# Tagged Questions

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

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### $\sin(\theta)= 2\sin(\theta/2)\cos(\theta/2)$. 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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### 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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### 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, ...
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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### 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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### 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 ...
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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### 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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### 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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### 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 ...
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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### 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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### 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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### 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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### 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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### 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 ...
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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### 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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### 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 ...
362 views

### Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
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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### 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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### 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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### 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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### 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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### 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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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. $... 1answer 354 views ### In$\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)!}$, Then prove that the value of ... 3answers 235 views ### Finding$\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} ... 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 ... 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 ... 3answers 81 views ### Need help solving \;\arcsin(\sqrt3\sin x)=1 I need help solving\arcsin\left(\sqrt3\sin x\right)=1$$I've tried substituting various x's in, but not exactly sure what it means to find x fitting to the arcsin. 1answer 71 views ### Exact value of \frac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)} Let \alpha\in\left(0,\dfrac\pi2\right). What is the exact value of$$\dfrac{\arccos(1-2\tan^2\alpha)}{2\arcsin(\tan\alpha)}$$Firstly, I tried to simplify 1-2\tan^2\alpha and got ... 2answers 84 views ### How to find all solutions of the equation \sin x+\cos x=0 which belong to (-\pi, \pi)? Could you please help me understand and answer this question? Find all the solutions of this equation$$ \sin x+\cos x=0 $$which belong to the interval (-π; π) Progress Divided by ... 3answers 72 views ### How to answer the question “what is the domain of this function”? Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is the domain of this function?$$ 2\sin\sqrt{2x-1}+1 $$... 2answers 97 views ### Trigonometric equation, find \sin \theta Find \sin \theta if a and c are constants$$ 1-\left(c-a\tan\theta\right)^2=\frac{\sin^2\theta\cos^4\theta }{a^2-\cos^4\theta } $$3answers 169 views ### Angle in a triangle within a circle. A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC \perp OB. AC = 12 cm. BC = 5 cm. Calculate the size of \angle AOB, marked \theta on the ... 2answers 34 views ### x-intercepts of secant function I have tried setting f(x) = 0 and solving for x by undoing the operations, and what I end up with is x= -\pi/6. The book gives the answer as B, however, and I haven't been able to obtain those ... 2answers 1k views ### The exact value of csc -420 degrees (Find the exact value of each trigonometric funtion) I'am very confused, I have looked all over google and I can not find out how too do this problem. I have the answer its number 14 since our teacher gives us the answer but we need to show work. I ... 2answers 275 views ### Determinant of a 4x4 matrix with trigonometric functions I am stuck with my homework from math. I should calcutate the determinant of a matrix:$$\begin{bmatrix} sin(x) & \sin(2x) & \cos(x) & \cos(2x)\\ cos(x) & 2\cos(2x) & ... 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 ...