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cosecant

cosecant Trig. (kəʊˈsiːkənt) [f. co- prefix 4 + secant. The L. cosecans was used a 1576 by Rheticus, Opus Palatinum (1596). F. cosécante.] The secant of the complement of a given angle. (Abbreviated cosec.)1706 in Phillips, Co-secant. 1807 Hutton Course Math. II. 3 The radius, cotangent, and cosecan... Oxford English Dictionary

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Cosecant squared antenna

A cosecant squared antenna, sometimes known as a constant height pattern, is a modified form of parabolic reflector used in some radar systems. The name refers to the fact that the amount of energy returned from a target drops off with the square of the cosecant of the angle between the radar and wikipedia.org

en.wikipedia.org

cosecant

cosecant/ˌkəuˈsi:kənt; ko`sikənt/ n(abbr 缩写 cosec) (mathematics 数) in a right-angled triangle, the ratio of the length of the hypotenuse to that of the opposite side 余割. 牛津英汉双解词典

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Integrating Cosecant by Multiplying it by $\csc(x)+\cot(x)$ From what I’ve seen you integrate cosecant by multiplying it by $\csc(x)+\cot(x)$: How do you know to multiply it by this?

$$\int\frac{1}{\sin{x}}dx=\int\frac{\sin{x}}{\sin^2x}dx=\int\frac{\sin{x}}{1-\cos^2x}dx=$$ $$=-\frac{1}{2}\int\left(\frac{1}{1+\cos{x}}+\frac{1}{1-\cos{x}}\right)d(\cos{x})=$$ $$=-\ln\frac{1+\cos{x}}{1-\cos{x}}+C=\ln\frac{1-\cos{x}}{1+\cos{x}}+C.$$

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Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$? I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never...

Those are the same thing. Either it's undefined as part of the definition (i.e. part of $\mathbb{R}$ is not in the domain) or it's undefined as a consequence of $\frac{1}{0}$ being undefined. It's just like how $f(x)=\frac{1}{2-x}$ is completely clear without a note saying "but x can't be 2!" If a f...

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How do you write the cosecant function of a negative number as the trigonometric function of a positive number? More specifically, how would you write $\csc(-2\pi/5)$ as the trigonometric function of a positive number?

**Hint:** Recall that $\sin(x)=\sin(x+2k\pi)$ for any integer $k$. Let $x=-2\pi/5$ and let $k=1$.

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Reference for "tangent of half angle equals cosecant minus cotangent" The identity $\tan(\frac{\theta}{2})=\frac{1-\cos \theta}{\sin \theta}$ is easily seen to be equivalent to $\tan(\frac{\theta}{2})=\csc \theta - \c...

It shows up in Schaum's Mathematical Handbook.pdf) as formula 5.43 (on p.16).

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absolute value in derivative of inverse hyperbolic cosecant derivative of inverse hyperbolic cosecant is: $$\frac {-1} {|x|\sqrt{1+x^2}}$$ i saw in some website the absolute value of $x$ (in denominator) obtained af...

The first one is because $$\text{csch}^{-1}(-x)=-\text{csch}^{-1}(x)$$ On the other side, in the real domain, $\text{sech}^{-1}(x)$ is undefined.

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Question about name-convention for secant and cosecant. Ok so if we take a right triangle and consider an angle $\alpha$ we get the following: !enter image description here From here we can define the fundamental tr...

Yes, this would be nice if the "co"s matched up! However, it is indeed all geometry related. No calculus necessary, all you need is to look at the unit circle. Check out this webpage. One thing to keep in mind is that a "secant line" is just a line that "cuts" through a figure. Secant has a root wor...

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How to solve ${\rm Ln}({\rm cosecant}(z)) = {\rm Ln}(2) + iz$? $\newcommand{\Ln}{\operatorname{Ln}}$ $$\Ln(\csc(z)) = \Ln(2) + iz$$ I need to know what complex number $z$ is

Hint $$\dfrac{\csc(z)}2=e^{iz}$$ If $e^{iz}=a,$ Using Intuition behind euler's formula $2a=\dfrac{2i}{a-1/a}$ as $a\ne0$ $a^2-1=i$ $a^2=\sqrt2e^{i\pi/4}$

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Derivative of inverse cosecant? I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this wou...

Your intuition is almost correct, but not quite. You can think about this in two ways, depending on how you define $\operatorname{arccosec}$ Firstly, note that $\operatorname{arccosec}{x}= \arcsin{\left(\frac{1}{x}\right)}$ (this is one possible definition). Then taking the derivative gives $$ \frac...

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Partial Integration $ \int \frac{x\cos x}{\sin^3x}dx $ The problem: $$ \int \frac{x\cos x}{\sin^3x}dx $$ Can someone give me a hint on how to solve this without using cosecant? The solution provided is: $$ -\frac{...

