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
27 views

finding unkowns in Trignometry equation

If $\tan(\theta + i \pi) = \tan(\alpha) + i \sec(\alpha)$, then what is the relation between $\theta$ and $\alpha$ $\theta = \frac{n\pi}{2}+\alpha$ $\theta = \frac{1}{2}(n\pi+\frac{\pi}{2}+\alpha)$ ...
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2answers
36 views

How to get A,B and C given XYZ?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{a+b+c}{3}$$ in other words How do i get $a$, $b$, and $c$ on the left ...
3
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1answer
50 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
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0answers
20 views

Get Vector From Angle In $3D$ Space

I have a camera angle in $x,y,z$ with values between $0$ and $360$, I'm trying to compute this into an 'aim vector' which would have values between $0,0,0$ and $1,1,1$ depending on what the angles ...
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4answers
32 views

Negative inverse trig functions

I was practising my integration earlier and for one question I got $-\arccos(x - 2)$, while the book has $\arccos(2 - x)$. The answers seems so close and I was wondering (a) am I right? But more ...
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1answer
33 views

How prove $ \frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4}) $?

For any $x,y\in[0,\frac{\pi}{2}]$ , how prove the inequality $\frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4})$?
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1answer
41 views

Summation and product over $k$ with $k$ prime to $n$ sought

I just come to a standstill with the following two formulas. If $$E_n=\lbrace k\mid 1\le k\le n\ \&\ (k,n)=1\rbrace$$ then I hope for a closed formula $f(n)$ for those $$\sum_{E_n}k$$ ...
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1answer
22 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
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2answers
45 views

Is this trigonometric identity correct?I tried by using compound formulae for cos8A but …

Prove that: $\sqrt{2+\sqrt{2+\sqrt{2\cos(8A)}}}=2\cos A$ Is this trigonometric identity correct?, I tried by using compound formulae for cos8A but couldn’t get it to RHS, give your try and if ...
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1answer
29 views

Given a trigonometric function of an angle, how to find half of the same angle.

Let us assume that we are given an unknown angle $x$. We dont know if it is negative or positive, we only know its in the 4th quadrant. Also, we're given a trigonometric function of $x$ (for example ...
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2answers
48 views

Need help with trigonometry [closed]

$$\sin\theta-\sin\theta\cdot\cos^2\theta=\sin^3\theta$$ I have to prove them. I dont know how.
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1answer
33 views

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$?

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$ for $x \in (0,\pi)$?
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1answer
54 views

Limits of trigonometric functions as $x$ approaches to a constant $a$

$$\lim_{x \to a} \sin{x} = ?$$ $$\lim_{x \to a} \cos{x} = ?$$ What are some ways of computing these limits? I'd appreciate if you could post different methods as well.
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1answer
45 views

Acute plane triangle with two sides coinciding with a right triangle

Below are the two questions: 1) If $T$ is a plane triangle with $x, y < z$ such that $x^{2} + y^{2} > z^{2}$ as side lengths, is $T$ necessarily acute? 2) Is an acute plane triangle $T$ such ...
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0answers
13 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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1answer
33 views

How to find limits involving trigonometric functions as $x\to 0$?

Problem: find the limit as $x\rightarrow 0$ of $\dfrac{\tan(3x)}{\sin(2x)}$ $\dfrac{(\sin(2x) + 3)}{(\cos(7x)-8)}$ Note I am able to solve the first one using l'Hopitals, but I really want to be ...
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1answer
25 views

Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
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1answer
38 views

Write the equation that gives you your velocity away from planet X.

Earlier, I posed the following problem: Suppose you are travelling through a planetary system. From your space ship you view planet X. The planet is known to be spherical. As you view planet X, ...
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0answers
36 views

trigonometric sums $ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$

can someone help me to prove that: $ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$ ...
2
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2answers
96 views

Does Niven's theorem apply to cosine function?

Niven's theorem says that if $\theta$ is a rational multiple of $\pi$ and $\sin \theta$ is rational then $\sin \theta = 0, -\frac12, \frac12, -1, 1$. But is this theorem applicable to cosine function? ...
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2answers
25 views

Substitution of rational values of cosine function

If $x$ and $y$ are integers $>0$ and $0 < \theta < \pi/2$ is a real number such that $y = x\cos \theta,$ can one conclude that $$\frac{y}{x} = \frac{1}{2}?$$ Under what conditions $\cos ...
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2answers
35 views

Any fast way of remebering trigonometric ratios of compound angles

This is a list of equations in my book: $$\sin(90-\theta)=\cos\theta$$ $$\cos(90-\theta)= \sin\theta$$ $$\tan(90-\theta)=\cot\theta$$ or this... $$\sin(180-\theta)=\sin\theta$$ ...
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2answers
38 views

Taylor series convergence for sin x

a. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}<\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!},$ b. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots-\frac{x^{4k-1}}{(4k-1)!}<\sin ...
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2answers
132 views

For small $\theta$, $\sin \theta$ is just $\theta$, WHY? [duplicate]

When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?
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2answers
57 views

Limit of $ (2^n)\sin(n) $ as $n$ goes to infinity

I'm stuck with the limit $\lim_{n\to\infty} (2^n)\sin(n) $. I've been trying the squeeze theorem but it doesn't seem to work. I can't think of a second way to tackle the problem. Any push in the right ...
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2answers
39 views

What is the radius of a planet, given some basic information?

