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

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26 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
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1answer
49 views

A Ferris Wheel Word Problem (Trigonometry)

Please help, will give me an idea how to do more of these. A Ferris Wheel called Colossus has a diameter of 158 feet. Using the figure as a model, find the distance traveled by someone starting at ...
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2answers
43 views

solve my trigonometry doubt?

According to trigonometry identities , $$\tan\alpha= \frac{1}{\cot\alpha}.$$ If $\alpha = 0$ or $\alpha = 90$, place the value of $\alpha$, get your result and by observing the result. Give your ...
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1answer
41 views

I need to do math from ground up, so what is a good workbook?

Can you guys recommend me a workbook that begins with arithmetic and ends with calculus. Or from pre-algebra to calculus. Like all "Master Math Series" books but in one complete book. It would really ...
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12 views

Determining distance to an object based on the distance from it to two objects on a perpendicular line and the angle between them

Is possible to determine length $d$, given I only know lengths $a$ and $b$ and $\Theta$ ($\angle$ ACB )? More importantly, how?
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41 views

Solve the triangle trigonometry.

Solve the triangle $PQR$ where. $q=2.9\text{ m}$ $r = 3.5\text{ m}$ $\angle LQ = 25^{\circ}$ Does the $\overline{LQ}$ mean right angle triangle? Should I use $3.5 \text{ m}$ as the hypotenuse?
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2answers
68 views

How to prove that $\frac{r}{R}+1=\cos A+\cos B+\cos C$?

How do we prove that for any triangle this holds: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$ I can use this beautiful identity to prove several geometric inequalities, but I have no idea how to prove the ...
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0answers
32 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
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45 views

Find this if $\mbox{cosec}( x) =$

Given that $csc(x) = 9$ without a calculator evaluate: i) $\cot(x)$ ii) $\tan(x)$ iii) $\cos(x)$ I know that $\csc(x) = \sin(x)$ divided by $1$. But I don't know what $x$ is. Not sure what to do ...
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97 views

An expression equal to its reciprocal (but not really)

"Everybody knows" that \begin{align} \tan\theta & =\frac{2\tan\frac\theta2}{1 - \tan^2\frac\theta2} = \frac{2\sin\frac\theta2\cos\frac\theta2}{\cos^2\frac\theta2-\sin^2\frac\theta2} \tag 1 ...
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40 views

Harder Trigonometry Identities

How do I prove: sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A I've tried expanding the brackets by multiplying sin A and cos A to the left hand side ...
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31 views

Shadow angle calculation for solar tracking application

Shadow Length Dear all, *I am looking for relationship Between Lmin and solar radiation angle.I know Here in above link they provided relation. But i don't know how to calculate it. x- modules ...
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0answers
35 views

Stuck with trig substitution

I am stuck with problem at my homework assignment. $$\int \sqrt{1+4x^2}dx$$ I try to apply trigonometric substitution $$x = \frac 1 2\tan{2u}$$ $$dx = \frac 1 {\cos^2{2u}}du$$ But after ...
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2answers
339 views

How to show that this real function is not periodic?

How can one prove that $$\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$ is not periodic? This question is motivated by the harmonic spectral representation of time series. Indeed, it is ...
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19 views

linear/nonlinear system definition

genmerally we know that linear system is defined by following two rule $1.T(x+y)=T(x)+T(y);$ $2.T(c*x)=c*T(x) $ or operation on sum of two income is equal to sum of operation on each input and ...
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1answer
41 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
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2answers
47 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
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1answer
64 views

Why is $\sqrt{\sin^2 x} = |\sin x|$?

Please thoroughly explain this as I am completely lost. I kind if understand but if $\sin x$ is negative, then $\sin^2 x$ is a positive number and then the square root would be $|\sin x|$for sure but ...
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1answer
39 views

Problem with solving a complicated Integral

I need to determine the $ \int \frac{\sin^3(x)}{8-\cos^3(x)} dx$. It's an indefinite integral.
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2answers
71 views

Let $0\,^{\circ}\mathrm{} < A < 45\,^{\circ}\mathrm{}$ . If $420(\tan A + \cot A) = 841$ then find the value of $(116 \cos A − 58 \sin A)$.

