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

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2
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1answer
16 views

Confusion with modeling a trigonometric function

I am studying trigonometry on Khan Academy and came across this problem: The daily low temperature in Guangzhou, China, varies over time in a periodic way that can be modeled by a trigonometric ...
0
votes
1answer
45 views

Trig identities dividing fractions

The question is : $\dfrac{\csc x}{\sec x} = \cot x$ After solving a bit I get $$\frac{\frac1{\sin x}}{\frac1{\cos x}} = \frac{\cos x}{\sin x}$$ $$\frac{\cos x}{\sin x} = \frac{\cos x}{\sin x}$$ Is ...
1
vote
1answer
26 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
2
votes
4answers
69 views

Evaluate $\int_0 ^{\pi}\left (\frac{\pi}{2} - x\right)\sin\left(\frac{3x}{2}\right)\csc\left(\frac{x}{2}\right) dx$

How would you evaluate the integral $$\int_0 ^{\pi} \left(\frac{\pi}{2} - x\right)\sin\left(\frac{3x}{2}\right)\csc\left(\frac{x}{2}\right) dx$$ The answer from Wolfram is $0$. Would you use a ...
1
vote
3answers
21 views

Calculate uncertainty of sine function result

I have an angle given in degrees: $$\theta_{\min} = 63^{\circ} \pm 0.5^{\circ}$$ I need to calculate it's sine and still know the uncertainty of the value: $$n = 2\sin(\theta_{\min}) = ...
2
votes
1answer
62 views

Solving messy integral with modulus and trigonometry.

If $$a\in \mathbb R,\int_{a-\pi}^{3\pi+a}|x-a-\pi|\sin(x/2)dx=-16$$ then a can be? I tried substituting $x-a=u$ and then breaking into two integrals removing modulus then used $\int \sin x=-\cos ...
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0answers
17 views
+50

Given three vectors involving trigonometric functions, how many $\theta$ satisfy a particular box product relation?

If $$\vec a =(1+\sin \theta )\hat i+\cos \theta \hat{ j}+\sin2\theta\hat k\\ \vec b =(\sin( \theta +2\pi/3))\hat i+\cos ( \theta +2\pi/3) \hat{ j}+\sin( 2\theta +4\pi/3)\hat k\\ \vec c =(\sin ( \theta ...
1
vote
0answers
44 views

Find all the triangles satisfying $\cos(A)\cos(B)+\sin(A)\sin(B)\sin(C)=1$ [duplicate]

I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$. In the case where the triangle has a right angle, ...
0
votes
4answers
86 views

How can I find the lenght of the third side of any triangle

I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how ...
0
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1answer
40 views

How to find the formula for these repeating sequences?

How to find formula for this number pattern? 3,6,5,2,3,... When plot this sequence into the graph, it is going to be the sine graph..
0
votes
1answer
20 views

Find the edges of a triangle from a vertex

If I have a series of three vertices that make up a triangle, how can I take one of these vertices and find the edges that go from that vertex to the other two vertices?
0
votes
1answer
24 views

Finding the limit of trigonometric functions

Find: $$\lim_{x\to\pi/2}\left(\frac{\cos x}{(1-\sin x)^{2/3}}\right).$$
0
votes
2answers
21 views

Find the $sin$ of an angle $B$ using law of sines given side angle side

this is giving me trouble, here's what I've tried: Q.A triangle has sides $a=2$, $b=3$, and $\angle C = 60^o$. Using the law of sines, find $\sin(B)$ OK so I know the law of sines is: ...
1
vote
1answer
36 views

$Si(x)\leq Si(\pi)$ for every $x>0$

I need to show that $Si(x)\leq Si(\pi)$ for every $x>0$ where $Si(x)=\int_{0}^{x} \frac {\sin (t)}{t} dt$. I see it graphically but i can't prove it. Thank you!
3
votes
3answers
50 views

How to find $\lim\limits_{x \to 0} \frac{\sin 2x}{\sqrt{1+\tan x} - \sqrt{1-\tan x }}$?

How to find $\;\;\lim\limits_{x \to 0} \dfrac{\sin 2x}{\sqrt{1+\tan x} - \sqrt{1-\tan x }}$ ?
2
votes
1answer
38 views

If $\tan\theta$, $2\tan\theta+2$, $3\tan\theta +3$ are in geometric progression then find …

Problem: If $\tan\theta$, $2\tan\theta+2$, $3\tan\theta +3$ are in geometric progression then find the value of $$\frac{7-5\cot\theta}{9+4\sqrt{\sec^2\theta -1}}$$ Solution : Since ...
0
votes
1answer
12 views

How can I parameterize these angles

My angle starts at 90, goes down to 0/360, then down to 270. This is a 180 degree range of motion. How can I express these angles from 0 - 180 instead of 90 - 270, where 90 gives 0 and 270 gives 180. ...
1
vote
2answers
58 views

How to find trigonometry function limit [closed]

What is the solution for trigonometry functions limit when we're in $\dfrac{0}{0}$ situation? $$\lim_{x\to 0} \frac{\sin^2 3x}{x^2}$$ for example
0
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2answers
43 views

How to combine two series of solutions into one?

