MHB Katlynsbirds' question at Yahoo Answers regarding inverse trigonometric identity

MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Prove the identity, pre calc!?

cot inverse= sin inverse of 1/sqr of 1+x^2

Here is a link to the question:

Prove the identity, pre calc!? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Re: katlynsbirds' question at Yahoo! Answers regarding inverse trignometric identity

Hello katlynsbirds,

We are given to prove:

$$\cot^{-1}(x)=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

Let's let $$\theta=\cot^{-1}(x)\,\therefore\,x=\cot(\theta)$$, and now please refer to this diagram:

https://www.physicsforums.com/attachments/765._xfImport

We see that $$\cot(\theta)=\frac{x}{1}=x$$ and we can also see that:

$$\sin(\theta)=\frac{1}{\sqrt{1+x^2}}\,\therefore\, \theta=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

and so we may conclude:

$$\theta=\cot^{-1}(x)=\sin^{-1}\left(\frac{1}{\sqrt{1+x^2}} \right)$$

Shown as desired.

To katlynsbirds and any other guests viewing this topic I invite and encourage you to post other trigonometry problems here in our http://www.mathhelpboards.com/f12/ forum.

Best Regards,

Mark.
 

Attachments

  • katlynsbirds.jpg
    katlynsbirds.jpg
    3.5 KB · Views: 86
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top