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

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20 views

can these two trig functions cross?

I have two functions; $$f(i) = \frac{1+A^2 \cos{\frac{2 \pi i}{N}}}{1+A^2}$$ and $$G(j,k) = \frac{(1+A^2 \cos{\frac{2 \pi j}{N}})(1+A^2 \cos{\frac{2 \pi k}{N}})}{(1+A^2 ) ^2}$$ and ...
0
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2answers
24 views

How to scale a 2D vector and keep direction

I want to take any vector in R2 and scale its length to 1 while keeping the original direction (ratio of x component to y). As an example of my goal, let's say I have the vector (1,1), it would become ...
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1answer
22 views

Do I need to use different trig functions in different quadrants?

I don't have any formal education in Trigonometry or Calculus, but I'm studying a book on Pre-calc before school begins this fall. I've completed College level Algebra too, so math isn't something ...
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2answers
30 views

Equation with sine and cosine - coefficients

I have some trouble with the conceptual understanding of the way we solve this kind of equations. Let's say we have: $$(3-3b^2)\sin(bx)+3a\cos(2x)=6\cos(2x)$$ The method employed on classes was ...
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0answers
15 views

Belt & Pulley Problem Using Trigonometry [on hold]

I found this problem in an old trigonometry book, but my answer is totally different from the one given. Can anyone work it out? Thanks. Two Pulleys, diameters 10 in. and 18 in., have their axles ...
3
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2answers
50 views

Solving an infinite series containing $\arctan$

I need to compute: $$\tan\bigg(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{2}{9}\right)+ ...
3
votes
3answers
31 views

Finding periodic (trigonometric?) function given points

It's been a while since I've taken a math class. I need a couple functions for a program I'm working on. I can tell they involve trigonometry, but I can't figure out how to derive the function ...
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1answer
87 views

I don't understand what my Calculus hw question is asking of me…not looking for answers, just guidence.

I really don't understand what I am supposed to do. I understand cos = opp/hyp...etc. But my book doesn't give me enough info to figure this question out, nor do I really understand what it is asking ...
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0answers
18 views

Trigonometry problem solve [on hold]

An expedition team decided to have a practice run prior to their North Pole trek. One team member started to walk due north. The other three travelled 70° east of north at a speed of 4 km/h. How far ...
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2answers
59 views

Solving for $x$ in $A=B\cdot \cos(x)+C\cdot \sin(x)$ [duplicate]

I´m working on a little paper, and I want to know if it´s possible in any way to solve this: $$A=B\cdot \cos(x)+C\cdot \sin(x)$$ $A$, $B$ and $C$ are known. I need a way to get the $x$ without using ...
3
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1answer
31 views

Explain why two right triangles, each with an acute angle of 17 degrees, must be similar.

Two right angles with an acute angle of 17 degrees must be similar because triangles that are similar share the same angles.Is this proper?
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0answers
31 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of 1/4 inch per foot and a maximum slope of 1/2 inch per foot for waste to drain properly. This means that for every horizontal ...
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4answers
44 views

Trigonometry equation. Not sure about solution.

The equation goes as follows: $$\sin x +\cos x = 1 + \sin x \cos x$$ and here is how I solved it: $$(\sin x+\cos x)^2=(1+\sin x\cos x)^2$$ $$\sin^2x+2\sin x\cos x+\cos^2x=1+2\sin x\cos ...
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2answers
32 views

(definite integral) area between two trig functions

I'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever ...
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1answer
25 views

Solving a mixture of exponential and trigonometric equation

Solve for $x$ $$e^{\cot ^2 x}+\sin ^2 x-2\cos ^2 2x+4=4\sin x$$ It looks like I need to find the ranges of both the sides... But i am getting left side greater than $3$ and right side less than $4$. ...
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1answer
20 views

What are the coordinates of a point given its distance from another point?

If the abscissa of a point is twice the value of the ordinate and has a distance of $2\sqrt{17}$ units from the point $(4,-5)$, what are the coordinates of the point?
2
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1answer
54 views

How do I solve this trigonometric equation?