Suppose you are travelling through a planetary system. From your space ship you view planet X. The planet is known to be spherical. As you view planet X, the angle from the centre of the planet to ...
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2answers
21 views

Circle Line segment intersection

I have a circle with radius r and center $(c_x, c_y)$. I have a line segment $(x_1, y_1)$ and $(x_2, y_2)$ given $(x_2, y_2)$ is always a point inside the circle. I am trying to find the ...
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0answers
35 views

Find the ratio of sides in a triangle, if they form an arithmetic progression and the largest angle is 90 degrees more than the smallest [duplicate]

The three sides of a triangle form an arithmetic progression. Given that the largest angle is 90 degrees more than the smallest angle, show that the sides are in the following ratio $$\sqrt{7}\, -1 : ...
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0answers
21 views

Calculating rotated relative positions on 2D plane

For a game project, I need to calculate positions of items on a 2D plane relative to the camera. Camera can be rotated, and it's coordinates refer to it's center. In the attached images, ...
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1answer
42 views

Find distance, given angles of elevation

Write an equation giving the distance d between the plane and observation post in terms of $\theta$ and $\phi$. Is this correct? when using the Law of Sines answer: $a/\sin\theta = c/\sin C$ ...
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1answer
55 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
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5answers
72 views

Solving an equation with arctan and arcsin

I am trying to do what I think a problem with a simple answer. Here are the two equations I have resolved the problem down to: $$\angle A = \arctan \frac{28}{x}$$ and $$\angle A = \arcsin ...
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3answers
71 views

How do I solve the trigonometric equation $\sec^3x - 2 \tan^2 x = 2$? [closed]

A friend asked to me how could she resolve this equation, but I don't know how to resolve it?? Could you help me?. The equation is : $\sec^3x - 2 \tan^2 x = 2$ Note: She told me that I can use ...
1
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1answer
27 views

Specific or Universal methods for proving trig identities

So this might seem like an elementary question to everyone here, but does anyone have any direction or method to follow when proving trig identities? For example, when proving LHS = RHS, sometimes the ...
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3answers
74 views

Is my textbook wrong? Simple trig equation

For the trigonometric equation: $$2\sin(3\theta)=-1$$ defined by $0 < \theta < 360$ I acquired the solutions of $75, 165, 255, 345, 435, 525$ Would anyone care to confirm if I am correct ...
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3answers
61 views

How to evaluate trigonometric function sine?

$$g = 357.528˚ + 0.9856008˚ n$$ $$\lambda = L + 1.915˚ \sin g + 0.020˚ \sin(2g)$$ My calculator is the Python Interpreter. How can I calculate this? What will the resulting lambda be? Degrees? Ratio? ...
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1answer
21 views

A problem which requires Goniometry, Latitude and Longitude.

Two observers are situated at the equator, but in different longitudes, respectively -43°55'48" and 45°7'12". In the same instant, the first observer sees the moon at its zenit, while the second ...
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3answers
51 views

What is the area of the parking lot?

Geometry A parking lot has the shape of a parallelogram. The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70° What is the area of the parking lot? ...
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0answers
72 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
2
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1answer
73 views

What is the distance from the boat to the shoreline? [closed]

A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see ...
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4answers
82 views

Simplifying a trigonometric identity

Simplify $1 + \tan^2x$ My attempt: $$\begin{align}1 + \tan^2x&\\ &= \frac{1}{1} + \frac{\sin^2x}{\cos^2x}\\ &= \frac{1(\cos^2x)}{1(\cos^2x)} +\frac{\sin^2x}{\cos^2x}\\ ...
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4answers
142 views

Manipulating a trigonometric equation involving $\tan^2(3\theta)$ [closed]

If $\tan^23\theta = 1$, how do I manipulate the equation so I can make $\tan\theta$ the subject? I forgot how to do these since it has been a long time. I tried searching before posting. My answer is ...
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1answer
24 views

Trig equations involving recip. ratios

what is the procedure for solving trig equations that involve a recip. ratio ? For example, if $\csc\theta = -1$, what exactly do I do, if I know how to solve ones that involve just sine, cosine and ...
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3answers
41 views

For $0 < \theta < 360^\circ,\,$ solve $\,\cos\theta = -\frac{\sqrt3}{2}$.

For $0 < \theta < 360^\circ,\,$ solve $\cos\theta = -\dfrac{\sqrt3}{2}$. I got 120 and 210 degrees. But this doesn't match the textbook's solutions. Did I do something wrong?
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0answers
41 views

$\frac{\pi }{2}-\tan^{-1}\left(x\right)$ = $\tan^{-1}\left(\frac{1}{x}\right)$?

It's said that $$\frac{\pi }{2}-\tan^{-1}\left(x\right)=\tan^{-1}\left(\frac{1}{x}\right)\tag{1}$$ Well what about this. Lets say $$\frac{\pi }{2}-\tan^{-1}x=\frac{\left(\pi ...
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0answers
31 views

Translate vertical movement into radial movement?

I've tried all sorts of things, but I'm no mathematician and I've conceded defeat. So I come here for help. I don't know if I really worded the question correctly since I don't even know what I should ...
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1answer
26 views

Trigonometry in projectile motion

I initially posted this question on Physics SE but got no responses probably because it's more related to maths than physics. A plane surface makes an angle $\bf X$ with the horizontal. From the ...
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1answer
29 views

Not understanding arc midpoint computation

I'm trying to find the midpoint of an arc, so I found this page which describes the midpoint formula. I pasted the formula & description from the site below. Let origin-centered arc of radius ...
0
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1answer
20 views

Plotting three variables on an XY plane, involves distance formula.

I have 3 dynamic constants with values of 0 to 1. Lets label them A,B and C. I want to be able to plot them on a 2 dimensional cartesian plane. so given all three constants I will be able to find the ...
1
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2answers
32 views

Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$

Evaluate Left And Right Limits Of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ At $0$ The graph of $f(x)=\frac{x}{\sqrt{1-\cos2x}}$ appears to have a jump discontinuity at $0$ and I want to calculate the left ...