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

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4
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3answers
36 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 ...
1
vote
5answers
93 views

Integrating $\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt$.

I am trying to compute $$ \int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt. $$ This is what I got so far: $t=\sec(x)$ and $dt=\sec(x)\tan(x)x\,dx$ So plugging this in gives me $$ \int ...
0
votes
0answers
10 views

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

How to prove that following identity $$\frac{\cos(5x)-\cos x}{\sin 5x-\sin x}=-\tan (3x)$$
2
votes
4answers
67 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.
3
votes
4answers
60 views

How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$?

I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear ...
1
vote
2answers
48 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.
2
votes
3answers
52 views

$x$-intercept of cosine graph

I am having problems understanding how to find the $x$-intercept of a cosine graph. Example: $10\cos(x/2)$ Answer:$((2n + 1)\pi , 0 )$ I have the answer just need help understanding the steps, ...
3
votes
2answers
119 views

Prove $2\cos^2(x)=1+\cos(2x)$

I need help to prove that: $2\cos^2(x)=1+\cos(2x)$. I know that $\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$, but I don't know how to get to this step without memorizing ...
3
votes
1answer
39 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 $4\pi$?
1
vote
2answers
21 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 ...
0
votes
1answer
1k views

Determine possible coordinates for point $P$ on the terminal arm of angle

a) If angle $\theta\\$ lies in Quadrant II and $\sin \theta ={3 \over {\sqrt {45} }}$. Determine possible coordinates for point $P$ on the terminal arm of angle $\theta$. b) Determine the Quadrant ...
0
votes
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 ...
-3
votes
1answer
41 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
votes
1answer
33 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
33 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 ...
3
votes
2answers
117 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: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
0
votes
1answer
35 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 ...
0
votes
1answer
275 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
-1
votes
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
4answers
41 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.
10
votes
3answers
9k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
3
votes
3answers
2k views

Useful trigonometry tricks/shortcuts

I'm curious as to any "tricks" or shortcuts that could help make verifying/solving trigonometric identities easier, for example one is: $$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$ ...
1
vote
5answers
195 views

Is $\cos(x^2)$ the same as $\cos^2(x)$?

I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
4
votes
1answer
420 views

Calculating equidistant points around an ellipse arc

As an extension to this question on equiangular fisheye distortion, how can I calculate equidistant points around an ellipse (or 1/4 segment of) given it's aspect ratio? When it's circular, I can use ...
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
29 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 ...
0
votes
0answers
113 views
+100

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
1
vote
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
votes
2answers
26 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 ...
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 ...
4
votes
2answers
296 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 $0 < q < 1$? ...
0
votes
1answer
40 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 ...
2
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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: ...
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
4answers
176 views

Complex numbers - Exponential numbers - Proof

Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6. For this problem I am stumped...how should I begin? Also there's a ...
0
votes
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 ...
0
votes
1answer
496 views

Determine sin cos and tan from slope? NON CALCULator

I am used to finding this by drawing the triangle and knowing the angle measurement. With the angle measurement i can find sin cos and tan. But i dont have angle all I have is line. I have put a ...
1
vote
5answers
79 views

Solving a trigonometric equation: $2 \sin(3a)=\sqrt{2}$

I have the following equation : $2 \sin(3a)=\sqrt{2}$ Not sure how to solve it (Because it's a transformed sin function, meaning 6 solution with 3 cycles in $2\pi$) after a moment I finally found ...
1
vote
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?
-2
votes
5answers
720 views

Solving the equation $\sin t = -\sqrt{2}/2$

Solving the equation $$ \sin t = -\frac{\sqrt{2} }{2} .$$ I know the solution is $1.25$ and $1.75$, but I do not know how to get there. An explanation would be GREATLY appreciated, thanks!
0
votes
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 ...
0
votes
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 ...
2
votes
1answer
43 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
vote
1answer
46 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
0
votes
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$ ...
1
vote
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 ...
7
votes
3answers
250 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.