Solve the equation $$\sin\left(\frac{3\pi}{10}-\frac{x}{2}\right)=\frac{1}{2}\sin\left(\frac{\pi}{10}+\frac{3x}{2}\right).$$ I tried applying some Ratio properties, but they just made the ...
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3answers
59 views

Peripendicular distance from a line segment

I have a line given by $Ax + By + C= 0$, and a point $x0,y0$. From that point $x0,y0$ in the direction of the line up to distance $d$, I want to find the perpendicular distance of the points from this ...
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3answers
29 views

Trigonometry - log/ln and absolute sign in equations

Will this equation still hold if the absolute sign is being used at different places For example, This trigonometry identity; ...
10
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4answers
446 views

Evaluating limit (iterated sine function)

The limit is $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$ where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times: $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$ For $n=1$ the ...
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0answers
9 views

What does 3D gaze direction contains? And how to convert it to yaw and pitch?

I am trying to use a dataset. But I am facing two problems or confusions in understanding it. Can anbody guide me what 3D gaze direction stands for or means (angles, (x, y, z) coordinates or what)? ...
2
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3answers
61 views

How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?

How do I solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x $? Note : I have tried using trigonometric transformation but it seems very complicated to get the result .. may ...
2
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3answers
48 views

Find the general values of $x$ satisfying the trigonometric equation

Find the general values of $x$ satisfying $$ \frac{\tan^2 x \sin^2 x}{1-\sin^2 x \cos2x}+\frac{\cot^2 x \cos^2 x}{1-\cos^2 x \cos2x}+\frac{2\sin^2 x}{\tan^2 x+\cot^2 x}=\frac{3}{2} $$ It ...
3
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1answer
65 views

How to use Chebyshev Polynomials to approximate sin(x) and cos(x) within the interval [−π,π]? [on hold]

I have approximated sin(x) and cos (x) using the Taylor Series (Maclaurin Series) with the following results How can I use Chebyshev Polynomials to approximate sin(x) and cos(x) within the ...
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1answer
34 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
2
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3answers
36 views

Trigonometry equation maximum

Given the equation: $\cos x + \sqrt3 \sin x = a^2$ find the maximum value for $a$ for which the equation has solutions and for this case solve the equation, $a \in \mathbb{R}$. I'm guessing I need ...
3
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1answer
63 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
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0answers
31 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
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1answer
40 views

When solving trigonometric irrational equations does the condition of existence of the radicand under an even root matter?

Hi everyone I would like to ask a thing about the following equation: $$\cos(x) + \sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = 0$$ It is trigonometric and irrational, the root's index is 4 (even ...
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1answer
29 views

linear or bilinear interpolation

I want to know how to use linear and bilinear interpolation in 2D. Specifically the pairs $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ are given in a quadrilateral. In this case how to ...
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2answers
24 views

Euler's formula for off-center circle [on hold]

A circle with radius $R$ and center at $(a,b)$ is given by the formula $(x-a)^2 +(y-b)^2 = R^2$. A circle with radius $R$ whose center is at the origin is given by Euler's formula: $R e^{i \theta}$. ...
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1answer
41 views

Derivative of a trigonometric function

What is the derivative of $$\cos^2 a (\tan a - \tan b)$$ Please anyone explain in detail. The differentiation is with respect to $a$. I tried to obtain the answer using chain rule, but didn't get it. ...
2
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1answer
63 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
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0answers
42 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
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2answers
34 views

Find the least positive angle satisfying the trigonometric equation

$\sin^3 x+\sin^3 2x+\sin^3 3x=(\sin x+\sin 2x+\sin 3x)^3$. I did solve the question, but my method is highly tedious. I combined the sin and then opened the cubic.... Is there some trick? Something I ...
2
votes
1answer
48 views

