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

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1answer
50 views

Is the count for graphing trigonometric functions always 1/4 of a period? [closed]

I saw this video in YouTube since I'm studying for a quiz and found out that the count used for graphing trigonometric functions is 1/4 of a period. Is it always like that?? Sine, cosine, tangent, ...
1
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1answer
53 views

Trig limit in Spivak's Calculus

$$\lim_{x\rightarrow 1} (x-1)^3 \sin\frac{1}{(1-x)^3} = 0$$ To prove that this is true, the chapter on limits has things like $\lim_{x\rightarrow a}(f\cdot g)(x) = \lim_{x\rightarrow a}f(x)\cdot ...
0
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1answer
29 views

Mind refresher on a few simple algebra-geometry problems

I feel silly for asking this, but I've completely forgotten some steps on how to do a few of these simple algebra/geometry problems. 1) Simplify $\sqrt{18x}-4\sqrt{x^3}$. I rearranged this to ...
1
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2answers
33 views

How to express $\phi$ in terms of $R\text{, }x\text{ and }\theta$

Let $S$ be a circle with radius $R$ and center at $O$. Let $P$ be any arbitrary point inside circle such that its distance from $O$ is $x$ and the ray $\overrightarrow{OP}$ cuts the circle $S$ at ...
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5answers
58 views

Trigonometric equation. [closed]

Please help in Solving the Trigonometric Equation: $$\cos^2x - \sin^2x = \cos3x$$
4
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6answers
103 views

Evaluate $\int e^x \sin^2 x \mathrm{d}x$

Is the following evaluation of correct? \begin{align*} \int e^x \sin^2 x \mathrm{d}x &= e^x \sin^2 x -2\int e^x \sin x \cos x \mathrm{d}x \\ &= e^x \sin^2 x -2e^x \sin x \cos x + 2 \int e^x ...
0
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1answer
17 views

Angle of view based on height and distance to a determined object

I'm trying to determine what angle of view is needed for a photo shoot so that I can determine which super telephoto lens to rent. I'm photographing an object thats 2,600 meters across from an ...
0
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1answer
83 views

Simplifying an algebraic equation

I stumbled across this specific question where there was a fractional equation and I did not know whether I should simplify it through canceling out the terms or not. However, I decided not to. ...
0
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0answers
38 views

Simplify addition of 4 sin()s into one term

Is it possible to turn $\sin(a)+\sin(b)+\sin(c)+\sin(d)$ into one term, such that there is no addition or subtraction? I've tried by using $$ \sin(a) + \sin(b) = 2 \sin \left(\frac{a+b}{2} \right)\cos ...
1
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2answers
34 views

Solve for “$x$” when, tan(x) = csc(x), and Domain of $x$ is $(-\pi ,\pi)$ .

I solved it and get four answers, but the book gives only two: $\pm 0.905$ radian. Since the domain is $(-\pi ,\pi)$, I thought there would be two more values. I checked with the calculator, and the ...
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1answer
24 views

Measure the slope of a triangle relative to a plane

Suppose I have a list of vertices that form triangles floating in space. How do I measure the slope of each triangle relative to a flat ground plane?
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2answers
49 views

Finding Trignometric Equation

$$3\sin x + 4\cos x = 5$$ then the value of $\tan(x/2)$ I should get the answer as $1/3$ but my answer is $(5\sec x-4)/3$. Can any one help me thanks in advance.
5
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3answers
50 views

Line for set of three-dimensional vectors

If there is a set for 3D vectors $v$ where $ v \times \begin{pmatrix} -1 \\ 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -27 \\ 8 \end{pmatrix}$ is a line, what is this line's equation? I'm not sure ...
3
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2answers
112 views

Integrating $ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$

I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?: $$ \frac{{ \int_{0}^{\infty} e^{-x^2}\, ...
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1answer
157 views
+50

This 1 innocent looking recurrence relation seems to have no closed form solution.

$$P(c \cdot x) = \cos(x) P(x)$$ For $c=2$, $P(x) = \sin(x)/x$ is a solution to this. I don't know if there's a closed-form solution for $c \ne 2$. Rather than add my own attempt at solution, which ...
1
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0answers
40 views

Standard properties of trigonemetric functions

You have $\sin(\frac{\pi}{6})= \frac{1}{2}$ and $\sin(\frac{5\pi}{6})= \frac{1}{2}$ and the interval [$\frac{\pi}{6},\frac{5\pi}{6}$] has length $\geq 1$ This is used as I'm sure most will be familiar ...
5
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2answers
67 views

Validity Michael Hardy's proof of Pythagoras Theorem using differentials

A proof of the pythagorean theorem has been published by Mike Hardy during 1988 in Mathematical Intelligencer (Hardy, Michael, "Pythagoras Made Difficult". Mathematical Intelligencer, 10 (3), p. 31, ...
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3answers
44 views

Clarification regarding a question

In the question in the link is it compulsory that $A+B+C=\pi$ ? If sin A +sin B+sin C = cos A+cos B+cos C=0 prove that sin 2A+sin 2B+sin 2C =cos 2A+cos 2B+cos 2C
2
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3answers
63 views

How to simplify elegantly $\arcsin(2t-1)+2\arctan\left(\sqrt{\frac{1-t}{t}}\right)$?

