# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?
### 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.