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

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33 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
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30 views

algebraic determination of the correct phase angle

Let's solve $A\sin x+B\cos x=C$. We know $A\sin x+B\cos x=R\sin(x+k)$ and we easily calculate $R = \sqrt{(A^2 +B^2)}$. We calculate angle $k$ to be the $\arctan(B/A)$. We get a result from the ...
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1answer
144 views

Show that if A is diagonalizable, then sin^2(A) + cos^2(A) = I. Does this identity also hold for nondiagonalizable matrices?

Show that if A is diagonalizable, then $\sin^2(A)+\cos^2(A)=I$. Does this identity also hold for nondiagonalizable matrices? This is what I got so far: $$ e^{iA}= \cos A +i\sin A \\ \cos A= \frac{...
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1answer
31 views

Trigonometric Identity Symmetry

I'm currently trying to prove the following trig identity. $\dfrac{\sin \left ( \frac{\alpha}{2} \right ) \cos \left ( \frac{\alpha}{2} \right ) + \sin \left ( \frac{\beta}{2} \right ) \cos \left ( \...
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2answers
58 views

Trigonometric ratios of compound angles

If $\alpha$ and $\beta$ be two different roots of equation $a\cos\theta+b\sin\theta=c$, prove that $\sin(\alpha+\beta)=\dfrac{2ab}{a^2+b^2}$
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125 views

how to calculate the phase angle

When we transform $a\sin x+b\cos x=c$ into $a\sin x+b\cos x=R\sin(x+k)$, we calculate the $k$ angle by $k=\tan(b/a)$. By using calculator, we get a positive or negative degree value for $k$. I know ...
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1answer
1k views

Calculating a perpendicular distance to a line, when using coordinates (latitude & longitude)

I'm trying to implement the Douglas-Peucker algorithm for simplifying a recorded GPS track (a list of coordinates). All implementations I can find assume a simple X/Y grid of squares, however ideally ...
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2answers
208 views

What method can i use to find the first 3 roots of y(t)=tan(t)+t?

Just by looking at the function: $$y(t) = \tan(t)+t$$ I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer ...
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2answers
56 views

Usage of law of sines

The vertex angle of an isosceles triangle is 35 degrees. The length of the base is 10 centimeters. How many centimeters are in the perimeter? I understand the problem as there are two sides with ...
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2answers
132 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since $|...
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1answer
75 views

Trigonometry (non right angled triangles)

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From ...
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3answers
31 views

Show Trigonometric Identities from Complex indentity

So the exercise says to show $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ By using the following identity: $e^{i(a+b)}=e^{ia}e^{ib}$ How do ...
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1answer
28 views

Is this solution about unit circle right?

If cos(x)=-2/pi Are these solutions for x right: X=129 and x=230 If they are not correct please correct them
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2answers
177 views

Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...
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2answers
246 views

Trigonometric Graphs - Point of intersection with the curve and line

The diagram shows the graph of y=(a)sin(b)x +(c) . 1)write down the value of a,b and c. 2)Find the coordinate of P an Q, the points of intersection with this curve and the line y=2.
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1answer
131 views

Given $\sin{A}:\sin{B}:\sin{C}$ how to find $\sin{(2A)}:\sin{(2B)}:\sin{(2C)}$?

Question: Given $x,y,z>0$, and the positive number $k$ such $$\dfrac{x^2}{x^2+k}+\dfrac{y^2}{y^2+k}+\dfrac{z^2}{z^2+k}=1$$ in $\Delta ABC$, $$(1):\sin{A}:\sin{B}:\sin{C}=\dfrac{x}{x^2+k}:\...
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1answer
34 views

Solving/simplifying a trig expression

My problem sheet says that $\tan a= 5/12$ and $a \in {\rm Q\,III}$ ($a$'s in quadrant III). Using this information, I am to solve/simplify the expression $\quad \quad \cos\left(\frac{1}{2}a\right)$ ...
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2answers
244 views

Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$

The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...
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563 views

Could trigonometry exist in one dimension?

Even though trigonometry is based on circles, and angles, both of which commonly exist in two dimensions, could it also exist in one dimension? This question probably sounds really weird to you, but ...
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34 views

Number of values that satisfy $2\sin ^2(x) - 3 = 3 \cos (x), \: 90^{\circ} < x < 270^{\circ} $

Graphing this function is difficult as many overlaps exist and finding a viewing window is hard. What's a good algebraic method to solve this problem?
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29 views

Finding the domain of this trigonometric function

How can I find the domain of this function? $$f(x)=\frac{x\sin(x)+\cos(x)}{1-\cos(x)} + \frac{|x|-2}{x^2-4}$$ I assume we don't want the denominator to be zero, but do we have to combine the ...
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3answers
161 views

Solving $\;2^{\large \cos x} = \sin x$

$$2^{\large \cos x} = |\sin x|$$ Solve the equation. I found just one solution $\cos x= 0$ and are there any other solutions. Right hand side is modulus $\sin x$.
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226 views

Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...
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3answers
281 views

Why doesn't $\arccos x = -\tfrac12\sqrt{3}$ have any solutions?

