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

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3
votes
1answer
55 views

When $\cos x$ is transcendental?

About the transcendence of trigonometric functions I know that: 1) if $x$ is an algebraic number $\ne 0$ than $\cos x$ is transcendental. 2) if $p=\dfrac{m}{2^n}$ with $m,n \in \mathbb{Z}$ than ...
0
votes
1answer
19 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
0
votes
0answers
15 views

How to calculate solid angle of a rectangular detector of 20cm x 10cm?

I have an detector of $20cm*10cm$, how can i calculate the solid angle subtended by the detector if the detector is placed at $30$cm apart? Because directly i cannot use the formula ...
3
votes
3answers
61 views

Integral of trig fraction using substitution?

I'm chewing on an integral problem and don't have a clue where to begin. If someone could assist by suggesting a good starting point, I'd really appreciate it! Not asking for anyone to solve the ...
2
votes
1answer
22 views

Generalization of $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$

I'm looking for a generalization of the following statement $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$ In particular, I want to find $\sup \limits_{\theta} (a \sin ...
3
votes
2answers
48 views

Differentiating the function $\arcsin(3x-4x^3)$

When I have to differentiate the function $\arcsin(3x-4x^3)$ which of the following methods is more appropriate ? Putting $x=\sin θ$,simplifying and then differentiating for certain ranges of $x$. ...
3
votes
5answers
124 views

Find the values of $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$

Calculate $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$. accurate upto two decimal places or in surds . $\begin{align}\sin 69^{\circ}&=\sin (60+9)^{\circ}\\~\\ &=\sin ...
-9
votes
1answer
42 views

Pre-Calc Trigonometry, multiple angle question [on hold]

$ tan \frac{x}{2}= \sqrt{3}$ Solve this(Edited because spam)
4
votes
3answers
498 views

A trigonometric proof of an inequality

We have $f(x) = \sin(\cos(\sin(\cos(...\cos x)...))))$, where $5$ $\sin$ and $5$ $\cos$ are side by side. Prove, that $|f(\frac15) - f(\frac{1}{10})| \le \frac{1}{10}$ I simply have no idea how to ...
0
votes
2answers
49 views

Solve for y in sin(y) = cos(y) using a fixed point procedure

I'm reading an programming book that uses a lot of math equations and formulas as coding examples. In one lesson, it demonstrates finding the fixed point for $\sin(x) + \cos(x)$ by repeatedly calling ...
6
votes
2answers
80 views

Are basic trigonometry functions ( sine, cosine, tangent ) intuitive or memorized?

First, I'm really sorry for this somewhat vague and possibly just silly question. I also apologize if the following context runs a bit long. But please trust me that I'm asking with total sincerity ...
0
votes
0answers
21 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
1
vote
3answers
43 views

What does it mean that $\sin(t) = 0$ for $t = 0$ or $\pi$ mod $2\pi$

My prof wrote $\sin(t) = 0$ for $t = 0$ for $\pi$ mod $2\pi$ on the board Is this some fancy pants way to say that $\sin(t) = 0$, at points $0, \pi, 2\pi...$ Can someone explain how $\pi$ mod $2\pi$ ...
-6
votes
1answer
42 views

Simple trigonometrical ratios and equations [on hold]

I'm getting stuck with this question. It is from a trigonometry chapter. In the diagram, triangle $ABC$ is right-angled and $D$ is the midpoint of $BC$. Angle $DAC=30°$ and angle $BAD=x°$. ...
0
votes
2answers
34 views

Simple trigonometrical equations

I'm having difficulties in solving the simultaneous equations $$ \begin{cases} \sin(x+y)=\frac{1}{\sqrt{2}}\\ \cos(2x+y)=\frac12 \end{cases} $$ for $0^{\circ}\le x,y\le 90^{\circ}$. The answer is ...
1
vote
2answers
39 views

$\cot^{-1}(x)=\pi+\tan^{-1}(1/x)$ when $x<0$

My book says $\cot^{-1}(x)=\pi+\tan^{-1}(1/x)$ when $x<0$ but when I made these two plots on wolfram alpha they look exactly same even when $x<0$. Why is this happening? 1) Plot of ...
-3
votes
2answers
48 views

Verify the identity : [on hold]

Verfiy the identity: $$\cos^2(x) = ( 1 - 2 \sin^2{x} + \sin^2{x} \cos{x} )$$ Please help me
0
votes
1answer
37 views

Solving trigonometric equation $\frac{1-\cos^{2} x}{\cos x} = \frac{3}{2}$

I'm having difficulties in solving the equation below: $\dfrac{1-\cos^{2} x}{\cos x} = \dfrac{3}{2}$, where $x$ is between $0$ degrees and $360$ degrees inclusive. The answers: $60^\circ$, ...
1
vote
1answer
68 views

Is $a \sin x + b \sin y \leq \sin(ax + by)$ true?