I have an equation, like $$\sin\left((\beta-1)\sqrt{\xi^2-\gamma^2}\right)\sin\xi=0,$$ with $\beta>1$ and $\gamma>0$. I've found two series of solutions, corresponding to roots of each of the ...
3
votes
2answers
50 views

Integration giving different answers (trig substitution)

Integrating $\sin^3x\cos^5x$, i get 2 different answers, using techniques that should both be valid.
1
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3answers
32 views

How to find roots of $\sin (x) - a$?

How to find roots of $\sin(x) - a$, where $a \in [0, 1)$ and $x \in [0, 2\pi]$?
3
votes
3answers
254 views

Proof that $\lim_{n\to\infty}{\sin{100n}}$ does not exist

How to prove that $$\lim_{n\to\infty}{\sin{100n}}$$ doesn't exist? Some possible approaches: It would be enough to find two subsequences $n_{k}$ that converge to two different numbers. But ...
0
votes
1answer
51 views

The Sine Law: A Simplified Criterion for the Ambiguous Case?

Here is my suggestion for an issue that doesn't seem to be handled well in any online notes that I have seen. Can anyone give a counter-example? If you are given $a,b,$ and $B$ in $\triangle ABC$ ...
0
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4answers
53 views

Limit of $\frac{1-\cos x}{\sin x}$

I don't understand the rewriting that's being done in this limit: $$\lim_{x\to0} \frac{1−\cos x}{\sin x} = \lim_{x\to0} \frac{\sin x}{\cos x} $$ Why doesn't this simplify to $\frac{\sin x}{\sin x}$? ...
0
votes
1answer
43 views

Modeling with trigonometric functions

I'm having trouble with modeling trig functions that include phase shifts on KhanAcademy. Please be aware that the answer would not help me. That is already available to me. I would prefer an ...
0
votes
0answers
40 views

Answer to Plane Trigonometry Ex XLIX Q16?

If $\alpha, \beta, \gamma, \cdots$ be the roots of the equation $sin(mx) - nx cos(mx) = 0$ prove that $\tan^{-1}\frac{x}{\alpha} + \tan^{-1}\frac{x}{\beta} + \cdots + \tan^{-1}\frac{x}{v} = 0$. The ...
0
votes
0answers
24 views

How to find the root of this non-linear equation?

I am trying to solve this non-near equation using Matlab but it doesn't give me the correct answer (as shown in the document that I am doing it from). The Matlab code gives me imaginary root. Could ...
3
votes
3answers
49 views

Finding $\lim_{x\rightarrow 0}\frac{x}{2}\sqrt{\frac{1+\cos(x)}{1-\cos(x)}}$

We know that $$\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\tan({x}/{2})$$ so we can change the above function to another form as follow ...
1
vote
1answer
61 views

For which positive integers $n$ does $P(n)$ fail to hold?

Let $n$ be a natural number and let $z$ be a complex number. Consider the following proposition: $P(n)$: If $\cos (nz)$ is bounded above by one in absolute value, then $\cos z$ ...
3
votes
0answers
26 views

Probability density function of $x$ in the unit circle?

I'm trying to work out how to find the probability density function (PDF) for $x$ values on the unit circle - not within the unit circle but on the edge. The reason for doing so is that I'm trying to ...
6
votes
1answer
141 views

How to evaluate the integral of $\sqrt{\sin\sqrt x}\cos \sqrt x / ( 1+x^2)$?

$$ \int \frac{\sqrt{\sin\sqrt x}\cos \sqrt x}{1+x^2} dx $$ I have tried combinations of $x=t^2$, integration by parts, $\tan\left(\dfrac u2\right)$ substitutions it got even more complicated. Is ...
0
votes
0answers
16 views

Split this integral

I need to split this integral if possible: \begin{equation} \int_{\mathbb{R}^d} e^{\sum_{i=1}^dx_iz_i}cos(\sum_{i=1}^dy_iz_i)d\mathbf{z} \end{equation} I wanted split into two part : one with $x_i$ ...
0
votes
3answers
57 views

Solving $\cos^6(2x)+\sin^6(2x)=\frac58$ [closed]

