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

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266 views

$\sin(\theta)= 2\sin(\theta/2)\cos(\theta/2)$. How?

$$\sin (\theta) = 2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)$$ How? Please help. Thanks in advance.
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1answer
61 views

Flawed law of cosines application

I've been studying basic trigonometry since I forgot most of it, and my problem is with the law of cosines. I already understood how I can get the cosines law in the simplified way to get sides from ...
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3answers
141 views

The area of intersection of an isosceles triangle with another triangle

I tried graphing the equations that form the two isosceles triangles and integrating the bounded area and got 7.456 as my answer after rounding. The answer key has the answer listed as 7.2 However, ...
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2answers
1k views

Find coordinates for third point of a triangle given the other two points and their angles

I think this is best described by the picture below Given coordinates for point A and B, and their angles (a and b), which formula can I use to get the coordinates for point C ?
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3answers
69 views

$\int_{-\pi/2}^{\pi/2} dx \, \sin^{2n} x $

While reading a physics paper I came cross the following set of integrals. $$\int_{-\pi/2}^{\pi/2} dx \, \sin^{2n} x $$ I tried using De Moivre identity but not sure about the conclusion: $$ ...
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2answers
52 views

Stuck on a 3d equation

I have three 3D points with an known Z value: point 1 = (0, 0, Z1) point 2 = (64, 0, Z2) point 3 = (64, 64, Z3) I need an equation to solve for the Z value given ...
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2answers
138 views

Simplification a trigonometric equation

$$16 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \cos\frac{8 \pi}{15} \cos\frac{14 \pi}{15}$$ $$=4\times 2 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \times2 \cos\frac{8 \pi}{15} \cos\frac{14 ...
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2answers
133 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
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1answer
74 views

Distance from a point to Voronoi hyperplanes

I'm in the process of implementing this paper and I'm running into an issue with the process listed in portion V-A (page 5). Specifically, the paper mentions that I should store the distance from each ...
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4answers
674 views

Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$ [duplicate]

I am working on $\displaystyle\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\,dx$ First: I use integrating by part then get $$ x\arctan(\sin x)\Big|_0^\pi-\int_0^\pi \arctan(\sin x)\,dx $$ then I have ...
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3answers
160 views

Convergence of the series $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
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4answers
266 views

Calculate $\sin{55}-\sin{19}+\sin{53}-\sin{17}$ without calculator

So here is a trigonometric series. $$\sin{55^\mathrm{o}}-\sin{19^\mathrm{o}}+\sin{53^\mathrm{o}}-\sin{17^\mathrm{o}}$$ Strange isn't it, and I have to calculate the total result of the series ...
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1answer
97 views

Length of A Diagonal Line of Square

After watching this video to calculate the length of diagonal of square , a question arises to me is : Why is length of diagonal of square $$\frac{\text{side}}{cos45^0}$$? Why $cos45^0$ ?If i ...
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4answers
256 views

Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$

I have a problem with this integral. It seems that solution has to be simple, but I couldn't find out. $$I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$$ I tried using integration by parts and ...
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2answers
261 views

Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

For $x\in\mathbb{R}$, define $\sin (x) = x - x^3/3!+x^5/5!-\cdots$ and $\pi = 4(1-\frac{1}{3}+\frac{1}{5} -\frac{1}{7}+\cdots)$. Then show that $\sin(\pi/2) = 1$ In the prologue of Real and Complex ...
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1answer
95 views

Why does $\lim_{x\to 0}x^{\cos 1/x}$ fluctuate between $0$ and $\infty$?

I graphed the function $$y = x^{\cos(1/x)}$$ in matplotlib and realized that much like $\displaystyle y = \sin(\frac1x)$, the function has no limit as $x\to0^+$. However, $y = x^{\cos(1/x)}$ ...
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1answer
42 views

Determining a formula to approximate a periodic error

I am working on a barn door tracker for taking astro photos. My drive train has a small periodic error that I'm trying to eliminate and I was hoping someone might be able to suggest a formula that ...
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1answer
93 views

If I have an angle theta, what does sin theta and cos theta returns?

does sin theta returns y? also, does cos theta returns x? I am confused because in my program sin theta returns dy and cos theta returns dx. dx and dy are very less than x and y.
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2answers
44 views

Where to put angle ending on right triangle, only using variables.

Let's say I have a triangle ABC, with side lengths abc. I need to draw a line from the angle connecting the base (c) and hypotenuse (b). I don't know the real angle, but I know it's sin-1. I need to ...
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3answers
227 views

Understanding the periodicity of a complex exponential function

In the reals, $e^{nx}$ explodes to infinity very fast. But, $e^{inx}$ is bounded and periodic. I am familiar with Euler's formula $e^{ix} = \cos x +i\sin x $. Yet, could you give me some intuition ...
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1answer
51 views

roots of functions involving several sines

Is it possible to find exact solutions (in $\mathbb{R}$) of equations of the type $$\alpha_1\sin(\beta_1 t)+\alpha_2\sin(\beta_2 t)+1=0$$ for $\alpha_i,\beta_i\in\mathbb{R}$? In a comment to this ...
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2answers
132 views

Sum of the sine of values from $0$ to $\pi/2$ with some distance between angles.