Studying math essay exam, I saw the following strange formula $$ a \sin x + b \sin y \leq \sin(ax + by), $$ where $x, y$ are arbitrary angles and $a + b = 1.$ Is the above inequality true, and can it ...
0
votes
3answers
35 views

Simplify $ \csc(65^{\circ} + \theta) - \sec(25^{\circ} - \theta) - \tan(55^{\circ} - \theta) + \cot(35^{\circ} + \theta) $.

Simplify $$ \csc(65^{\circ} + \theta) - \sec(25^{\circ} - \theta) - \tan(55^{\circ} - \theta) + \cot(35^{\circ} + \theta). $$ I tried $ \csc(90^{\circ} - (65^{\circ} + \theta)) $. I tried the same ...
0
votes
1answer
15 views

Functions - Trig - Determine [on hold]

The vertical displacement of the end of a robot arm (in cm) at time t (in [16 marks] seconds) is given by y = 8 + 7 cos 3t + 7 cos 6t: (a) Find all times, t > 0, (in exact form i.e. in terms of ...
2
votes
4answers
53 views

Physics based trigonometry question

I am working on solving the problem stated in this image: I understand almost everything about this problem. I solved for the magnetic field along the axis of a circular loop, and now I need to ...
0
votes
5answers
54 views

Proving a complicated identity

Prove I know how to solve it, yet I can't! first I join fractions (Easy) then I "express" tans in sines and cosines after it everything turns black!
0
votes
2answers
38 views

Find the height of statue.

Standing on one side of a 10 meter wide straight road, a man finds that the angle of elevation of a statue located on the same side of the road is X. After crossing the road by the shortest possible ...
1
vote
2answers
67 views

A theoretic question about cosine general solution.

I have to find the extremas of: $f(x)=x-\tan({x\over 2})$ .$(\pi\le x\le\pi)$ Last result is $\cos({x\over 2})=\pm{1\over \sqrt{2}}$. I get that: ${x\over 2}=\pm{\pi\over 4}+2\pi k$ which derives: ...
-1
votes
1answer
25 views

Approximation for the Summation of Sequence of Powers of Sines Functions. [on hold]

Let $z_1,z_2,...,z_m$ be real numbers such that $0<z_1,z_2,\ldots,z_m<\pi/2$, $z_1>z_2>...>z_m$ and $n$ an integer such that $n>0$. Prove that: ...
1
vote
1answer
25 views

why is the domain of $\sec^{-1} x$, $\mathbb{R}- (-1 ,1)$? why can't $x$ take a values like 0.2, 0.3, etc?

Why is the domain of $\sec^{-1} x$, $\mathbb{R}- (-1 ,1)$? why can't $x$ take a values like $0.2, 0.3$ or $0$?
-4
votes
2answers
52 views

pls help to simpify [on hold]

pls help to simpify: $\sqrt{\frac{1+\cos x}{1-\cos x}}$
0
votes
3answers
62 views

Given $\frac{\sin(A - B)}{\sin(A+B)} = \frac57$, show $\tan A = 6 \tan B$

I can expand the sine using compound angle formula but then I can't continue to make it become a tangent.
1
vote
1answer
23 views

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$]

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$] ? How should the graph be ?
-3
votes
2answers
57 views

Need confirmation that the following problem is correct. [on hold]

I have to prove the following identity but as I couldn't do it I wonder if it is true? $$\frac{\cot^2\frac{α}{2}-\cot\frac{3α}{2}}{\cos^2\frac{α}{2}\cosα(1+\cot^2\frac{3α}{2})}=8$$
0
votes
3answers
57 views

Solving trigonometric expressions in $x$

I am having problems understanding how to write an algebraic expression in $x$ for: $$\sin\left(\arcsin(x)-\arctan\left(\frac{2}{x}\right)\right)$$
0
votes
1answer
43 views

The period of a non-linear pendulum

The period of a non-linear pendulum is $T = \sqrt{2} \cdot \int_{-\theta_0}^{\theta_0} \frac{d{\theta}}{\sqrt{\cos(\theta) - \cos(\theta_0)}}$. My problem: what will happened with the period $T$, ...
0
votes
1answer
50 views