Let $0\,^{\circ}\mathrm{} < A < 45\,^{\circ}\mathrm{}$. If $$420(\tan A + \cot A) = 841$$ then find the value of $$(116 \cos A − 58 \sin A)$$ One way to solve this is by usual method , that is ...
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29 views

How can I write $\frac{(4k-15)\pi}8$ as $n+2k\pi$ where $k\in\mathbb Z$ and $n\in(-\pi,\pi]$

How can I write $\frac{(4k-15)\pi}8$ as $n+2k\pi$ where $k\in\mathbb Z$ and $n\in(-\pi,\pi]$ $\boxed{\bf My\,try::}$ $$\begin{align} \frac{(4k-15)\pi}{8}&=\frac{4k\pi-15\pi}{8}\\ ...
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1answer
23 views

Trigonometry Word Problem Diagramming Pictorially

A footbridge is to be built across a small lake from a gazebo to a dock. From a tree 100 yards from the gazebo the bearing is S 66° E. From the tree to the dock the bearing is S 15° E. The bearing ...
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3answers
71 views

Limit of infinite loops of sin x as n tends to infinity [duplicate]

Show that $$lim_{n\to\infty} \text {sin sin ... sin x} = 0 $$ for all x. Note that the n here refers to the number of sin in the expression above.
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2answers
80 views

Why is $\arcsin(\sqrt 2/2) = 45^{\circ}$?

Just found this out by playing around with my calculator. Does that mean that $ \arcsin(\sqrt 2) = 90^{\circ}$? And then i wonder how you show that $\cos(90^{\circ}-v) = \sin(v)$ mathematically?
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131 views

Solve the following trigonometric exercise…

Can you please give a hint how to solve this exercise: $$\log(\tan 22)+ \log(\tan 68)=?\\ \tan22\cdot\tan68=?$$ Thank you!
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1answer
19 views

cos$^{-1}$ and units

So I got this problem: So let ABC be a right triangle at A such that BC=2AB. Find the angle $[\hat{ACB}]$ So I found that cos(ABC)=1/2 But when I want to find ABC, should I do cos$^{-1}$ in ...
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0answers
11 views

Shadow tilt angle relation

I am looking for formula to relate postion of angle between D,D' and tilt angle. here below they not mention relation ship between tilt angle and D,tilt angle and D' height=1200;meter all legth value ...
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1answer
55 views

Does the expansion of $\sin x$ contradict the normal formula $\sin x = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$?

Lets say I have a right angled triangle with sides $3, 4$ and $5$ units. They form a perfect Pythagorean triplet. One of the angles in the triangle, say $\alpha$ must have the following condition: ...
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1answer
42 views

Why is the cos, sin definition of the unit circle true?

So imagine we have a unit circle and there a point $M$ on it. Then the $x$ coordinate of $M$ is $\cos(\theta) $ ($\theta$ is the angle ${IOM}$ as you know) and its $y$ coordinate is $\sin(\theta)$. ...
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4answers
33 views

Trignomometry Identity Question

I need help showing that $(4\sin\theta)(\sin(\theta-\frac{\pi}{3}))(\sin(\theta-\frac{2\pi}{3}))=\sin(3\theta)$. Cheers.
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1answer
50 views

Someone can explain me why $\tan(-\frac{\pi}4+\arctan x)=\frac{x-1}{x+1}$

Someone can explain me why $tan(-\frac{\pi}4+\arctan x)=\frac{x-1}{x+1}$?? I try to understand it, bot I don't understand how to came from one side to the other... Thank you!
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1answer
23 views

How, in simplest possible terms, does one determine the speed an object is traveling along a given vector?

Say I have a two-dimensional vector v, which is the velocity of a rocket. I also have the vector f, which is a direction vector (with magnitude 1), that represents the direction the rocket is facing. ...
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1answer
60 views

How to solve this trigonometric equation / geometric problem

Is there any way to solve this type of equation exactly for x, where a-h are precalculated constants: $a\cos(g x)+b \sin(g x)+c\cos(h x)+d\sin(hx)+ex+f=0$ Or is my only/best option some sort of ...
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1answer
43 views

Trigonometry graph question.

