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

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0
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1answer
31 views

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? [on hold]

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? I need help on this one. I don't know how to approach this problem.
1
vote
1answer
22 views

Compound angle formula

I understand how to use the compound angle formula when solving $\sin(\pi/12)$. However I dont understand how I can use a compound angle formula to show $$\arctan(3)-\arctan(1/2)=\pi/4$$ Thankyou Any ...
1
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0answers
31 views

Mentally approximating an inverse sine?

Is there a method to approximate the inverse of a sine function in ones head? I know one can approximate the inverse of a cosine with the following equation: $\cos^{-1}(x) = ...
2
votes
3answers
28 views

how to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$

How to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$ ? I tried using taylor expansion: $$((x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^5))(\frac{1}{x}-\frac{1}{6x^3}+\frac{1}{120x^5}+O(x^{-5})))^2$$ but ...
2
votes
2answers
69 views

Compute $(\sin4^\circ)^2 +(\sin8^\circ)^2+(\sin12^\circ)^2+\cdots+(\sin176^\circ)^2$

Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu. I know that $\sin4^\circ=\sin176^\circ$, ...
0
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0answers
12 views

Bearing of a line

Please help me to find the bearing. I've attached the image.I've tried by drawing a North a B and C, and D but couldn't figure out the angle that give me bearing of C from D. Thanks for all your ...
6
votes
4answers
895 views

What is meant by a 'pure' wave?

What is meant by a 'pure' wave? I know it might sound like a basic question, but I've never been taught this. I saw that a sine wave is a pure wave. I tried Googling what a pure wave is, but all I ...
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4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
0
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1answer
38 views

Plane rotation: range of angles to produce all posible x'y' planes

Given an $(x, y, z)$ system I create a new system $(x', y', z')$ by applying two rotations $\theta$ and $\phi$. In the new system the $(x',y')$ plane, i.e.: the $z'=0$ plane, can be written as: $$ ...
0
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1answer
24 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
2
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3answers
63 views

$\sin2(x) - \tan(x) = 0$ , solve for $-180\le x\le 180$

I have been unable to solve the following question, If $$\sin(2x) - \tan(x) = 0$$ Find $x$ , $-\pi\le x\le \pi$ So far my workings have been Use following identity: $$\sin(2x) = ...
1
vote
1answer
49 views

Find the minimum value of $\frac {\sin \alpha \sin \beta}{\sin^2 \frac {\gamma}{2}}+…$

Find the minimum value of the following expression $$ \frac {\sin \alpha \sin \beta}{\sin^2 \frac {\gamma}{2}}+ \frac {\sin \beta \sin \gamma}{\sin^2 \frac {\alpha}{2}}+ \frac {\sin \gamma \sin ...
1
vote
0answers
36 views

Proving cosines product identity [duplicate]

Prove that $\cos\left({\pi \over 11}\right)\cdot\cos\left({2\pi \over 11}\right)\cdot\cos\left({3\pi \over 11}\right)\cdot\cos\left({4\pi \over 11}\right)\cdot\cos\left({5\pi \over 11}\right)={1 ...
2
votes
1answer
54 views

Joining the Midpoints of the Sides of a Quadrilateral

$ABCD$ is a quadrilateral. $P$, $Q$ and $R$ are the midpoints of $AB$, $BC$ and $CD$ respectively. If $PQ = 3$, $QR = 4$ and $PR = 5$; find the area of $ABCD$. Since, $5^2 = 3^2+4^2$, So, ...
1
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0answers
29 views

Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
1
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1answer
42 views

How to calculate $\lim_{x \rightarrow 0} \frac{\int_0^{G(x)} \arctan(s+2s^2) ds}{x^2}$ based on the following assumption?

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. To start this question, we need to integrate $\arctan(s+2s^2)$ but how do you do that? Then, what do we do ...
1
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0answers
28 views

How to prove that there a constant $C$ such that $\arcsin \frac{1-x}{1+x}+2\arctan\sqrt{x}=C$? [duplicate]

How to prove that there a constant $C$ such that $\arcsin \frac{1-x}{1+x}+2\arctan\sqrt{x}=C$? I have no idea using which theorem to prove. Could someone show me how to start the problem?
0
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0answers
22 views

Find the rotation angles of a 2-D rotation matrix between two vectors

I am trying to solve the following to find $\theta$. I was given two vectors $\begin{bmatrix}-4.95 \\ -.7\end{bmatrix}$ and $\begin{bmatrix}3 \\ 4 \end{bmatrix}$ and asked to compute the rotation ...
1
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2answers
21 views