Solving double angle trigonometric equities [on hold]

I am having problems understanding how to solve the equation: $\sin{(2x)} = \sin{(x)}\cos{(x)}^{1/2}$
-1
votes
1answer
34 views

Inverse trigonometric function problem [on hold]

Please prove this easy way. I tried it by solving right hand side but it didn't go well. $$ 2\cos^{-1} \frac45 = \sin^{-1} \frac{24}{25} $$
-2
votes
0answers
30 views

Trigonometric equation of 4 answer

$3\tan(2x+22)= -6.339$ $\tan(2x+22)= -2.113$ $2x+22=-64.67$ $x=-43.33$ $x=180-43.33$, $x=360-43.33$ $x=136.67$, $x=316.67$ How to find another two answer ?
0
votes
3answers
34 views

Inverse trignometric proof? [on hold]

Please help me prove the following : $$ \cos^{-1}\frac{63}{65} + 2\tan^{-1}\frac15 = \sin^{-1}\frac35 $$
1
vote
3answers
56 views

Solve these equations simultaneously (trig)

Solve for $ x,y: $ \begin{equation}\cos x -\cos(x+y) = 0 \end{equation} \begin{equation}\cos y -\cos(x+y) = 0 \end{equation} The answers are $(0, 0), (\frac{2\pi}{3}, \frac{2\pi}{3})$. I get ...
1
vote
2answers
50 views

Proving trigonometric identity $\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4\sec x \tan x$ [on hold]

This is a trigonometry question that i did not quite understand very well. Show that $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4\sec x \tan x$$
1
vote
1answer
32 views

The graph of tan(sec(x))

A lot of the trig_function(trig_function(x)) look something like this, with asymptotes that have infinite (?) oscillating (?) lines infinitely approaching them ...
-1
votes
2answers
63 views

How to Calculate csc(2.85) in Calculator?

In my calculator (TI-84), there are only $sin, cos,$ and $tan$ commands (and inverse sin, inverse cos, inverse tan). I had a question that was as follows: Calculate $csc(2.85)$ in which I was ...
2
votes
1answer
54 views

How we can find the sign for trigonometric functions without graph

For $\sin(x)$ or $\cos(x)$ etc. how we can show that it is negative on $\left[\pi ,2\pi \right]$ ? without graph? So if we have $\sin(2x)$ or $\cos(2x)$ how we can find the sign on $\left[0,2\pi ...
1
vote
3answers
51 views

Integral of a tangent function

$$ \displaystyle {\int_{0}^{z}} \sqrt {1 + \tan^2(\dfrac{\pi}{4} \dfrac{z}{H} )} dz $$ _ $$ gives $$ _ $$ \dfrac{4H}{\pi} {\sinh^{-1}} ( {\tan \dfrac{\pi}{4} \dfrac{z}{H} } ) $$ Please advise ...
0
votes
1answer
34 views

Why this formula used ? angle = (2π- 2πθ/255.0); [on hold]

i got a source code regarding reading a minutia from an iso standard, now i need to create another code to save the minutia in the same ...
1
vote
1answer
43 views

Infinite summation of a trigonometric series

$\sum_{n=1}^{n=\infty}\sin(\frac{n\pi x}{L})\sin(\frac{n\pi y}{L})\surd(k^2+\frac{n^2 \pi^2}{L^2})$ I am trying to solve the above summation. I still could not figure out if this summation converges ...
0
votes
1answer
43 views

Finding speed of snowballs given initial velocity and angles

You and a friend stand on a snow-covered roof. You both throw snowballs from an elevation of $14$ m with the same initial speed of $12$ m/s, but in different directions. You throw your snowball ...
1
vote
1answer
12 views

How can I find the inner limit of a line passing through a lune?

I have a crescent defined by two offset circles with different radii: a small one (let's call it outer circle) centered at (0,0) with radius ...
1
vote
2answers
53 views

Is $\log_{\cos x}(1)$ defined at $x=0+2k\pi$? [duplicate]

I have an equation like this: $\cos(x) ^ {\sin(x)} = 1$ I thought I would solve it like this: $\cos(x) ^ {\sin(x)} = 1$ $\sin(x) = \log_{\cos(x)}(1)$ $\sin(x) = 0 $ $x = 0+k\pi$ But I'm ...
8
votes
4answers
357 views

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. Why?

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please ...
2
votes
2answers
63 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...