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

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Finding coordinates of position given three coordinates and distances: 3D

I'm hoping to determine the x, y, z coordinates of a 4th position (D) given the coordinates of three other positions and their distances: $A(0.25, 0.25, 0.25), B(0.4663, 0, 0.25)$, and $C (0.3912, ...
-2
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3answers
47 views

What is the angle between these unit vectors? [closed]

Let $a$ and $b$ be two unit vectors and $p$ is the angle between them. If $a+b$ is also a unit vector, then: $p = \pi/3$ $p = \pi/4$ $p = \pi/2$ $p = 2\pi/3$
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1answer
15 views

Prove of right regular pyramid that a perpendicular line in the middle of its height intersects its edge exactly in the middle

Prove of right regular pyramid that a perpendicular line in the middle of its height intersects its edge exactly in the middle. It seems for me to be counter intuitive that this is the case. How ...
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0answers
25 views

zeroes of tan function

Currently in preparation for an exam on Separation of variables for PDEs. And I might have conveniently forgotten the zeroes of tan. Can I confirm that the zeroes are located at $n\pi$ where $n$ is ...
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2answers
39 views

How would you find the length of a side of a triangle where 2 sides are known and the length of a line in the middle is also known?

How would you find the length of a side of a triangle where the other 2 side lengths are known and the length of a another line that meets at the same point is known? I know there has to be an answer ...
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3answers
4k views

Do “other” trigonometric functions other than Tan Sin Cos and their derivatives exist?

I remember my physics teacher mentioning that other trigonometric functions exist apart from the Sin Cos and Tan, he mentioned a few and they did not sound familiar, nothing like Sec Csc and Cot. I ...
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1answer
42 views

Optimizing trigonometric equation

I've come across a problem from an old calculus textbook which goes like A tool shed, $250\space cm$ high and $100\space cm$ deep is build against a wall. Calculate the shortest ladder length that ...
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1answer
29 views

EM waves - orthogonality - amplitude/phase angle

A plane electromagnetic wave has the shape: $\vec{E}(\vec{r},t)=E_0\cdot cos(\vec{k}\vec{r}-\omega t)\cdot \vec{e}_y$ $\vec{B}(\vec{r},t)=(B_1\cdot cos(\vec{k}\vec{r}-\omega t)+B_2\cdot ...
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1answer
14 views

Value of sin-product only depended of argument difference

I have the product of $$Y_1 = A \sin (\omega t_1 + \phi_1)$$ and $$Y_1 = A \sin (\omega t_2 + \phi_2).$$ I know, since I plotted it with python, that the product $$X = Y_1 \cdot Y_2$$ is indepened ...
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2answers
101 views

Trigonometric identities — a parallel RLC circuit connected to an AC-supply [closed]

An RLC-circuit is connected to an AC-supply as in the figure below. $I_{tot}(t)=I_0sin(\omega t+\phi)$ (denoted as $I_{ges} ( t)$ in the picture), $\phi$ is the phase angle between ...
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56 views

How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
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3answers
42 views

Math trigonometry transformation

Hi, I haven't done math in a while, and stumbled upon this thing. The angle ($\arccos 7/25) is given, and i have to calculate the cosine of it's half. I've used the basic formula for cosine of an ...
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5answers
72 views

Value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$

How can we find the value of $x$ in $\sin^{-1}(x)+\sin^{-1}(1-x)=\cos^{-1}(x)$? Note that $\sin^{-1}$ is the inverse sine function. i'm asking for the solution x for this equation Pls workout the ...
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0answers
44 views

Length between two circles intersection area?

How do I know the (smallest) length of the intersection area between two circles of different sizes? We know both circles radii and the overlapping area. ...
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1answer
23 views

Taking it a step further with a sum

So I was watching an "old" video from numberphile about the three square problem. https://youtu.be/m5evLoL0xwg Here is also an image: ...
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0answers
46 views

how to prove this nice trigonometric identity [duplicate]

I was working on a complex analysis problem in "Berkeley Problems in Mathematics", which was asking me to prove that some product is equal to $n$. I had reduced the problem to proving a trigonometric ...
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2answers
71 views

Value of expression of $8\sin^280^\circ - 2\sqrt{3}\sin40^\circ - 2\cos40^\circ$ is equal to? [closed]

This should be done by using identities and double angle formulas I guess, but I need hint? Solution in my booklet is 4.
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3answers
45 views

Sum of all solutions of equation $\sin{2x} = 1 + \sqrt{2}\cos{x} + \cos{2x}$ on interval $(0,2\pi)$?

