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

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99 views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...
3
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2answers
175 views

Triangle with two constraints, each corner on a given line

Given: - 3 3-dimensional straight lines: a, b, c. All lines are coninciding in a single point S. - Point B on b. I'm now looking for a point A on a and a point C on c, where AB and BC have the same ...
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1answer
695 views

Physics Vector Problem - Airplane

Heres the question: A plane leaves the airport in Galisto and flies $140$km at $68.0^∘$ east of north and then changes direction to fly $255$km at $48.0^∘$ south of east, after which it makes an ...
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1answer
40 views

Help expressing Logs in natural log and then simplifying

I need some help changing regular Logs->Ln, and then simplifying it. I made a post last night, but must of made a mistake. Here are the problems, Express as ratio's of natural logs and simplify 3: ...
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2answers
83 views

Any reason beyond academics to represent a known constant as variable?

In biomechanics, for calculating joint angles there is a research paper most often referenced and which most of the algorithms are based on ...
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2answers
68 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...
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1answer
35 views

How to show that $ 0< a\leq\cos^2(\theta)\leq b<1$ in this problem?

The inequality $2\cos^4(\theta/2)-2\cos^2(\theta/2)+1/4\leq 0$ means that $\cos^2(\theta/2)$ lies between the roots of $2x^2-2x+1/4$ i. e., we can conclude that $$ ...
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1answer
47 views

Finding the following limit: $\lim\limits_{x\to 0}\frac{\sin -1x}{-7x}$

I'm studying calculus at a online course and the practice exercise came up like this: $$\lim_{x\to 0}\frac{\sin -1x}{-7x}$$ Now, I know that $$\lim_{x\to 0} \frac{\sin{x}}{x} = 1$$ But I don't ...
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1answer
109 views

value of $|2a+5b+5c|$ if

If a, b, c are unit vectors satisfying $|a-b|^2+|b-c|^2+|c-a|^2=9$ then find the value of $|2a+5b+5c|$ Options are: A: $1$, B: $2$, C: $3$, D: $4$ considering $|2a+5b+5c|^2$ as $k$ then ...
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3answers
226 views

$\triangle ABC$ has an angle $\angle A=60^{\circ}$. Also $AB=c$, $BC=a$, $AC=b$ and $2\cos B-1=\frac{a+b}{a+c}$. Find all the other angles.

$\triangle ABC$ has an angle $\angle A=60^{\circ}$. Also $AB=c$, $BC=a$, $AC=b$ and $2\cos B-1=\frac{a+b}{a+c}$. Find all the other angles. And we can't use calculus, logarithms, limits or ...
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2answers
51 views

Simple way to compute an integral

Is there a simple way to show that $$\int_{\sqrt{n\pi}}^{\sqrt{(n + 1)\pi}}\sin(x^2)x \mathrm dx= 1$$ if $n$ is even. We don't know how to integrate a multiple of functions ($\int{f(x)g(x)}$), but ...
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3answers
115 views

Find the general solution of $\sin(4t) + \sqrt3 \cos(4t)$

How to find the general solution of the equation $\sin 4t + \sqrt{3} \cos 4t$ in exact form. so what I know is $\sin 4t + \sqrt3 \cos 4t = 2 \cos(4t - \frac{\pi}{3})$ The answer is $t = ...
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2answers
1k views

prove this identity $\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y$ [duplicate]

prove this identity : $$\sin(x+y)\sin(x-y)=\sin^2 x - \sin^2 y$$ I tried solving it with additional formulas but I can't get the right answer. I get $$\sin^2 x \cos^2 y-\cos^2 x \sin^2 y$$
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2answers
530 views

Find the cartesian equation of: $r=2\cos\left(\frac {3\theta}{2}\right)$

I've managed to use identities to simplify it down to: $$r = 2\left(\cos^3\left({\theta\over2}\right)-3\sin\left({\theta\over2}\right)\cos\left({\theta\over2}\right)\right)$$ using trig identities, ...
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1answer
42 views

What is a method to figure out what x is in the equation “1 = secx”?