I want to find the sum $$\sum_{n=0}^{\pi/(2\delta)}\sin n\delta$$ Where we're summing over all numbers from $0$ to $\pi/2$, with some $\delta$ descriping the distance between them. For example with ...
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2answers
362 views

Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
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4answers
133 views

Integrating $\int \frac{dx}{x^2+x+1}$

I am trying to evaluate the following integral: $$I=\int \frac{dx}{x^2+x+1}$$ I am not supposed to do it with complex numbers so it's kind of hard. I checked the answer on WolframAlpha. It gives ...
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1answer
44 views

Transposing when finding hypotenuse

Background: I have just about high school knowledge of math, I am sorry if this is a stupid question. In school, we learned that when transposing, this: ...
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2answers
45 views

Simplification ideas

Looking for a neat simplification idea to be able to solve for $x$ analytically in the expression below: $$S=k\tan x-Bk^2\frac{1}{\cos^2x}$$ where $\{S,k,B\}\neq0$ and $\in \mathbb{R}^+.$ Of ...
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0answers
48 views

How to add a sum of sines with general form $\frac{(4n-2)\pi}{23}$ between $1$ and $11$?

The question is to solve for $x$: $$\tan x = \sum_{n=1}^{11} \sin\frac{(4n-2)\pi}{23}$$ How do I express the above as a tangent of one angle without using a calculator In general what is the ...
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2answers
138 views

Definite integral $\int_0^{2\pi}\frac{1}{\cos^2(x)}dx$

I encountered this very simple problem recently, but I got stuck on it because I think I am missing something. It is easy to see that indefinite integral $\int\frac{1}{\cos^2(x)}dx$ is $\tan(x)+C$. ...
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1answer
42 views

Is this Trig identity derived correctly?

Is the derivation of the following trigonometric identity correct? I accept the conclusion is true and the author reached the correct equation, but look at the 2nd to last line of the page I scanned ...
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2answers
58 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
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1answer
21 views

MAPLE- Developed in a Fourier Basis - Simplifying commands

After a succession of simplifying commands, I am trying to have a truncated serie Fourier of the expression T3. I get this kind of result T33 : [1] : ...
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2answers
87 views

Transforming linear combination of the cosine and sine function

In the proof of Transforming $a\cos\left(\, x\,\right)+b\sin\left(\, x\right)$ to $r\cos\left(\,\phi - x\,\right)$ \begin{align} a\cos\left(\, x\,\right) + b\sin\left(\, x\,\right) ...
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1answer
48 views

Deriving angle from sin or cos

How can I derive the value in degrees of an angle starting from either the cos or sin value? $$ \cos(t) = c_{1} \quad \text{or} \quad \sin(t) = c_{2} $$
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1answer
78 views

Finding the angle value given 1 point and the centre of a circle

I got the coordinates of the center of a circle $(a,b)$ as well as one other point $(x, y)$. From those I can derive the radius by applying square root to the result of following formula. $$ (x-a)^2 ...
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4answers
165 views

$\sin 4x +\sqrt{3} \sin 3 x + \sin 2 x=0$

This question is from a 2012 VMK entrance exam I was trying to solve it first by expanding $\sin 4 x = 2 \sin 2 x \cos 2x$, then by noticing that if divided by 2, one can get, e.g. $ ...
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1answer
354 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value of ...
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3answers
235 views

Finding $\lim_{x\to 0}\frac{\sin(x+x^3/6)-x}{x^5}$

I'm trying to find the limit of this expression: $$\lim_{x\to0}\frac{\sin\left(x+x^3/6\right)-x}{x^5}$$ My solution is as follows: $$ \begin{align} ...
3
votes
3answers
855 views

Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!)

I'm doing a test about limits and continuity and got these two wrong. $\mathbf{Q1}$: The function $f(x) = \tan x$: $\hspace{1em}\mathtt{a)}$ is continuous $\hspace{1em}\mathtt{b)}$ is ...
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3answers
243 views

Equation of a tangent to the graph of a function parallel to a line

Please help me find the answer to this question. Thanks. What is the equation of a tangent to the graph of a function $y=x-\frac{1}{x^2}$ which is parallel to the line $y=3x$? Update: found the ...
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3answers
81 views

Need help solving $\;\arcsin(\sqrt3\sin x)=1$

I need help solving $$\arcsin\left(\sqrt3\sin x\right)=1$$ I've tried substituting various x's in, but not exactly sure what it means to find x fitting to the arcsin.
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1answer
71 views

Exact value of $\frac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$

Let $\alpha\in\left(0,\dfrac\pi2\right)$. What is the exact value of $$\dfrac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$$ Firstly, I tried to simplify $1-2\tan^2\alpha$ and got ...
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2answers
84 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
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3answers
72 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
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2answers
97 views

Trigonometric equation, find $\sin \theta $

Find $\sin \theta $ if $a$ and $c$ are constants $$ 1-\left(c-a\tan\theta\right)^2=\frac{\sin^2\theta\cos^4\theta }{a^2-\cos^4\theta } $$
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3answers
169 views

Angle in a triangle within a circle.

A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC $\perp OB$. AC = 12 cm. BC = 5 cm. Calculate the size of $\angle AOB$, marked $\theta$ on the ...
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2answers
34 views

$x$-intercepts of secant function

I have tried setting $f(x) = 0$ and solving for $x$ by undoing the operations, and what I end up with is $x= -\pi/6$. The book gives the answer as B, however, and I haven't been able to obtain those ...
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2answers
1k views

The exact value of csc -420 degrees (Find the exact value of each trigonometric funtion)

I'am very confused, I have looked all over google and I can not find out how too do this problem. I have the answer its number 14 since our teacher gives us the answer but we need to show work. I ...
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2answers
275 views

Determinant of a 4x4 matrix with trigonometric functions

I am stuck with my homework from math. I should calcutate the determinant of a matrix: $$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ...
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3answers
122 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
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1answer
92 views

Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...