I have this exercise with an unclear answer. The question is this: $$\arccos x = -\frac{\sqrt3}{2}\,.$$ The answer is this: $$\begin{gather*} \varphi(x)= \arccos x\\ V_\varphi = [0,\pi]\\ -\frac{\...
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3answers
147 views

Trigonometry General Solution

I've been working out some general equations recently, whilst the simple ones are fairly easy to work with, the more advanced ones (for me ofcourse) seem to somewhat confuse me since I have no ...
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0answers
70 views

Expectation of product of cosines

I am reading a paper that starts with $$ E[ \cos( a(x-y) ] = E[ \cos(a x) \cos(a y) + \sin(a x) \sin(a y) ] $$ where the expectation is over $a$, then converts it into something of the form $$ = 2 ...
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1answer
34 views

Finding the domain of this trigonometric function

how can I find the domain of this function? f(x) = (xsin(x) + cos(x) / 1 - cos(x)) + (|X| - 2 / x^2 -4) I assume we don't want the dominator to be zero so f(x)1 ...
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67 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
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44 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
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255 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
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1answer
55 views

Solution to trigonometric derivative

Version 2 For \begin{align} &x(t)\text{:=}\cos (t)+\cos (2 t)+1&\\ &y(t)\text{:=}\sin (t)+\sin (2 t)&\\ \end{align} how would I go about proving that the solutions to \begin{align} ...
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2answers
20 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
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78 views

Trigonometric Expression Simplification

Could someone explain how to simplify $(\cos(x)-\csc(x))/(\sin(x)-\sec(x))$? Any help would be appreciated.
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88 views

Why is $x^5 \sin x$ an odd function?

Why is $x^5 \sin x$ an odd function? Is the result just wrong? Because $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
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1answer
54 views

Given the matrix $A^k$, how to get $A^{k+1}$?

Given: $$A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$$ $$A^{k+1} \overbrace{=}^? \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \...
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6answers
619 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
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2answers
36 views

Simplfy trigonometric functions by only considering integer inputs?

I have the below function which only takes integer input, $$ 2 \sqrt{3} \sin \left(\frac{\pi t}{3}\right)+\sqrt{3} \sin \left(\frac{2 \pi t}{3}\right)-\sqrt{3} \sin \left(\frac{4 \pi t}{3}\right)+...
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3answers
30 views

finding and angle and coordinate point

"For a given angle $θ$ and a circle of radius $r$ and center $(h,k)$, recall that we can determine the Cartesian coordinates $(x,y)$ of the point on the circle determined by $θ$ and $r$, where $x=h+...
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1answer
48 views

Trigonometry graphs sinusoidal waves

i need help on this questions. I couldn't figure how to determine for both question A and B. But i have the answers for them, i just don't understand how the amplitude is 3 and so on.
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43 views

Exact value of polynomial at trigonometric argument

Given that $$\cos 8\theta= 128\cos^8 \theta −256\cos^6 \theta +160 \cos^4 \theta −32\cos^2 \theta +1$$ Find the exact value of: $$4x^4 −8x^3 +5x^2 −x$$ where $x=\cos^2 (\frac{\pi}{8})$ ...
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1answer
55 views

Finding the square roots of a complex number.

Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram. So far I've worked out the mod/arg form of the ...
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1answer
66 views

Is there a quicker way to write $\cos (n\theta)$ in terms of $\cos \theta$?

Im writing $\cos 8\theta$ in terms of $\cos \theta$ using De Moivre's Theorem $$\cos 8\theta= \Re {(\cos\theta+ i \sin \theta)^8}$$ Let $s=\sin \theta$ and $c=\cos \theta$ $$=c^8 -28c^6(1-c^...
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2answers
55 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
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1answer
53 views

Inverse cosine of a complex number, take $\cos z=\sqrt{2}$ for $z$

If I am given $\cos z=\sqrt{2}$ for $z$ and asked to solve it using the following: $$ \cos^{-1} z =-i \log\sqrt{z+i(1-z^2)} $$ I've only gotten as far as taking $\cos z=\sqrt{2}$ and changing it to $...
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2answers
32 views

Product to sum formulas

Write the product as a sum. cos 4x cos 2x this is what i tried 2{cos2xcosx} = 2[1/2 cos(2x+1x)+ cos(2-1)] = 1[cos(3x)+cos(1x)] = cos 3x + cos x
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145 views

Why “$\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does not exist” is not an acceptable answer?

Find the limits: $\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ Since the numerator and denominator tends to infinity as $x$ tends to infinity, then applying Lhopital's rule: $\lim\limits_{...
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155 views

Is a trigonometric function applied to a rational multiple of $\pi$ always algebraic?

Specifically, just to talk about cosine, is it true that $\cos(\frac{a\pi}{b})$ is algebraic for integers $a$ and $b$? Looking at this post and the link to trigonometric constants in the comments, it ...
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3answers
127 views

How to prove that $\tan 55^\circ<\pi/2$

How to prove that $\tan 55^\circ<\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
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1answer
37 views

Which identity is being used to get $\sin(wa)\cos(wt)=\sin(w(a+t))+\sin(w(a-t))$?

Which identity is being used to get $\sin(wa)\cos(wt)=\frac{\sin(w(a+t))+\sin(w(a-t))}{2}$? Couldn't find it among the trigonometric identities.
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0answers
30 views

Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...