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

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3
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1answer
53 views

Solution to a trigonometric system

Find the solutions of the system $$\sin a-\frac{\sqrt{3}}{3}\sin b=0$$ $$\frac{\tan 2a-2\tan a}{\tan 2b}\cdot\frac{\tan 2b-2\tan b}{\tan 2a} =1$$ How to work with them ? Thanks
1
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2answers
37 views

Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$

yesteryday was my class test and I found this question. Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$ I have read the question for arctanx as $1/1 + x^2$. ...
1
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1answer
122 views

question about intermediate step in trig identity

In reading this article on sinusoid operations, I would like to know why you need equation 1) as an intermediate step and how you derive it. It seems like you would apply the sum formula directly to ...
0
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1answer
51 views

Find the area of these 2 specific regions

Here is the question.... When I put it into wolfram to just get a good visual of it...it looks like this... I believe this is the formula we use The problem is, I don't quite understand what they ...
-1
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1answer
108 views
0
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1answer
119 views

Approximate to the nearest degree, the two solutions of the equation $\cos^2x - 3\cos x -1=0$ in the interval [$0$ degrees, $360$ degrees].

Approximate to the nearest degree, the two solutions of the equation $\cos^2x - 3\cos x -1=0$ in the interval [$0$ degrees, $360$ degrees]. How do I solve the question above? I tried factoring ...
5
votes
2answers
78 views

A trigonometic integral with complex technicals

Let $a,b \in \mathbb{R}^+$. Show that $$\int_0^{2\pi}\frac{1}{a^2\cos^2(t)+b^2\sin^2(t)}dt=\frac{2\pi}{ab}$$ Help please! Thanks.
4
votes
2answers
58 views

Solving $\sin(t+am)=a$ where $-1 \le m \le 1$

What theory or algorithm would I need to research to solve equations such as $\sin(t+am)=a$ (knowing that $-1 \le m \le 1$) for the value of $a$? My equations may become more complex but have similar ...
1
vote
1answer
949 views

Area of a square in polar coordinates?

I was attempting, for the exercise of it, to find the area of the a simple square with an infinite number of infinitesimal circle sectors. Let us say this square is $[5 x 5]$. Alas, it's been ...
0
votes
1answer
52 views

Calculating the Apollonius Circle

This is a followup to a question I asked earlier. I have looked for an example on Google and StackExchange, but I have yet to see a clear example of the formula to determine the equation of an ...
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3answers
75 views

Solving trigonometry identities by simplifying terms

Verify the identity by simplifying the left side. $\sin^2x-\sin^2y=\cos^2y-\cos^2x$
2
votes
2answers
93 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
1
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3answers
1k views

Prove for $\cos (x+iy)$

I know it is such a foolish thing to do to ask this dumb question in this site. Please prove that $\cos (x + iy) = \cos x \cosh y - i\sin x \sinh y$ and $\cos (x - iy) = \cos x \cosh y + i\sin x ...
0
votes
1answer
461 views

Equation of a line maintaining equal ratio distance between two points.

Say I have two points. One at $A = (0, 0)$ and another at $B = (0, 10)$. I wish to derive an equation for a line that, for any point $P$ on the line, would equal a set ratio for the distance $P$ to ...
7
votes
5answers
744 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
1
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5answers
83 views

Solving a trigonometric equation: $2 \sin(3a)=\sqrt{2}$

I have the following equation : $2 \sin(3a)=\sqrt{2}$ Not sure how to solve it (Because it's a transformed sin function, meaning 6 solution with 3 cycles in $2\pi$) after a moment I finally found ...
2
votes
1answer
95 views

What's the minimum value?

What's the maximum and the minimum value of $x$? $$\frac{(\sqrt{100-x^2}+\sqrt{99+x^2})}{40} = \cos \frac{\pi}{x^2-2|x|+4}$$ I've done all what I could do but I failed. Any ideas? Thanks
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1answer
84 views

How to find $\arccos(\cos15\pi/11)$?

How to find $\arccos(\cos15\pi/11)$? I'm completely lost. $15\pi/11$ isn't on the unit circle so how do I find the cosine of it?
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6answers
135 views

How do you find the exact value of $\cos(\operatorname{arcsin}(\frac{-4}{5}))$?