HINT: $$\int x\cdot\frac{\cos x}{\sin^3x}dx=x\int\frac{\cos x}{\sin^3x}dx-\int\left(\frac{dx}{dx}\int\frac{\cos x}{\sin^3x}dx\right)dx$$ For $\int\dfrac{\cos x}{\sin^3x}dx$ write $\sin x=u$

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Trigonometric identity expressing $\sec \theta+\text{cosec } \theta$ in terms of sine and cosine > $\large{\text{cosec }\theta+\sec{\theta}=\dfrac{\sin\theta+\cos\theta}{\sin\theta\,\cos\theta}}$ I know that cosecant...

Perhaps most algebraically natural is to go from right to left. We have $$\frac{\sin\theta+\cos\theta}{\sin\theta\cos\theta}=\frac{\sin\theta}{\sin\theta\cos\theta}+\frac{\cos\theta}{\sin\theta\cos\theta}=\frac{1}{\cos\theta}+\frac{1}{\sin\theta}=\sec\theta+\csc\theta.$$ However, going from left to ...

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Verify a trigonometric relation Suppose we have the ratio $\frac{a}{\sqrt{a-b}}$, and we have that $b=a\cos(c)$. Then, do we have $\frac{a}{\sqrt{a-b}}=\csc(\frac{c}{2})$? Or at least some some thing with cosecant?

$$\frac{a}{\sqrt{a-b}}=\frac{a}{\sqrt{a-a\cos c}}= \frac{a}{\sqrt{a} \times \sqrt{1-\cos c}}$$ Now use $$1-\cos c=2 \sin^2 \left( \frac c2 \right)$$ To get $$ \frac{a}{\sqrt{a} \times \sqrt{1-\cos c}}= \frac{\sqrt a}{\sqrt{2 \sin^2 \left( \frac c2 \right)}}=\color{blue}{\sqrt{\frac a2} \Bigg|\csc \l...

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graphical representation of trig functions I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at khan academy. !enter image descriptio...

While this figure is elegant in its way, it has a little bit too much going on for the beginning student. ![enter image description here]( Lets start with a simpler figure. ![enter image description here]( Our two triangles are similar. The smaller triangle has side lengths $(\cos x, sin x, 1)$ Mult...

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cosecant

Answers

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cosecant

Cosecant squared antenna

cosecant

Integrating Cosecant by Multiplying it by $\csc(x)+\cot(x)$ From what I’ve seen you integrate cosecant by multiplying it by $\csc(x)+\cot(x)$: How do you know to multiply it by this?

Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$? I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never...

How do you write the cosecant function of a negative number as the trigonometric function of a positive number? More specifically, how would you write $\csc(-2\pi/5)$ as the trigonometric function of a positive number?

Reference for "tangent of half angle equals cosecant minus cotangent" The identity $\tan(\frac{\theta}{2})=\frac{1-\cos \theta}{\sin \theta}$ is easily seen to be equivalent to $\tan(\frac{\theta}{2})=\csc \theta - \c...

absolute value in derivative of inverse hyperbolic cosecant derivative of inverse hyperbolic cosecant is: $$\frac {-1} {|x|\sqrt{1+x^2}}$$ i saw in some website the absolute value of $x$ (in denominator) obtained af...

Question about name-convention for secant and cosecant. Ok so if we take a right triangle and consider an angle $\alpha$ we get the following: !enter image description here From here we can define the fundamental tr...

How to solve ${\rm Ln}({\rm cosecant}(z)) = {\rm Ln}(2) + iz$? $\newcommand{\Ln}{\operatorname{Ln}}$ $$\Ln(\csc(z)) = \Ln(2) + iz$$ I need to know what complex number $z$ is

Derivative of inverse cosecant? I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this wou...

Partial Integration $ \int \frac{x\cos x}{\sin^3x}dx $ The problem: $$ \int \frac{x\cos x}{\sin^3x}dx $$ Can someone give me a hint on how to solve this without using cosecant? The solution provided is: $$ -\frac{...

Trigonometric identity expressing $\sec \theta+\text{cosec } \theta$ in terms of sine and cosine > $\large{\text{cosec }\theta+\sec{\theta}=\dfrac{\sin\theta+\cos\theta}{\sin\theta\,\cos\theta}}$ I know that cosecant...

Verify a trigonometric relation Suppose we have the ratio $\frac{a}{\sqrt{a-b}}$, and we have that $b=a\cos(c)$. Then, do we have $\frac{a}{\sqrt{a-b}}=\csc(\frac{c}{2})$? Or at least some some thing with cosecant?

graphical representation of trig functions I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at khan academy. !enter image descriptio...