Solving $\sin x=2\sin(2\pi/3-x)$

How can I solve the equation: $$\sin x=2\sin\left(\frac{2\pi}{3}-x\right)$$ Without using the formula: $$\sin(a-b)=\sin a \cos b-\sin b \cos a$$? Thanks.
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6answers
36 views

Trigonometrical inequation problem

Solve the inequation: $\sin^4x+\cos^4x \geq 1/2$. I did this: $(1-\cos^2x)^2+\cos^4x \geq 1/2$ $-2\cos^2x+2\cos^4x \geq -1/2$ $-2(\cos^2x-\cos^4x) \geq -1/2$ $\cos^2x(1-\cos^2x) \leq 1/4$ ...
0
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3answers
34 views

Find the roots of equation involving $\arctan x$

I try to find the roots of the equation: $$y=x-2\arctan\left(x\right)$$ I know that one of them is $(0,0)$ but there are two others that should solve $$\dfrac{x}{2}=\arctan\left(x\right).$$ Is ...
0
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2answers
73 views

Integration problem: $\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $

I need help in solving the following problem: $$\int \ln\left(\sin(\sqrt{x})+\cos(\sqrt{x})\right)dx $$ I really don't know how to start solving this problem; any tips or solutions will be greatly ...
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2answers
87 views

Evaluate the indefinite integral $\int \frac{\cos \theta}{ \sqrt{2 - 9 \sin^2 \!\theta}} \mathrm{d}\theta$

I want to evaluate $$\int \dfrac{\cos \theta \, \mathrm{d}\theta}{ \sqrt{2 - 9 \sin^2 \theta}}$$ but I can't seem to get the answer, my working is as below:
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0answers
32 views

Point in the circle [on hold]

I have a circle with center $(x, y, z)$. Circle lies in plane which normal also known $(n_1, n_2, n_3)$. How calculate points on circle for given angle?
5
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2answers
95 views

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer

Prove that $\tan^6 20°+\tan^6 40°+\tan^6 80°$ is an integer. Doesn't this problem seem a little out of the box? It seems beautiful, but I don't have an idea on how to start. Calculating the value does ...
6
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2answers
49 views

Convergence and divergence depending on whether $n$ is odd or even

It is part of one problem I am working on: I want to prove the following conjecture ($x\ne q\pi$ where $q\in\Bbb Q$) $$\sum_{n=1}^{\infty}\frac{\sin^{2k-1}(nx)}{n^{\alpha}}\quad\text{converges ...
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2answers
46 views

Integrating inverse trigonometric functions

I want to find the integral of $$\frac {\sin^{-1}(\ln x)}{x}$$ I know the best way to find th integration of trigonometric shirt substitutions is to substitute to eliminate the inverse trigonometric ...
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2answers
43 views

Is this always true that if the angle in degrees is negative, its radian counterpart will also be negative and vice-versa?

I want to know : if the angle in degrees is negative, will its radian counterpart will also be negative, or can it be anything(positive/negative)? I know that 180 degrees = pi radians. but, does ...
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0answers
24 views

initial height = 60“. There is 5 degree decline over 163.5”. What is ending height?

I'm building a roof for a structure and need to get the ending height correct. The initial height is 60". The adjacent length (the ground) is 163.5". The decline is 5 degrees. I have gotten the ...
3
votes
4answers
111 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
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2answers
48 views

$A\sin X +B\cos X=c,\,B\sin X -A\cos X =d.$ Eliminate $X$

$A\sin X +B\cos X=c,\,B\sin X -A\cos X =d.$ Eliminate $X$ How can we eliminate the angle? I can't understand the question. How to try these type of questions?
5
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4answers
99 views

$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $

$$ \cos ^2\left(x\right)+\cos ^2\left(2x\right)+\cos ^2\left(3x\right)=\frac{3}{2} $$ How can I solve this one, I mean I get something like this: $-3+\left(-1+2\cos ...
0
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2answers
29 views

Is the arctan of a negative number always negative?

Is the $\text{ arctan}$ of a negative number always negative and $\text{ arctan}$ of a positve number always positive ?