As the title suggests I am doing Trigonometric substitution. I am okay with everything until we have to change the boundaries of the integration. The answer I believe is pi/3, but I don't know how ...
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4answers
118 views

indirect measurement triangle

A person positions a 8ft vertical pole so that the top of the pole and the top of a distant tree are aligned in the person's line of sight. The person's eye level is 5.7 ft above the ground, the ...
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2answers
439 views

Radian and the length of a chord of a circle

Question In a circle of radius $r$, an arc of it is $2S$ long. Find the length of the chord corresponding to that arc (AB in the diagram below) . Details I got this question in a math test. And ...
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2answers
101 views

I need to solve equation, I really need some help

I really need to solve this equation, but my knowledge is not enough to figure it out: $$\cos(-55.82) = (0.6893\cos(-70) + 0.3381\sin(-70)) \cdot (-\frac{-0.4206\cos f + 0.6423 \sin f}{\sin 67.33}) - ...
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1answer
16 views

Choose constants to limit max value and satisfy condition

I have a function $$ f(x)=A\sin(n\pi x/3)+B\cos(n\pi x/3) $$ where $n$ is integer. For any value of n, I want to find $A$ and $B$ such that $f(2)=0$ and amplitude of $f(x)$ is 1. What is the cleanest ...
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3answers
53 views

Can't show that equations are the same

This is the problem i cant solve: Show that: $$ \frac {\tan^2(x-1)}{(\sin x+\cos x)} = \frac {(\sin x-\cos x)}{\cos^2x} $$ I can't get any longer than: Left-hand side: $$ \frac{ {\sin^2(x-1) ...
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1answer
64 views

Strange trigonometric equation

I have to solve the following trigonometric equation: $$2\sin(x)+x\cos(x)=0.$$ This problem comes from an analytical geometry one. I have tried to find the exact solutions, but I didn't succeed. Is ...
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2answers
201 views

Coordinates rotation by $120$ degree

If I have a point on a standard grid with coordinates say: $A_1=(1000,0)$ $A_2=(707,707)$ Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and ...
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3answers
57 views

Show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $

Hello everyone I have been trying to show that $dx = \frac{2}{1 + u^2} du$ where $ u = {\tan(\frac{x}{2})} $ but I keep ending up with something like this: $2d{\sin(\frac{x}{2})}\cos(\frac{x}{2}) $ ...
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3answers
72 views

Comparing $\sin 1$ with $1$

I don't quite understand how this is done. If possible, please give a full explanation since I'm new in trigonometry. $\sin 1$ and $1$ I've thought that 1 stands for radians , is that true?
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1answer
97 views

Solving equation: $\cos x-\sin x (2\cos x-4)=0$ [closed]

Solve the following trigonometric equation $$\cos x-\sin x (2\cos x-4)=0$$ Thank you very much.
8
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1answer
326 views

Prove trig identity: $(\sin \theta + 1)(\sin \theta − 1) = −\cos^2 θ$

This is my attempt: $(\sin \theta+1)(\sin \theta-1) = \sin\theta^2 - \sin\theta + \sin\theta - 1$ $= \sin^2\theta - 1$ $= -\cos^2\theta$ Is it correct, and can it be improved? Thanks!
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4answers
158 views

Finding $\lim_{x \to 0} \frac {a\sin bx -b\sin ax}{x^2 \sin ax}$ witouth L'Hopital, what is my mistake?

I was working on this question. $\lim_{x \to 0} \dfrac {a\sin bx -b\sin ax}{x^2 \sin ax}$ $\lim_{x \to 0} \dfrac {1}{x^2} \cdot \lim_{x \to 0} \dfrac { \frac {1}{abx}}{\frac {1}{abx}} \cdot \dfrac ...
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1answer
112 views

Prove trig identity: $\csc \tan - \cos = \frac{\sin^2}{\cos}$

I keep hitting seeming dead-ends. \begin{align*} \csc\ x \tan\ x - \cos\ x &= \left(\frac{1}{\sin\ x}\right)\left(\frac{\sin\ x}{\cos\ x}\right) - \cos\ x \\ &= \frac{\sin\ x}{(\sin\ ...
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2answers
61 views

Trig Substitution with integrals can't figure it out.

The question is to find the indefinite integral of: $$\int {x^2\over(x^2+a^2)^{(3/2)}}\;{{\rm d}x}$$ if anyone can help me out that would be awesome! I went through a few steps but I get stuck. So I ...
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2answers
949 views

Prove trig identity: $\tan + \cot = \sec \csc$

I appreciate the help. My attempt: $$ \begin{align} \tan + \cot &= \frac{\sin}{\cos} + \frac{\cos}{\sin} \\ &= \frac{\sin^2}{\cos \sin}+\frac{\cos^2}{\cos \sin} \\ &= ...
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3answers
36 views

Find the unit normal $N$

of ${\bf r}=14 \mathrm{e}^{-10 t}\cos(t){\bf i}+14 \mathrm{e}^{-10 t}\sin(t){\bf j}$ The answer should be in vector form. I can't find an easy way to do this. I end up with T being something ...
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2answers
127 views

Trigonometric Equation,

I was studying for my trigonometry test, a subject where I am usually very strong at, however I came across this questions and simply have no idea how to go about solving for m and n. The question is ...
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1answer
67 views

Proving An Equation [duplicate]

I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. A first step, or point of ...
2
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1answer
60 views

Solving a particular trigonometric equation

I am wondering if it is possible to solve the equation \begin{equation} \sin(x) = 0.4. \end{equation} If it is possible to solve this, how does one do so?
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2answers
79 views

The triangle must be obtuse if two of its medians are perpendicular?