I currently try to simplify the following trigonometric expression: $$ \arcsin(2t-1)+2\arctan\left(\sqrt{\frac{1-t}{t}}\right) $$ where $t\in(0;1]$. I know that ...
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3answers
58 views

$4\cosh \left(x\right)-3\sinh \left(x\right)=5$ where to start?

The question is: Solve this equation giving your answer to 3d.p $4\cosh \left(x\right)-3\sinh \left(x\right)=5$ I have no idea what to use for this one
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votes
1answer
55 views

Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it. The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) ...
0
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2answers
56 views

How do I isolate/solve for $\theta$ in $\sin (2\theta) = 4 \cos (2\theta)$

Isolate the variable/solve for $\theta$: $$\sin (2\theta) = 4 \cos (2\theta)$$ Like which $\cos$ double angle formula would I use? Because there are three of them. Thanks in advance.
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1answer
117 views

Prove that $\sum_{k=1}^{n} \frac1{\sin^2 \frac{\left( 2k-1\right)\pi}{4n+2}}=2n\left( n+1\right)$

Prove that $$\frac{1}{\sin^{2}\frac{\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{3\pi }{4k+2}}+\frac{1}{\sin^{2}\frac{5\pi }{4k+2}}+\cdots+\frac{1}{\sin^{2}\frac{(2k-1)\pi }{4k+2}}=2k(k+1)$$
0
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2answers
40 views

Inverse Trigonometric Function: Find the Exact Value of $\sin^{-1}\left(\sin\left(\frac{7\pi}{3}\right)\right)$

$$\arcsin\left(\sin\left(\frac{7\pi}{3}\right)\right)$$ I cannot use this formula, correct? $f(f^{-1}(x))=x$ The answer in the book is $\frac{\pi}{3}$ How do I approach solving a problem such as ...
1
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1answer
36 views

$\int_{0}^{x_{x>0}} \left(\sin^2(x)\cos(x)\right) \text{d}x=\frac{\text{d}}{\text{d}x}\left(\sin^2(x)\cos(x)\right)$

The original problem is: $$\int_{0}^{x_{x>0}} \left(\sin^2(x)\cos(x)\right) \text{d}x=\frac{\text{d}}{\text{d}x}\left(\sin^2(x)\cos(x)\right)$$ With the work I've done I'm only left to this ...
3
votes
2answers
33 views

Area of shaded region circle help

Find the area of the shaded region Area of the sector is $240^\circ$ or $\frac{4\pi}{3}$ Next find $\frac{b\cdot h}{2}$ which is $\frac{2\cdot2}{2}$ which is $2$. Then subtract the former ...
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2answers
59 views

a integration with two constants

I am trying to solve below integration $$\int\frac{dx}{(x^2+a^2)(x^2+a^2+b^2)^{\frac12}}$$ I tried substituting $x=a \,tan\,u$. Then I ended up with ...
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2answers
17 views

How would I find a point on a sphere with a UV coordinate?

I'd like to do the opposite of the example specified here: https://en.wikipedia.org/wiki/UV_mapping Can somebody explain to me how to do it? Thanks, For any point $P$ on the sphere, calculate $\hat ...
0
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3answers
76 views

Show $\tan(x)+\tan(y)+\tan(z) = \tan(x) \tan(y) \tan(z)$ [duplicate]

I am not able to show that: If $x+y+z=\pi$, show that $\tan(x) + \tan(y) + \tan(z) = \tan(x) \tan(y) \tan(z)$.
0
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3answers
49 views

How to prove the trigonometric identity $\frac{\cot x}{1- \tan x} + \frac{\tan x}{1 - \cot x} - 1 = \sec x \csc x$

I am doing some practice questions for a Math class and I was told that similar questions would be in the exam. So I need to learn this but I have no idea where to even start with this question: ...
0
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1answer
36 views

Find triangle $ABC$ satisfies $1+2\sqrt{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos B+\cos C$

$\color{Red}{\texttt{Find all the triangle ABC}}$ whose angles satisfies $$2\left (1+tan^2\frac{C}{2} \right )\left [ cos^2\left (\frac{13\pi }{2}+\frac{B}{2} \right ...
2
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1answer
43 views

Proof of the law of reflection without calculus

I am working on some optimization problems, and I am aware of the method of proving that the "angle of incidence equals the angle of reflection" using Fermat's principle and calculus. However, my ...
1
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2answers
31 views

Intersection between two three-dimensional planes

The intersection of the planes defined by $x \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $x \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. Find an equation of ...
0
votes
1answer
23 views

A triangle has an area of $12$ in$^2$, and two of the sides of the triangle have lengths $5 in.$ and $7 in$

A triangle has an area of $12$ in$^2$, and two of the sides of the triangle have lengths $5 in.$ and $7 in$. Find the angle included by these two sides. (Assume the angle is acute) How do I do this?? ...
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2answers
76 views

Why does $\int \sin y\;dx = x \sin y + C$?