Integral of $-4\sin(2t - (pi/2)) $ weird behavior on wolfram alpha

I'm confused by what Wolfram Alpha is doing with my function: $$-4\sin{(2t - (\pi/2))}$$ on why the it gets replaced by $$4\cos{(2t)}$$. Is it equal? Link: See behavior here
5
votes
1answer
119 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!
1
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2answers
35 views

Problem integrating (substitution)

can you help me identify the mistake I'm making while integrating? Question: $$\int{\frac{2dx}{x\sqrt{4x^2-1}}}, x>\frac{1}{2}$$ my solution ...
0
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1answer
31 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
0
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0answers
21 views

Find the third angle

Three planes are orthogonal to each other. I have found the rotation about the 2 axis (x and y). Is there a way to find the third angle around z provided the angles around x(70 degrees) and y(-1 ...
0
votes
2answers
40 views

Elementary integral for square roots of trig functions?

What's an easy way to calculate something like $\int \sqrt{1+\cos x} \text{ d}x$?
4
votes
1answer
63 views

Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]

The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta ...
1
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1answer
50 views

Express this expression in terms of $x_1$ and $x_2$?

We have the following definition: $$ x_1=A \cos(\omega t_1 +\phi) \\ x_2= A \cos(\omega t_2 +\phi) $$ The expression we want to simplify is: $$ S=A^2\omega \left[\sin 2(\omega t_2+\phi)-\sin 2(\omega ...
0
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0answers
26 views

Expand a $\arctan(x)$ function [duplicate]

I want to expand a function $\arctan(x)$ as a polynomial form. I know that I can use Taylor expansion in the case of x <1. But in my case, the x can be pretty large. Is there any way to expand or ...
2
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0answers
27 views

Fundamental Theorem of Algebra for Trigonometry

The Fundamental Theorem of Algebra states that "Any polynomial of degree n ... has n roots." Is there anything analogous for trigonometric equations? I've been solving some trigonometric equations, ...
1
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0answers
31 views

Number of Distinct Roots of Trig Equation [on hold]

How do we find the number of distinct roots of a trigonometric equation? For example: Consider the trigonometric equation. $$2\cos(2x+\frac\pi3) = \sqrt2,\quad x \in [-\pi, \pi]$$ This equation has ...
1
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1answer
89 views

Proving $\int_0^{\pi } f(x) \, \mathrm{d}x = n\pi$

I've been asked to show $$ \displaystyle \int_{0}^{\pi} \dfrac{2(1+\cos x) - \cos((n-1)x) - \cos((n+1)x) - 2\cos nx}{1-\cos 2x} \ dx = n\pi $$ The integrand simplifies nicely to $$\frac{\cos nx - ...
2
votes
5answers
87 views

What angle is $\sin^{-1}(3/2)$?

So i have this trigonometric equations: $$2\cos^2(x)+4\sin(x)+\cos(2x)=0$$ I have rewritten the expression and came up with $$(2\sin(x)-3)(1+2\sin(x))=0$$ Then i split the equation in two and got ...
3
votes
2answers
47 views

Representation of roots of unity.

How to represent solutions of $\sqrt[26]{1}$ with solutions of $\sqrt[26]{-1}$? I know that $$w_{k}=\cos\left(\frac{0+2k\pi}{26}\right)+i\sin\left(\frac{0+2k\pi}{26}\right), \; \; ...
1
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2answers
88 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
0
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1answer
30 views

Let $f(x)=p\cos x+q\sin x,|p|+|q|\ne0$ and $|f(x)|\leq 1$.Let $\alpha,\beta$ be the roots of the equation $f(x)=1,|\alpha-\beta|=k\pi,k\in R,$

Let $f(x)=p\cos x+q\sin x,|p|+|q|\ne0$ where $p,q\in R$ and $|f(x)|\leq 1$.Let $\alpha,\beta$ be the roots of the equation $f(x)=1,|\alpha-\beta|=k\pi,k\in R,$then the find the possible values of $k.$ ...
0
votes
0answers
14 views

range of $\phi$ in given trigonometric equation

Let $\theta,\phi\in[0,2π]$ be such that $2\cos(\theta)(1-\sin(\phi)=\sin^2\theta(tan(\theta/2)+\cot(\theta/2))\cos\phi-1,\tan(2π-\theta)>0,-1<\sin(\theta)<\sqrt{3}/2$ then $\phi$ cannot ...
1
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4answers
45 views

How to prove this? $\lim_{h \to 0} \frac{\sin(\theta+h)-\sin(\theta)}{\cos(\theta+h)-\cos(\theta)} = -\frac{1}{\tan(\theta)}$ [closed]