In problem #577679, the question says if two of the medians of triangle ABC are perpendicular, then .... ($5a^2 = b^2 + c^2$, the result). In the course of solving it, I drew the following picture ...
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2answers
58 views

Help with basic trigonometric (Physics) problem

I am re-learning basic Physics and I would like to know if I followed the correct steps, so I can continue doing more exercises. The problem says: "A person kicks a ball from the surface of a playing ...
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1answer
94 views

Engineering Mathematics

I have been revising basic compound angles and I am struggling to understand the following question from the examples I have previously studied on such topic. I cannot envisage the compound angle ...
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1answer
36 views

When can the length of a line be equal to a circular function?

So, I'm having a bit of trouble trying to grasp this concept. I understand that a circular function like cosine is a ratio of two sides of a triangle in reference to an angle, however, one of my ...
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2answers
78 views

Convert phasors to sinusoidal waveform

I'm trying to convert this phasor into a sinusoidal waveform. $$ j6e^{-j\pi/4} $$ Here's what I have so far: $$ 6\sin(\omega t-\pi/4) = 6\cos(\omega t-\pi/4 - \pi/2) = 6\cos(\omega t-3\pi/4) $$ ...
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1answer
280 views

Proving the fundamental period of tangent

I'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that PI is a fundamental period of the tangent function. I need ...
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2answers
241 views

Find this limit without using L'Hospital's rule

I have to find this limit without using l'Hôspital's rule: $$\lim_{x\to 0} \frac{\alpha \sin \beta x - \beta \sin \alpha x}{x^2 \sin \alpha x}$$ Using L'Hôspital's rule gives: ...
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2answers
120 views

Graph of $\sin(x)$ along the line $y=x$

I want the equation of $\sin(x)$ which has the line $y=x$ as its axis. Basically, I want the $\frac\pi4$ rotation of the curve $y=\sin(x)$. I already attempted differentiating the curve and adding ...
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1answer
29 views

How to convert decimal quantity to a fractional quantity using sin(x)?

I'm embarrassed to say I never properly learned how to do this, but in my Calc Physics class I need to find the hypotenuse of an angle give the $sin(40)$ or $.642788$ using only 3 significant figures. ...
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2answers
1k views

Definite integral involving trigonometric functions and absolute values

Solve the following integral: $$ \int_{0}^{4\pi}\frac{x|\sin x|dx}{1 + |\cos x|} $$ I tried variable substitution, but nothing seemed to work. Could you give me some clues?
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3answers
199 views

Why does this recurrence relation generate a sinusoidal curve?

I came across the following coupled recurrence relation while watching this video called Media for Thinking the Unthinkable: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
3
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4answers
91 views

Limit involving $(\cos x)^{1/x^4}$

I am having trouble calculating the following limit. $$\lim_{x \to 0}(\cos x)^{1/x^4}$$ In Problems in mathematical analysis by Demidovich there is a hint that in case of $1^{\infty}$ indeterminate ...
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2answers
73 views

Evaluate $\int_o^{\pi ^2\over4}\cos^2(\sqrt{x})\, \operatorname d\!x$?

I want to integrate following: $$\int_o^{\pi ^2\over4}\cos^2(\sqrt{x})\,dx.$$ I try solve instead: $$\int_o^{\pi ^2\over4}\frac{\cos(2\sqrt{x})+1}{2} \,dx.$$ but I can't integrate $\cos(2\sqrt x)$
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1answer
75 views

Angle that car is at after angular acceleration

A car starts from rest on a curve with a radius of $150m$ and tangential acceleration of $\displaystyle 1.5\frac{m}{s^2}$. Through what angle will the car have traveled when the magnitude of its ...
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0answers
935 views

3D Game: Pitch Yaw Roll of a point

I have a flat elliptical plane and I'm trying to figure out how to represent it based on its direction. So I basically need to calculate its pitch, yaw, and roll. I have a camera at $C$, and a point ...
0
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1answer
43 views

how can i find the angle of a reflecting line?

i am trying to make the game pong ( my math is really bad and i am working on it) and trying to do the bouncing ball part. I am trying to calculate the out going(reflecting) angel of the ball after ...