How do I solve the following equation for $x$? $$\cos^6(2x)+\sin^6(2x)=\frac58$$ Thanks
0
votes
2answers
24 views

Calculate sum of angles if you know their tan value

$$\tan(u) = 2, \ \ \ 0 < u < \frac{\pi}{2}\\ \tan(v) = 3, \ \ \ 0 < v < \frac{\pi}{2}$$ What is $u + v$? I know that both angles are in the first quadrant in the unit circle. How do I ...
0
votes
0answers
8 views

Averaging of a product of fast oscillating function and slow oscillating one

Let's have a product of fast oscillating function and some slowly oscillating function: $$ F(\mathbf r) =f(kz)g(\mathbf r), \quad f(kz) = cos^{2}(kz),sin^{2}(kz). $$ I want to average this quantity ...
4
votes
2answers
241 views

Using an Integral to Solve for a Variable a

I am struggling to use the following equation: $$ \int_0^a \sqrt{a^2-x^2}\,\,\text{sgn}(|x|-1)\, dx = 0 $$ where $a > 1$, to deduce that $a = \text{cosec}(\frac{\pi}{4} - \frac{\alpha}{2})$, ...
3
votes
5answers
144 views

Why is $(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000\cdots$?

Apparently, $$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$ Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$. Why is that?
1
vote
1answer
48 views

How to find this limit without l'Hospital's rule: $\lim_{x\to 0} \frac{\cos(2x)-1}{1-\cos(7x)}$ [closed]

How to find this limit without l'Hospital's rule? $$\lim_{x\to 0} \frac{\cos(2x)-1}{1-\cos(7x)}$$ Note: Taylor expansion is not available.
0
votes
0answers
7 views

Proving a specific case of reduction

Here is the problem: Let I(m) = integral(0-pi/2)(sinx)^m (dx) Prove that I(m) = ((m-1)/m)(I(m-2)) I used the reduction formula for (sinx)^m, which is: Doing this, you find that the integral ...
3
votes
1answer
59 views

Why is trigonometry important in calculus? [closed]

I need to write short note why trigonometry is important is calculus and engineering mostly for presentation. I am not focusing on on what topic it specifically it appears (because I am guessing the ...
0
votes
1answer
24 views

Given that $\cos a= 24/25$ and $\sin a<0$, find $\cos(a+\pi/6)$

Given that $\cos a= 24/25$ and $\sin a<0$, find $\cos(a+\pi/6)$ I graphed $24/25$ in the $4$th quadrant and then did Pythagorean theorem. After that I don't know what to do.
0
votes
1answer
23 views

find image and inverse image of function

I have function $f:R\to R^2 , \ \ f(x)=<\cos 3x, \sin 3x>$ and I have to find image on the interval $(0, \pi]$ and inverse image $[0, +\infty) \times[0, +\infty)$ I think the image will be ...
0
votes
2answers
47 views

Trigonometry specific problem

This was all the information given $$\sin^2{2 x} - \sin x-1 = 0, \ x \in [0,2\pi)$$ I did the quadratic formula and ended up with two answers which was a positive and negative. I canceled the ...
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8answers
68 views

Proving $\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + \cot A$

I got this question from a paper but can't solve it and the question paper has no solutions section.How do you prove this? $$\displaystyle\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \csc A + ...
0
votes
1answer
30 views

How to prove this, a sin(B-C) + b sin (C-A) + c sin (A-B) = 0 [on hold]

I used Sin rule and I couldn't solve rest of the part.
2
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2answers
32 views

Trigonometry : Find the length of side

Can someone tell me how to calculate the length 'd' from the below figure? It is from Lecture 06 - Optical flow : ...
2
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0answers
65 views

Prove that $a^2(p-q)(p-r)+ b^2(q-r)(q-p)+ c^2(r-p)(r-q) =4(\delta)^2$

If $p$,$q$,$r$ are the perpendiculars drawn from the vertices of a triangle ABC upon any straight line meeting the sides externally in D,E,F. where a,b,c are the sides opposite to angles A,B,C in ...
0
votes
4answers
76 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [closed]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
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vote
4answers
42 views

What is the approximation of trigonometric function by simple function

for $f(x)=\sin x$, $g(x)=\cos x$, $h(x)=\tan x$, What is the approximation of each function by using simple function?
5
votes
4answers
82 views

How to simpify $\cos x - \sin x$

How does one simplify $$\cos x - \sin x$$ I tried multiplying by $\cos x + \sin x$, but that just gets me $$\cos x - \sin x = \frac{\cos 2x}{\cos x + \sin x}$$ which is worse. Yet ...