# 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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### Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
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### Round 2: Derive this trig identity from the common ones? $\;\cos^2(3x) = \frac{1+\cos(6x)}{2}$

$$\cos^2(3x) = \frac{1+\cos(6x)}{2}$$ Just came across another wacky identity today. Is this an easy derivation from the more popular identities, or is this one you just take it at face value and ...
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### Trigonometric equation tangent

I want to solve trigonometric equation $\tan x=\pi-x$. one of these solutions in interval [0,$\pi$] is $x=\pi$. Find another root please.
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### How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
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### Find the area of a circle that is inscribed in a circular sector with a radius $R$ and an angle $2x$.

The circle within the sector touches the radii R and the arc. So what is the area of the inscribed circle? The answer is actually $$S = \pi R^2\frac{\sin^2x}{(1+\sin x)^2}$$ How can I derive this?
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### Representing a function as a real part of a complex variable?

To represent a simple sinusoidally varying function $V(t)$ let us use $V(t)=Re(\hat V e^{i\omega t})$ where $\hat V$ can be a complex constant.Let $\hat V =-iV_o$ Therefore, $V(t)=V_o \sin(\omega t)$. ...
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### Why can't I use trigonometric functions here?

I tried solving this answer by using trig functions to get the answer (E). First I calculated angle FBC which was 30 and then i used sin 30 to get the length of FC. then I got the length of line BF ...
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### Is $\cos(90 - 2x) = \sin (2x)$?

What does $\cos(90 - 2x)$ equal? I know $\cos (90 - x) = \sin x$ but does this equal $\sin 2x$? Thanks.
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### maximum of $f(x)=\sin{(\pi Ax)}\left(\,\csc{(\pi x)}+\csc{(\pi (\frac{1}{A}-x))}\,\right)$

Let a function $f(x)$ be $f(x)=\sin{(\pi Ax)}\left(\,\csc{(\pi x)}+\csc{(\pi (\frac{1}{A}-x))}\,\right)$ where $A\geq 2$ is a positive integer and $\csc{(x)}=\frac{1}{\sin{(x)}}$. I want to prove ...
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### Solve $x - 2\arctan(x)= 0$

$x - 2\arctan(x)= 0$. I can find one root (0) from the equation $\tan(x/2) = x$ but there are two others, namely ($-2.3312, 2.3312$) that I don't know how to find. Looking for help! Thanks :) ...
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### Simplify the trig function $\frac{2\cos{2x}}{\sin{2x}}$

How would I simplify the trig function $\frac{2\cos{2x}}{\sin{2x}}$? Is this a trig identity of can I simplify it down further?
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### How to prove that $\lim_{x \to 0} \sin(x) = 0$ using the epsilon-delta definition?

How do I prove that $\lim_{x \to 0} \sin(x) = 0$ using the episilon-delta definition of a limit? Do I have to divide the domain of $x$ into 4 cases for each quadrant? Update: Based on the ...
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### Prove that $\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is an increasing function

Could anybody show that $f(x)=\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is increasing on the interval $x\in[0, 0.25]$? Of course, $\csc{(x)}=\frac{1}{\sin{(x)}}$. Here is ...
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### How find this $\sum_{i=1}^{5}\tan^4{\frac{i\pi}{11}}$

show that:$$\tan^4{\dfrac{\pi}{11}}+\tan^4{\dfrac{2\pi}{11}}+\tan^4{\dfrac{3\pi}{11}}+\tan^4{\dfrac{4\pi}{11}}+\tan^4{\dfrac{5\pi}{11}}=2365$$ my try: I think first we can find this ...
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### Verifying trig identities

$$\sec(2x)=\frac{\sec^2x+\sec^4x}{2+\sec^2x-\sec^4x}$$ I have no idea how to verify this. I've tried changing it into cosine but it doesn't work.
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### Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$$
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### Assuming *only* that $\sin^2{\theta}$ + $\cos^2{\theta} = 1$, show that $\sin{\theta}\cos{\theta}\le1/2$

Assuming only that $\sin^2{\theta}$ + $\cos^2{\theta} = 1$, show that $\sin{\theta}\cos{\theta}\le1/2$. I only know how to do it using calculus!
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