my experience the teaching of calculus is normally delayed until after learning basic trigonometry. Now that I've started learning ...
-1
votes
0answers
28 views

How to manually calculate Arctan and Arctan2

If I can calculate $ tan(\Theta) $ with this formula: $$ tan(\Theta) = {sin(\Theta) \over cos(\Theta)} $$ So if I've an angle of 45° I can do: $$ tan(\Theta) = {\sqrt2 \over 2} * {2 \over \sqrt2} = ...
0
votes
3answers
39 views

Trigonometric identity with powers of sine and cosine.

Prove that $3(\sin x -\cos x)^4 +4(\sin^6 x +\cos^6x) +6(\sin x +\cos x)^2 =13$ I cannot get it to an expression which does not have any trigonometric function I have tried this problem several ...
1
vote
3answers
67 views

Area of the triangle formed by circumcenter, incenter and orthocenter

Lets say we have $\triangle$$ABC$ having $O,I,H$ as its circumcenter, incenter and orthocenter. How can I go on finding the area of the $\triangle$$HOI$. I thought of doing the question using the ...
0
votes
2answers
21 views

Finding Radius Of Circle From Circle's Equation

For basic equations like:- $$ x^2 + y^2 = 4 $$ we can find out that the radius of the circle is 2. But for an equation like:- $$ x^2 + (y+1)^2 = 1 $$ What will be the radius of the circle?
0
votes
2answers
28 views

CAST Method for finding value of $\theta$.

Hi, So I'm having little trouble with the above question. So what I've done here is firstly found the $\sin^{-1}$ and $\cos^{-1}$ for these two. They sin = 30 degrees and cos = 150 degrees ...
1
vote
0answers
48 views

Trigonometric Roots of a Polynomial

After wondering on this question, I wondered how would you be able to find the roots of a polynomial, in the form $y=x^3+ax^2+bx+c$ if they are the sums of cosines? I'm wondering if it can, too, be ...
0
votes
0answers
56 views

How does one calculate supplementary trigonometric functions (e.g.: susin, sucosin, suversin)

Summary: I'm looking for data regarding the execution of trigonometric sufunctions (beyond their basis being the completion of supplementary angles of π, rather than complementary angles of π/2 [as is ...
1
vote
2answers
60 views

why are some trigonometric substitutions not explained in books?

In some books I find trigonometric substitution explained for only 3 cases in which one is: $$\sqrt{a^2-x^2}=acos\theta$$ but so far i've never seen this one $$\sqrt{a^2-x^2}=asin\theta$$ I believe ...
1
vote
1answer
33 views

Composition of trigonometric functions

$M$ denote the set of functions $y=\sin x, y=\arcsin x, y=\cos x, y=\arccos x, y=\tan x, y=\arctan x, y=\cot x, y=arccot x$. Is there a $n \in \mathbb N$ and such functions $f_1\in M, f_2\in M, ....
6
votes
1answer
67 views

Iterating a multiple of sine function makes a square wave

So, I found something curious playing around with a graphing calculator. Say we start with a function, $f_1(x) = 2\sin(x)$ and we define a constant, $C$,to be the positive fixed point for $f_1(x)$. ...
1
vote
3answers
103 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
1
vote
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
vote
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
votes
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
votes
2answers
228 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
votes
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=\...
10
votes
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 \...
1
vote
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=\...
-1
votes
1answer
94 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
vote
3answers
123 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\\ \...
1
vote
0answers
78 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 $ ,...
0
votes
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
votes
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\...
0
votes
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
votes
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
votes
5answers
77 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 ...
1
vote
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!
0
votes
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 $...
-1
votes
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?
-1
votes
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?
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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?
-1
votes
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)
-1
votes
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
137 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
votes
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
vote
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 ...