Solution of this task is ${7\pi\over2}$ but I don't know how to get to this solution.I used the formula for double angle for $\sin{2x}$ and $\cos{2x}$ and moved everything on one side of the equation ...
2
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3answers
44 views

Maximum value of trigonometric expression [closed]

If $r=3+\tan c \tan a, \quad q=5+\tan b \tan c, \quad p=7+\tan a \tan b$ Provided $a,b,c$ are positive and $a+b+c=\dfrac{\pi}2$ Find the maximum value of $\sqrt p + \sqrt q + \sqrt r$ .
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1answer
44 views

Find x in degrees: (there souls be 2 answers from 0 degrees to 360 degrees) 5cos(x)=2sec(x)-3 [duplicate]

$2(1/\cos(x))-5\cos(x)-3=0$ $(2/\cos(x))-5\cos(x)-3=0$ $2-5\cos(x)-3=\cos(x)$ $-5\cos(x)-1=\cos(x)$ $-6\cos(x)=1$ $\cos(x)=-1/6$ $x=99.6$ Reference angle$=80.4$ $180+80.4=260.4$ $x=99.6^o$, ...
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3answers
43 views

Find $x$: $5\cos(x) = 2\sec(x)-3$

This was the last question on my exam and it's driving me nuts the answer has to be in degrees $5\cos(x) = 2\sec(x)-3$ $5\cos^2(x)/2 = -3$ $\cos^2(x) = -6/5$ Since $-6/5$ is negative The answer ...
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1answer
83 views

Finding the area between $F(x) = 4\sin(x) + \cos(2x) -4x$ and the and $x$- and $y$-axes

I'm kind of stuck while studying for my AP Math final (11th grade). I need to find the area between $$F(x) = 4\sin(x) + \cos(2x) -4x$$ and the and $x$- and $y$-axes. So I did $$F(x) = 0$$ To ...
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5answers
89 views

Sum of real solutions on equation $\sqrt{\sin^2{x} + {1 \over 2}} + \sqrt{\cos^2{x} + {1 \over 2}} = 2$ in interval $[0,2\pi]$ is?

I know that solution is $4\pi$ but I do not know how do they get to this solution. I always get that $x \in R$ and that $-1 < \cos 2x < 1$ when converting it to double angle. EDIT : So ok, I ...
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4answers
53 views

Geometry problem, find lengths

Here's my problem: I know the angles $\alpha$ and $\beta$, and the line length $a + b$. I need to find the lengths $a$ and $b$ and the height of the triangle. I came up with the identity ...
3
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3answers
83 views

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I've been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles ...
3
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0answers
51 views

When was the unit circle formalised

I am wondering about the origins of the Unit Circle. Of course it is part of trigonometry, which goes back many centuries. But since it uses Cartesian coordinates, it should be after Descartes. So, ...
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8answers
1k views

Variation on Pythagoras: If $a^2 + b^2 = c^2$, then $a + b \leq c\sqrt{2}$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of the Pythagorean Theorem using calculus, trigonometry, ...
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1answer
43 views

Simple question: Intuition for x and 1-x

In Trigonometry for Dummies (2014 edition), the author writes, "If you look at Figure 10-7, you see that two right triangles are formed. The two triangles share a side — the one opposite the measured ...
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1answer
49 views

Math Problem Help || Trig (Updated picture) [closed]

So I got the following math problem I solved but I'm not confident with my answer. My answer was 32. The question is to find the area of the following: ...
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5answers
120 views

How do I solve $\int\sin^5 x$?

Should I transform $\int\sin^5 x$ into $\int (\sin^2 x)^2 \sin x \; \Bbb d x$ to solve it? Or should I use a different trigonometric identity?
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3answers
32 views

How do I find if the roots of a quadratic equation are real and equal or real and unequal using the discriminant?

If I have an equation such as $x^2+3x+4=0$, how do I find out whether the roots are real, rational, and equal or real, rational, and inequal using the discriminant?
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1answer
37 views

How to calculate $E(\sin^2X)$

If $X \sim N(0,1)$ then calculate $E(\sin^2X)$ I understand that $0 < \sin^2x<1$. So the expectation exists. I proceed as $E(\sin^2X)= \int_{-\infty}^{\infty}\sin^2xf(x)\,dx=2 ...
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1answer
73 views

If $(a , b , c)$ is a Pythagorean triple, then so is $(ka , kb , kc)$ [closed]

From trigonometry text: Show that if $(a , b , c)$ is a Pythagorean triple then so is $(ka , kb , kc)$ for any integer $k > 0$. How would you interpret this geometrically? Can someone please ...
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1answer
42 views

Prove that the hypotenuse is the longest side in a right triangle. How to write a formal proof for something so obvious?