How to prove this? $$\lim_{h \to 0} \frac{\sin(\theta+h)-\sin(\theta)}{\cos(\theta+h)-\cos(\theta)} = -\frac{1}{\tan(\theta)}$$
3
votes
1answer
33 views

Is is possible to simplify the expression $\arctan(y)-\arctan(x)=c$

Is is possible to simplify the expression $\arctan(y)-\arctan(x)=c$. I tried writing the expression in the form $\frac{\arcsin(y)}{\arccos(y)}-\frac{\arcsin(x)}{\arccos(x)}=c$ but it does not lead to ...
2
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1answer
38 views

Integral of a trig function divided by the square root of a polynomial: $\int_a^b\frac{\sin x}{\sqrt{(x-a)(b-x)}}dx$?

I was trying to help some physics students with an integral on their homework and they've presented me with something that has me stumped. The integral they are working on is: $$\int_a^b\frac{\sin ...
3
votes
2answers
62 views

How do I prove that sin is not defined implicitly by an algebraic equation?

How do I prove that sin is not defined implicitly by an algebraic equation? In essence, there does not exist rational functions $f_0,\ldots,f_{n-1}$ that satisfies ...
1
vote
2answers
27 views

number of distinct values of x …

Number of distinct values of $x$ where $\sin x=\frac x2$ where $x\in [0,\pi]$ degrees ? Is it only at $0$ degrees? I would like to know the answer with a suitable explanation. That would be very ...
0
votes
1answer
29 views

find angle given point the trajcetory passes through and inital velocity

I'm currently studying M1 for A level maths and we've derived the equation to prove that the trajectory is a parabola. $y=x\tan\theta - \sec^2\theta \dfrac{gx^2}{2u^2}$ I am curious as to how to ...
0
votes
0answers
15 views

PARABOLA : How to Get Equation (general form) with the Given Vertex (-2,-1) Latus Rectum 12 & Opens to the left [closed]

How to Get Equation (general form) with the Given Vertex (-2,-1) Latus Rectum 12 & Opens to the left. Help me Please
2
votes
1answer
39 views

${2\pi \over 3} = 2u + \sin {2u}$ (intersections of circles)

So, I was browsing the internet today, when I saw an interesting problem: Two circles, each with radii of one, are intersecting. If the area enclosed by the intersection of the two circles is equal to ...
1
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1answer
24 views

How to find the most general value of $\cos(A-B) = 1/2$ and $\sin(A+B) =1/2$?

I'm learning Trigonometry right now with myself and at current about general solution. I have a question which is confusing me from some time. The question is - $If \cos(A-B) = 1/2$ and $\sin(A+B) ...
1
vote
2answers
17 views

Manipulating Complex Exponentials

I am trying to show that $$ sin(5\pi t) = \frac 12 e^{-j \frac {\pi}2}e^{j5\pi t} - \frac 12 e^{-j \frac {\pi}2}e^{-j5\pi t} $$ I am aware that $$ sin(\theta) = \frac {e^{j\theta} - ...
0
votes
1answer
54 views

Put $A\cos(x) + B\sin(x)$ into form : $A\sin(x+ \theta)$ [closed]

The task is to manipulate $$\cos(x) + \sqrt{3}\sin(x)$$ into the form $$A\sin(x+ \theta)$$ My question is : why is $\pi$ in the numerator and denominator both divided by $6$? I am familiar with ...
1
vote
2answers
64 views

Common factor in $2\sin(x)\cos(x) + \sin(x) = 0$ [closed]

I am stuck on part of a question : The 1st line of work is : $$2\sin(x)\cos(x) +\sin(x) = 0$$ The next line is : $$\sin(x) \cdot (\sin(x)\cos(x) +1)$$ I see that $$\sin(x) \cdot 1 $$ gives ...
0
votes
1answer
31 views

Can a denominator of fraction be multiplied by -1 without affecting the numerator ? and if so why?

I have been presented with a solution for solving trigonometric identities. However I would like to see further proof that one of the lines of work are valid. \begin{align*} \frac{2\sin x\cos x}{1 + ...
1
vote
2answers
60 views

Solve $f'(x)+\int_{\pi/4}^{x}f(t)dt=0$

So far what I have is that I set $F(x)=\int_{\pi/4}^{x}f(t)\,dt$ so that the equation satisfies $F(x)+F''(x)=0$, and I have $F(0)=-\int_{0}^{\pi/4}f(t)\,dt$ and $F'(0)=f(0)$. From there, I was able ...
1
vote
1answer
19 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...