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

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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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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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35 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
39 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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189 views
+100

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

Can someone help me to prove that: $$ \sum_{k=0}^{n-1}\sin\frac{2k^2\pi}{n}=\frac{\sqrt{n}}{2}\left(\cos\frac{n\pi}{2}-\sin\frac{n\pi}{2}+1\right)$$ ...
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2answers
98 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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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
36 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
40 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
158 views

Why is $\sin \theta$ just $\theta$ for a small $\theta$? [duplicate]

When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?
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2answers
58 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
23 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
36 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
59 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
73 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
73 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 ...
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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
76 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
62 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
75 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 ...
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1answer
78 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
145 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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43 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
33 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
28 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
30 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 ...
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1answer
21 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 ...
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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 ...
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143 views

Prove that $\sin(\sqrt x)$ not periodic

$\sin\sqrt x$ is not a periodic function. How can one prove this?
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0answers
24 views

General solution of $\sin x$ and $\tan x$ when equal to $-1$.

Why don't we write the general solution of $\sin\left(x\right)=-1$ as $x=2n\pi +\left(-\frac{\pi }{2}\right)$ but rather $x=2n\pi +\frac{3\pi }{2}$. And if this is correct why do we write the ...
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3answers
60 views

Find $\cos(x-y)$ if $\cos x + \cos y =2$

I had a question in my Math mcq test. If $\cos x + \cos y = 2$ find the value of $\cos(x-y)$. I couldn't get a way to calculate the value. So I just substituted $x = y = 0$. (It seemed obvious ...
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30 views

In what quadrant are the reciprocal ratios of tan, cos and sine positive?

If tan is negative in the third quadrant, would its reciprocal ratio, cot, be positive?
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25 views

Evaluate $3\tan30^\circ$, leaving answer rational denominator

I keep getting $3$, instead of $\sqrt 3$. Can someone show me solution to this basic problem please? thank you.
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58 views

Is my textbook wrong ? (Trig identities)

The question is to evaluate $\cot 30^\circ + \cot 60^\circ$. I got $\dfrac{4}{\sqrt{3}}$. The correct answer is $\dfrac{4\sqrt{3}}{3}$. I have no idea how. Can someone please show me why I am wrong?
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Bearings question

Kim leaves his house and walks for $2$ km on a bearing of $155^\circ$. How far south is Kim from his house now, to $1$ decimal place? I don't know where to start at all, the correct answer is ...
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3answers
40 views

Prove that $\frac{\tan^2\theta(\csc\theta-1)}{1+\cos\theta}=\frac{(1-\cos\theta)\csc^2\theta}{\csc\theta+1}$

Question: $$\frac{\tan^2\theta(\csc\theta-1)}{1+\cos\theta}=\frac{(1-\cos\theta)\csc^2\theta}{\csc\theta+1}$$ Prove that L.H.S.=R.H.S. My Efforts: ...
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1answer
35 views

Trigonometric Identity problem involving cot

Simplify $\displaystyle\frac{\cot25 + \tan65}{\cot25}$ My attempt is: $$\frac{\cot25 + \tan65}{\cot25}=\frac{\cot25 + \cot(90 - 65)}{\cot25}=\frac{\cot25 + \cot25}{\cot25}= \frac{\cot50}{\cot25}$$ ...
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110 views

How prove that $\max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$ if $f(x) = \cos(Ax)+\cos(Bx)$?

Let $ A, B$ be real numbers and $ f(x) =\cos(Ax) + \cos(Bx)$. How prove that $ \max(|f(1)|,|f(2)|,|f(3)|,|f(4)|)\geq \frac{1}{2}$?
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1answer
27 views

Find the value of $27\csc^2\theta+8\sec^2\theta$

$10\sin^4\theta+15\cos^4\theta=6$, then find the value of $27\csc^2\theta+8\sec^2\theta$ I don't know how to do it have just tried by converting sin and cos into csc and sec. But can't get the ...
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2answers
53 views

The limit of Riemann sums $\sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$

Find the limit of Riemann sums $$\lim_{n\rightarrow \infty} \sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$$ on the interval $$[0,\frac{\pi}{2}]$$ Progress All I have managed to do is ...
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2answers
50 views

Transforming $\tan3x=k\tan x$ into $(3k-1)(\tan x)^2=k-3$

Given that $\tan3x=k\tan x$, by first expanding $\tan(2x+x)$ show that $$(3k-1)(\tan x)^2=k-3$$ I have tried just about everything that I could think of, unable to do it. Expanded once, then expanded ...