From trig text. Given hint: is $ a^2 + b^2 > a^2 $. Pythagorean theorem,obviously. What would be an acceptable proof?
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1answer
21 views

How to compute sine function within certain precision

I know that there exists CORDIC algorithm, but CORDIC algorithm contains component $\arctan 2^{-i}$, which needs to be looked up. I do not know how this leads to the precision that is advertised by ...
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3answers
44 views

If $x_i \gt 0$ for $1\leq i\leq n$ and $x_1+…+x_n=\pi$ Then the greatest value of $\sin x_1+…+\sin x_n$ is $n \sin \left(\frac{\pi}{n}\right)$.

If $x_i \gt 0$ for $1\leq i\leq n$ and $x_1+x_2+x_3+...+x_n=\pi$ Then the greatest value of $\sin x_1+\sin x_2+...+\sin x_n$ is $n \sin \left(\frac{\pi}{n}\right)$. Prove it. I have no idea how to ...
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2answers
42 views

Special triangles

I have this question that I have the answer to but no working how to get it, is it by pure memorization of angles or there some steps? Without a calculator, determine, in radians, the angles of a ...
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3answers
57 views

I need intuition as to how trig substitution works.

Let's start with an example: $$ \int \frac{1}{\sqrt{4-x^2}}dx; $$ using a reference triangle, I find that $\sqrt{4-x^2}$ can be expressed as $2\cos\theta$ in polar coordinates. However, I don't ...
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3answers
28 views

Point $P$ is in the middle of the interval that surrounded in curve

Let $c$ be a close curve such that $c$ does not intersect itself, $c\in \mathbb{R}^2$ (in the plane), show that for all point $P$ that surrounded in $c$ there are two points $A,B$ on $c$ such that $P$ ...
3
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2answers
70 views

Solve $\sin(90^\circ-x) = 2\sin(x)$ [closed]

I am trying to solve a high-school (that I graduated $12$ years ago) question, but with no success :( . I totally forgot how to solve the following formula (Law of sines): ...
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4answers
37 views

Prove that $(G, *)$ is abelian group, for $ x * y = \tan^{-1}(tan(x) + tan(y))$

I have some troubles solving this problem. In order to prove that $(G, *)$ is an abelian group I have to find the identity element of the group, first; $\exists \ e \in \ G \ and \ x \in G$ such that ...
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4answers
31 views

get length of line connecting sector of a circle

What's the formula for getting the length of a line (in this case the red one) connecting starting point and end point of an arc, given the circle's radius R and angle A?
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4answers
48 views

Interpolation between 2 points on the perimeter of a circle?

I'm trying to produce movement on a unit circle from one point to another in equal increments, but I'm having trouble doing this without the use of angles (which isn't an option). Given 2 points on a ...
7
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2answers
73 views

Imaginary $\cos^{-1}$ value significance?

When I was bored in AP Psych last year, I jokingly asked myself if there was a cosine inverse of $2$. Curious about it, I tried calculating it as follows: $$ \begin{align*} \cos (x) &= 2 \\ \sin ...
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1answer
26 views

Transform the following cartesian equations in polar equations

$$4y^2-20x-25=0$$ The answer given by the textbook is $r=\frac{5}{2(1-\cos \theta)}$ and I couldn't get to this result. I have done $x=r\cos\theta$ and $y=r\sin\theta$ and it leads to ...
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3answers
38 views

Find out all solution of trigonometric equation $\tan \theta = - \frac{\sqrt 3}{3}$

$\tan \theta = - \frac{\sqrt 3}3$ I thought it was $2\pi\over3$ and $5\pi\over3$ but I was wrong please help
3
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1answer
30 views

Sine of an obtuse angle

In the figure above, $\angle MOP=\theta , \angle POP'=90^o$ $$\sin (90^o+\theta)=\sin \angle MOP'=\frac{M'P'}{OP'}(\text{how?})=\frac{OM}{OP}=\cos \theta$$ Sine is opposite side/hypotenuse, then ...
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5answers
80 views

$\sin \pi/6 =-1$? From sum of angles formula.

I was solving this problem from Apostol's calculus book and encountered the problem of getting the value of $\sin(\pi/6)$ with the aid of the equality $\sin3x=3\sin x-4\sin^3x $ (from sum of angles). ...
2
votes
4answers
227 views

Trigonometry and Proportions

Numbers are given $a = {\sin1\over \sin2}$, $b = {\sin2\over \sin3}$ and $c = {\sin3\over \sin4}$. Then: $a < b < c$ $c < b < a$ $c < a < b$ What is the solution and can someone ...
3
votes
2answers
101 views

Find $a$, when $\tan a$ is given in terms of $\tan1^{\circ}$ and $\tan2^{\circ}$.

If $\tan\alpha = {(1+\tan1°)(1+\tan2°)-2 \over (1-\tan1°)(1-\tan2°) - 2}$ and $\alpha \in (0°, 90°)$ then $\alpha$ is equal to? This is task from my faculty entrance exam workbook. This is mostly ...