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

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7 views

Find L for $r = \cos 3 \theta$.

Pictured above is the graph of $r = \cos 3 \theta$ for $0 \le \theta \le L$. Find the smallest value of $L$ that still produces the entire graph of $r = \cos 3 \theta$. I am having trouble starting ...
0
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1answer
16 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx ...
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3answers
52 views

What do we know about $\sin^{2} n$?

We all know that $-1 < \sin(n) < 1$. What about $\sin^2(n)$? What can we say about it? The main question is find the limit of $$\lim_{n\to\infty }\frac{\sin^2 n}{2^n}.$$
2
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2answers
38 views

Polar Plots and square roots

When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to ...
5
votes
1answer
83 views

Why is $\mathrm{arctan}(0)$ not infinity?

$\arctan x$ is defined as: $$\arctan x = \frac{1}{\tan(x)} = \frac{1}{\frac{\sin(x)}{\cos(x)}}$$ if I now have $x = 0$ I should get: $$\frac{1}{\frac{\sin(0)}{\cos(0)}} = \frac{1}{\frac{0}{1}} = ...
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1answer
35 views

Identity $\frac{\cos(5x)-\cos x}{\sin 5x-\sin x}=-\tan (3x)$

How to prove that following identity $$\frac{\cos(5x)-\cos x}{\sin 5x-\sin x}=-\tan (3x)$$
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votes
4answers
57 views

Partial fractions and trig functions

A long time ago I wrote down a silly problem. It starts with Attempt to write $$\frac{1}{\sin(x)\cos(x)}$$ using partial fractions. and then goes on to prove a trig identity. I was wondering if ...
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5answers
86 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
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2answers
50 views

Euler formula, trigonometry.

Prove with Euler formula that $$ \cos(x-y) = \cos(x)\cos(y) - \sin(x)\sin(y). $$ I know how to find $\cos(x+y)$, but as for $\cos(x-y)$, I'm clueless. Thanks.
4
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1answer
42 views

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$ [duplicate]

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$? The period of the first term is $\pi$ and that of the second is $4\pi$. Does that mean that the period of the whole is $4\pi$?
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2answers
24 views

Rewrite $\sin(\omega t)$ in terms of exponentials

Could someone please give me a pointer or two. I am trying to rewrite $\sin(\omega t)$ and it should be something similar to $\dfrac{e^{2j\omega t}-e^{-2j\omega t}}{2j}$ but I can't quite seem to get ...
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1answer
44 views

Find exact value of $\theta$ if $\tan\theta = 4\sqrt{5}$ [on hold]

$$\begin{align} \tan \theta &= 4\sqrt{5} \\ \theta &= \arctan 4\sqrt{5} \end{align}$$ What's the exact value of $\theta$?
0
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1answer
29 views

Closed Form of n(mod7) [on hold]

For an integer n,what is the closed form as a function of n, if it exists, of n(mod7)={0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,1,2,3,4,5,6,0,...,n(mod7)}? The closed form of n(mod8) uses trigonometric ...
0
votes
2answers
45 views

How to understand sinus?

In $\Delta PQR$ we have $\angle PQR=60^\circ$, $QR=4$ and $PR=a$. For which values of $a$ are there 0, 1 and 2 triangles matching the description? I think I'm supposed to use the law of sines, ...
0
votes
1answer
41 views

Generalized angle sum identity for $\arctan$?

The angle sum identity for arctan is: $$\arctan (\alpha)+\arctan(\beta)=\arctan\left(\frac{\alpha+\beta}{1-\alpha\beta}\right)$$ I was wondering if there exists a relationship for any linear ...
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5answers
48 views

What is the integral of $\frac{\sqrt{x^2 +4}}{x}dx$

I use trig substitution then get to this step but then I get stuck: $\int \frac{2\sec ^3\theta}{\tan \theta}d\theta$ anything I do seems to further complicate it. Thanks in advance.
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1answer
35 views

Finding third vertexes of any triangle where 2 vertex known and all sides length known

I am working with a CAD engine in the head but i working on code only. I have a rectangular tube that need to be put at an angle. I so have the diagonal of the tube where it has to start and stop but ...
-1
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1answer
39 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
0
votes
0answers
15 views

Point on ellipse after walking a distance on the perimeter [duplicate]

I've the equation of an ellipse. Given a point (x,y) on the ellipse and a length L , I want to find the coordinates (x1,y1) of the point where I'd end up after taking a walk of length L from (x,y), ...
2
votes
1answer
30 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
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0answers
19 views

Calculate vertical lines intersection hexagon at regular interval

I would like to calculate the total size of vertical lines that dissect an hexagon regular, like the one on the image. I would like to know the internal size of the blue lines inside the hexagon, ...
0
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2answers
30 views

Evaluating trig functions for a point that passes through…

I have the question "Evaluate the trig functions for angle a in standard position whose terminal side passes through (3, 4): Sec a, csc a, and cot a. For cot a the answer given is 3/4, which makes ...
4
votes
2answers
297 views