I've just started to learn calculus on my own. I don't get how $$\int \sin y\;dx = x \sin y + C$$ I've tried to search on Google, but I couldn't find a clear answer.
2
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1answer
71 views

How to show that $1/\cosh(x) < \sinh(x)/x < x/\sinh(x)$?

I was going through an old exam paper and I saw this question. How to show that for $0 < x < \pi/2$, $$\frac{1}{\cosh(x)} < \frac{\sin(x)}{x} < \frac{x}{\sinh(x)}\;?$$ I can see ...
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2answers
41 views

Proof of the identity: $c\sin \frac{A-B}{2} \equiv (a-b) \cos \frac{C}{2}$

Trigs is not my strongest apparently... I need to prove $c\sin \frac{A-B}{2} = (a-b) \cos \frac{C}{2}$ for a general triangle $ABC$. Here is what I do, or rather, here is how I fail at proving it: ...
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2answers
40 views

Prove that the following question based on trignometric identities [closed]

Prove that: $$\frac{1 + \cos A + \sin A}{1 + \cos A - \sin A}=\frac{1 + \sin A}{\cos A}$$
3
votes
3answers
27 views

Alternative area of a triangle formula

The problem is as follows: There is a triangle $ABC$ and I need to show that it's area is: $$\frac{1}{2} c^2 \frac{\sin A \sin B}{\sin (A+B)}$$ Since there is a half in front I decided that ...
1
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0answers
24 views

Cosine formula to show if an angle is obtuse or acute

Keeping in mind the cosine formula: $a^2 = b^2 + c^2 -2bc\cos A$, or rearranging $\displaystyle{\cos A = \frac{b^2+c^2-a^2}{2bc}}$, I need to show when $A$ is acute and when it is obtuse. Consider ...
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2answers
41 views

How to solve trigonometry question [closed]

How do I solve: Solve for $0° \leq \theta \leq 360°$. $$ \tan{\theta} = -\frac{1}{\sqrt{3}} $$
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4answers
50 views

Solve trigonometric inequality $\cos x \geq \sin^2 x - \cos^2 x $

Solve trigonometric inequality $$\cos x \geq \sin^2 x - \cos^2 x $$ My incorrect solution: $$\cos^2 x-\sin^2 x \geq -\cos x $$ $$\cos 2x \geq \cos (\pi - x) $$ which means: $$ 2x \geq -(\pi + x)$$ ...
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2answers
39 views

Solving the equation $\tan(x)=\cos(x+33.44)$

Please show a method of solving the equation $\tan(x)=\cos(x+33.44)$. I tried several methods (half-angle, cosine of sum, multiply cosines,etc...), but nothing worked. How should one solve such ...
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votes
3answers
37 views

Solve trigonometric equation $ \cot x + \cos x = 1 + \cot x \cos x $

Solve trigonometric equation: $$ \cot (x) + \cos (x) = 1 + \cot (x) \cos (x) $$ I tried to multiply both sides with $\sin x$ (which I'm not sure if I can multiply with sin).
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votes
1answer
40 views

Writing an expression for a change in angular velocity of an angle

Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$. So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference. By ...
0
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1answer
36 views

What's wrong with my version of this integral?

This is the steps to find the correct answer:[][]2 I used trig identities instead and got a different answer. I checked it multiple times, and I'm not sure what I did wrong:
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2answers
65 views

Find the value in the following question [closed]

Find the value of $$\left(\cos \frac{5\pi}{14}-\frac{1}{2016}\right) \left(\cos \frac{15\pi}{14}-\frac{1}{2016}\right) \left(\cos \frac{ 45\pi}{14}-\frac{1}{2016}\right)$$
2
votes
1answer
67 views

Calculate angle on bent bar based on height

I'm writing a small piece of software that shows a preview of a bent rebar. I am however unable to figure out how to calculate the angle so the shape fits within given height $(B)$ requirements. $A, ...
2
votes
1answer
40 views

Equation of the form $tan(\alpha)=cos(\alpha+C)$ where $C\in\mathbb{R}$

I have seen the following math problem posed online by a high school student (knowing their material, most likely it wasn't given as an exercise): Find the solutions for the equation ...
0
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1answer
7 views

How many pair solution satisfy both conditions [equation and inequation]

How many pair $(x,y)$ solutions satisfy these conditions: $|\tan(\pi . y)|+\sin^2(\pi x) = 0$ and $x^2+y^2\le 2$? Answer: 9.