Step by step explanations would be helpful. It's my understanding that the above is equal to $\cos (\theta )$ but I don't know why. Thanks in advance!
5
votes
2answers
59 views

Triangle and Maxium value

Given any triangle ABC with $a \ge b \ge c$ such that $\frac{a^3+b^3+c^3}{\sin^3(A)+\sin^3(B)+\sin^3(C)}=7$, what is the maximum value of $a$?
1
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1answer
1k views

Finding indicated trigonometric value in specified quadrant

If I have csc $\theta$ = - $\dfrac{10}{3}$ and have to find tan $\theta$ in quadrant III, would I use 1 + $\cot^2\theta$ = $\csc^2\theta$ then find reciprocal which would be tan $\theta$? If so, I ...
2
votes
2answers
90 views

Angle between vectors of the form $(\cos A,\cos B,\cos C)$

The question: Two vectors $S=(\cos A,\cos B,\cos C)$, $S'=(\cos A',\cos B',\cos C')$, What is the angle between them? The answer is $\cos(\theta)$ = $\cos A.\cos A'+ \cos B.\cos B'+ \cos C.\cos C'$. ...
0
votes
2answers
48 views

How was this distributed? (Trig equations)

How did this: $2(1-\sin^2x)=1+\sin x$ Become this: $2\sin^2x+\sin x-1=0$ Wouldn't it be: $2 -2\sin^2x-1+\sin x=0 ?$
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1answer
51 views

what does the ar in arcosB stand for?

regarding to the cosine formula solving with all sides and no angles? arcosB in the formula given where is it derived?
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0answers
38 views

Finding identities for $\cos(a+\frac{\pi}{4} \mod \frac{\pi}{2})$

The question is rather self-explanatory, but I can't find good answers online. The reason for wanting identities for this is that $b$ and $c$ are constants, and this formula is being used on a device ...
0
votes
1answer
71 views

Sum with sine function

Can anybody help me in solving the following sum $$\sum\limits_{n=1}^\infty \frac{\sin(b\log n)}{b}$$I tried using the expansion of sine function but got stuck there.
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1answer
48 views

Finding the other trig. functions with given values

If $\sec\theta = \frac{5}{2}$ and $\csc\theta < 0$, find the other five trig. functions The "$\csc < 0$" is confusing me. How do I know which quadrant that is in? What if it was "$\cot < ...
3
votes
2answers
1k views

Is it possible to express $\sin \frac{\pi}{9}$ in terms of radicals?

So, yes, this is a math homework question. I've done some research on it and I know that the actual value for $\sin \frac{\pi}{9}$ cannot be expressed without using imaginary numbers. ...
1
vote
2answers
38 views

Easy Trig question

Easy question here. I ran across this trig identity while doing a problem: $\tan(x)\cdot\tan(y) =-1$ implies that $y = x \pm \pi/2$. Why is this?
0
votes
1answer
45 views

Bode phase calculus with frequency to $0$ and $\infty$ in a quadratic term

The tangent to calculate is: $$\phi=-\tan^{-1}\left[\frac{2\zeta\frac{\omega}{\omega_n}}{1-(\frac{\omega}{\omega_n})^2}\right]$$ Where $\omega_n$ is constant. If $\omega=0 \to\phi=0$ ...
0
votes
1answer
46 views

Use Osborn's Rule to suggest an identity involving cothx^2 and cosechx^2 then to solve

i know 1+cothx^2=cschx^2 but how to suggest the identity then use to solve part b?
3
votes
3answers
121 views

To show inverse of tan x

It quite confuses me. Where do I start? Please help.
0
votes
1answer
111 views

Angle between two rectangles rotated around a point with a gap inbetween

I am trying to find the angle between two rectangles when there is a known gap between them. See this diagram: I have simplified the problem into three triangles, two of which are the same. Here ...
0
votes
3answers
737 views

Method of proving trignometric identities

How do we prove trigonometric identities? When my teacher did it in school today all I could see was him doing random steps. I didn't really understand his method. I am basically asking whether ...
0
votes
1answer
386 views