'Rational' solutions of sine

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q$$ Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$? ...
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1answer
42 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
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1answer
33 views

Help With Solving Trigonometric equations

$(\sin x)^2 - 5\sin x \cos x=0$ What would be the first atep to solve this. I normally get the equation into a quadratic one but I cannot seem to spot the first step here. What I mean by $(\sin ...
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1answer
21 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
2
votes
1answer
30 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
0
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1answer
47 views

How to solve: y'' + 9y = sin(3t)

I need to find the particular solution to the equation: $$y'' + 9y = \sin(3t)$$ I thought we were looking for a trigonometric forcing term on the form: $$y = a\cdot\cos(3t) + b\cdot\sin(3t)$$ But ...
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1answer
26 views

Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
2
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1answer
63 views

Calculationg the angle of a triangle

I am trying to find a specified angle of a triangle. In triangle $ABC$, $\angle A = 20^\circ$. $D$ and $E$ are points on $AB$ and $AC$, where $AB=AC$. $\angle EBC = 50^\circ$ and $\angle DCB = ...
0
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1answer
14 views

Calculate perimeter of rhomboid

I am trying to solve the following problem but I got stuck In a rhomboid with an area of $48 \space cm^2$, the major diagonal is $4$ cm shorter than the double of the minor diagonal. Calculate the ...
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3answers
31 views

Find base of isosceles triangle with side length and angle

I would like to calculate the length of the side in red on the image. I tried the Law of cosines, but maybe i haven't applied the formula right, because for a side "a" and "b" of size 64 and a angle ...
4
votes
3answers
48 views

Complex hyperbolic Trigonometry

When faced with the equation $\cos{z}=\sqrt{2}$ I want to solve for z so I break it up into a sum $z=x+iy$ and get: $\cos{z}=\cos{x}\cosh{y}-i \sin{x} \sinh{y}$ equating real and imaginary parts I ...
3
votes
2answers
118 views

Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
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1answer
46 views

Proving standard properties of sine and cosine defined by their power series

Definition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n ...
1
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2answers
91 views

Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$. I know that these numbers add up, but is there any ...
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4answers
38 views

Prove the inequalities without calculating the integrals

$$ \int_{0}^{\frac{\pi}{2}} \sin^4x dx \le \int_{0}^{\frac{\pi}{2}} \sin^3xdx$$ I have tried to define 2 functions $ f, g:[0, \frac{\pi}{2}] \rightarrow \mathbb{R}$ and say that $ f(x) = \sin^4x$ ...
0
votes
1answer
81 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [closed]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
1
vote
1answer
27 views

Square Wave Intuition

As I understand it, a square wave can be produced as follows: $$y = \cases{ 1 & \text{if } \sin(x) > 0\cr 0 & \text{if }\sin(x) = 0\cr -1 & \text{if } \sin(x) < 0} $$ What I'm ...
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2answers
70 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
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2answers
40 views

Why does the following equality hold? $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$?

Why is $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$ true?
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1answer
30 views

Find all angles that satisfy $6\cos^2(x)+5\cos(x)-6=0$ [closed]

Find all angles that satisfy: $$6\cos^2(x)+5\cos(x)-6=0.$$
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1answer
26 views

angle $0$ to $2\pi$ between two 3Dvectors

Ok this is for a computer game I'm learning to program with. How do you find angle between two normalized 3D vectors so that you get the resulting angle in the range $[0,2\pi]$ or $[-\pi,\pi]$? Using ...
6
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3answers
96 views

How prove $\sin \left( \alpha+\frac{\pi }{n} \right) \cdots \sin \left( \alpha+\frac{n\pi }{n} \right) =-\frac{\sin n\alpha}{2^{n-1}}$?

How prove $$\prod_{k=1}^{n}\sin \left( \alpha+\frac{\pi k }{n}\right) =-\frac{\sin n\alpha}{2^{n-1}}$$ for $n \in N$?
2
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1answer
60 views

Why is arcsin represented with the ^(-1) notation?

So in trigonometry, we have sin, secant (which is one over sin) and arcisn. Why is arcsin sometimes represented with sin^-1? sin^2 means sin to the second power, but sin^-1 explicitly does not mean ...
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3answers
251 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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1answer
28 views

How to divide trigonometric ratios using identities?

$$\frac{1-\tan^2x}{1+\tan^2x}$$ We know: $$\frac{1-\frac{\sin^2x}{\cos^2x}}{1+\frac{\sin^2x}{\cos^2x}}$$ Now what? Flip denominator and times numerator? Which equals ??? Please help - Thanks
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2answers
36 views

Find Coefficients from already fourier function

Hello I have this function and I'm asked 1.Find the period for $f(t)$ 2.Find the coefficients $a_n$ and $b_n$ $$f(t)=2(cos(2t+\frac{\pi}{4})-sin(6t-\frac{\pi}{2}))$$ I know that the period for ...
0
votes
1answer
34 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
1
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0answers
24 views

What is the best trigonometry book available free?

I am not a rich person but I really want to have a look on the trigonometry book