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

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18 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
1
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1answer
20 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
0
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2answers
17 views

Problems identifying harmonic motion

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that ...
2
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3answers
52 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
7
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1answer
79 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
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2answers
42 views

general solution to trigonometric equation, help!!!

if $$\sin\left(\frac {π}{4} \cot\theta\right)=\cos\left(\fracπ4\tan\theta\right)$$ then find general solution of $\theta$ I am new to solving trigonometric equations so it would be helpful if some ...
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2answers
66 views

How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?

Is there someone show me how I evaluate this integral:$$\int\frac{\mathrm{d}x}{1+\sin x−\cos x} $$ I used $t=\tan\frac{x}{2}$ but i didn't succeed . Thank you for any help .
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0answers
20 views

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...
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6answers
51 views

Does the equation $2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$ have real solution?

Do the equation $$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$$ have any real solutions? Please help. This is an IITJEE question. Here $x$ is an acute angle. I cannot even start to attempt ...
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0answers
32 views

Questions of multiple angles [on hold]

$2\sin A/\cos3 A+2\sin3A/\cos9A+2\sin9A/\cos27A= \tan27A-tanA $
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2answers
26 views

How are arc components of a spherical system derived?

I am studying a flight dynamics book (see Flight Dynamics by Stengel) and am rusty on spherical coordinates. Commonly, aerospace coordinates use a North/East/Down right-hand system. So $z=-h$, ...
3
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4answers
144 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
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1answer
20 views

Find the measurement of line BD

So I was trying to find the measurement of $BD$ I drew green lines to make myself some angles, the measurement $3$ is from the point A to C, If only I can line $AE$ or $CE$ then I will just use the ...
4
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1answer
96 views

Trigonometric ratio of multiple and sub multiple angles

Given that $a$ lies in 1st quadrant and $$ \sin a +\cos a +\operatorname{cosec} a+\sec a+\tan a+\cot a=7$$ then we have to prove that $\sin(2a)$ is a root of $$x^2-44x-36.$$ I have tried to break all ...
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4answers
42 views

trigonometry expression simplification with inverse cosine

While working on a problem, I ended up with this expression for y: $$ y=x\sin\left(\arccos\left(\frac{\sqrt{x^2-y^2}}x\right)\right) $$ Is there any way to express $y$ in terms of $x$ only, with no ...
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2answers
36 views

What angle does the board need to be cut at?

If someone has a 2'' wide board and a 1 1/2'' wide board, and they want to cut the narrower board at an angle so the cut is 2'' long and the boards will fit together, what angle do they need to cut ...
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2answers
38 views

If a 16' ladder is placed correctly on a level surface, how high up will the ladder reach?

So i have just began learning about sin cos and tan, and i came across this problem and for some reason I'm having trouble figuring it out. *** When using a straight ladder, it is recommended that ...
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2answers
28 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
31 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
31 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 ...
3
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2answers
52 views

Explanation of derivation made at wikipedia.

in this wikipedia article A deriviation to convert true and eccentric anomaly. I am however quite stunned by a single line - trying to reproduce but after half a dozen sheets of paper I can't find how ...
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0answers
16 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
52 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)+ ...
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3answers
36 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
62 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
35 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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34 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
21 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
56 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
73 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
30 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
458 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 ...
0
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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
votes
3answers
49 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
67 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 ...
2
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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 ...
-1
votes
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: $R e^{i \theta}$. ...
0
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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
64 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 ...