Proving $\left(\tan^2x +\frac{1}{\tan x}\right) \left(\sin x+\cos x\right)=\left(\frac{1}{\cos x}\right)+\left(\frac{1}{sin x}\right)$

$$ \left(\tan^2x +\frac{1}{\tan x}\right) \left(\sin x+\cos x\right)=\left(\frac{1}{\cos x}\right)+\left(\frac{1}{sin x}\right) $$ Can someone show me how to solve this identity and also explain the ...
0
votes
2answers
259 views

Find point on line, given starting point and angle

I have a line and I know the starting point (0,0) and I know the angle between the line and the x-axis (30 degrees). How would I be able to find another point (any other point, doesn't matter which) ...
1
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1answer
60 views

How to calculate v from $\Theta \:=\:\arctan \left(\frac{v^2\pm \sqrt{v^4-g\left(gx^2+2yv^2\right)}}{gx}\right)$

$\Theta \:=\:\arctan \left(\frac{v^2\pm \sqrt{v^4-g\left(gx^2+2yv^2\right)}}{gx}\right)$ What will be the formula to calculate the value of v when values of $\Theta$, $g$, $x$ and $y$ is already ...
2
votes
2answers
92 views

Maximum of |sin x| + |sin y| + |sin z|

If $x$, $y$ and $z$ are real numbers with the property $x+y+z= \pi$, then the maximum of $\sin x+\sin y+\sin z$ is $3\sqrt{3}/2$. Now, if $x+y+z=0$ then is the maximum of $|\sin x| + |\sin y| + |\sin ...
2
votes
2answers
99 views

solve trigonometric equation $\cos(2x) + \cos\left(x\right) -2 = 0$

Hi I can't figure out how to solve this equation: $$\cos\left(2x\right) + \cos\left(x\right) - 2 = 0$$ I think I'm supposed to rewrite $\cos\left(2x\right)$ into something else and then go from ...
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0answers
38 views

How to prove geometrically the limit $\lim_{x \to 0}\frac{1-\cos{x}}{x}$ using squeeze theorem [duplicate]

Here, I ask how to geometrically prove the limit \begin{equation}\lim_{x \to 0}\frac{1-\cos{x}}{x}\end{equation} using squeeze theorem. I see so many proofs for the other important limit ...
1
vote
1answer
29 views

3 equations with 3 unknowns

The Little Town Arts Center charges $\$23$ for adults, $\$12$ for senior citizens, and $\$8$ for children under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue ...
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0answers
80 views

Law of Cosines Manipulation

I'm supposed to use law of cosines on $S_1S_2P$ in the following diagram: To arrive at the following equation: $$ \frac{r_2}{r_1} = [1 - 2(\frac{a}{r_1})sin(\theta) + ...
3
votes
1answer
597 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
0
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1answer
682 views

Use fundamental identities to write the first expression in terms of the second, for any acute angle θ. sec(θ); sin(θ)

Can someone please help me with the question above? I don't understand what it's asking. A thorough explanation would be much appreciated.
3
votes
4answers
75 views

Question of trig formatting

Is there a difference between the following: $$\sin^2x$$ $$x\sin^2$$ How about: $$\sin(x)$$ $$\sin x$$ I'm new to trig and I've been confused on the formatting involved in trig, whether something ...
0
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1answer
141 views

Simplifying and Expanding trig identities

simplify expand $$8 \sin^2(3 \theta)-4$$ I'm stuck on this question can someone help me and show solution.
2
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1answer
71 views

Trigonometric Identities HW

Learning trigonometry right now. I have a question that asks: Write the trigonometric expression in terms of sine and cosine, and then simplify: $$(\cot^2\theta + 1) \sin^2\theta$$ I ...
4
votes
3answers
85 views

A limit with trigonometric functions

$$\lim_{x \to 0} \frac{1-\cos2x}{x\sin(x+4\pi)}$$ .... and then I got ... $$= \lim_{x \to 0} \frac{2\sin^2x}{x \sin x}$$ and I can't insert 0 in that calculation as long as there's x in the ...
2
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3answers
142 views

Prove that $2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$ [closed]

Prove that $$2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
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vote
3answers
101 views

Can someone help me simplify this trig expression?

$$( \tan x+ \sec x )( \cot x-\cos x ) $$ I got stuck after a few steps of converting and adding and what not.