(a) Sketch the graph of $\tan x$ for values of $x$ between $0$ and $360$. (b) Solve the equation : $\cot x = 0.15$ for values of $x$ between $0$ and $360$. Express your answer(s) in degrees. (c) ...
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3answers
48 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
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1answer
30 views

Radial velocity

I need to calculate the radial velocity ($v_r$) of an object to another. For this I have the cartesian coordinates ($X_n$,$Y_n$,$Z_n$) and cartesian velocities, ($\dot{X_n}$,$\dot{Y_n}$,$\dot{Z_n}$) ...
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2answers
61 views

How to prove this trig identity?

If $A+B+C=\pi$ then prove:$$\sin^2A+\sin^2B+\sin^2C=2-2\cos A\cos B\cos C$$ I am completely lost on this, please help.
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42 views

Why sin and cos do not get the same result that as tan, in this case?

Let $x=3$ and $y=3$. Then $\tan\theta = \frac{3}{3} = 1$. However if we use pythagorean theorem to find the value of hypotenuse and $sin$ or $cos$ function we get very different value. Here it is: ...
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1answer
18 views

Calculate/estimate the value of trigonometric functions in the unit circle by the value of their nearest angles

Given the fact that the values of trigonometric function (especially $sin$, $cos$, $tan$) for standard angles is easy to remember, is there a way that you can find / estimate the value of an angle in ...
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4answers
56 views

Linear combinations of sine and cosine

If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + ...
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27 views

Planes going in directions at different heights

One plane is flying due east straight and level at 30000 feet and at 420 mi/h. a second plane flies due north at 40560 feet at 480mi/h. The second plane crosses above the flight path of the first ...
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42 views

$4\cos x^2 - 4\cos x = 2$, find all solutions in the interval $0^º\leq x\leq 360^º$

$4\cos x^2 - 4\cos x = 2$, find all solutions in the interval $0^º\leq x\leq 360^º$ I'm not sure what I'm overlooking or not doing right but I can't seem to figure it out. I've tried factoring ...
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4answers
73 views

Why $\cos^3 x - 2 \cos (x) \sin^2(x) = {1\over4}(\cos(x) + 3\cos(3x))$?

Wolfram Alpha says so, but step-by-step shown skips that step, and I couldn't find the relation that was used.
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0answers
14 views

Find semimajor axes of ellipse from two points and normals

I have two points on an ellipse and normals perpendicular to the ellipse for each. I know where the vectors intersect and the angle between them. How can I compute the length of the ellipse axes? ...
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2answers
32 views

$\sin(x/2) + \cos(x) = 0$, Solve for x if $0^\circ \leq x < 360^\circ$

I have a pretty good grasp on what we're learning right now, but this question came up and I'm stumped. I would assume I need to use half-angle identities but would someone mind walking me through it ...
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0answers
10 views

Relation of Product Rule to sine sum identity

I was taking a look at the sine sum identity and noticed a resemblance to the Product Rule for derivatives. Applying this led to the following simplification: $$\begin{align}\sin(x + y) & = ...
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2answers
43 views

Integration Problem with a Trig substitution

Okay I am a little stuck on this problem. $$\int \tan^5(x)\sqrt{\sec(x)} \; dx$$ What should be my first step for a u sub or a trig sub? I have tried to use $u=\sec(x)$ and then $u=\tan(x)$, but I ...
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3answers
115 views

$\sin^2(x)+\cos^2(x) = 1$ using power series

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...
2
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3answers
105 views

$\cos(x)+\cos(3x)+…\cos(2n-1)x=\sin(2nx)/2\sin(x)$

I need to somehow show that $$\cos(x)+\cos(3x)+...\cos(2n-1)x=\frac{\sin(2nx)}{2\sin(x)}$$ for some integer n>0. This seems impossible to me since if I consider the Left Hand Side, I don't know any ...
3
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3answers
87 views

How to write $x=2\cos(3t) y=3\sin(2t)$ in rectangular coordinates?

How would I write the following in terms of $x$ and $y$? I think I use the inverse $\cos$ or $\sin$? $$x=2\cos(3t)\,, \quad y=3\sin(2t)$$